text:
\begin{document}

\title{The conformal-Killing equation on $G_{2}$- and $\mathrm{Spin}_{7}$-structures}

\author{Liana David}

\maketitle

{\bf Abstract.} Let $M$ be a $7$-manifold with a $G_{2}$-structure
defined by $\phi\in \Omega^{3}_{+}(M).$ We prove that $\phi$ is
conformal-Killing with respect to the associated metric $g_{\phi}$
if and only if the $G_{2}$-structure is nearly parallel.
Similarly, let $M$ be an $8$-manifold with a
$\mathrm{Spin}_{7}$-structure defined by
$\psi\in\Omega^{4}_{+}(M).$ We prove that $\psi$ is
conformal-Killing with respect to the associated metric $g_{\psi}$
if and only if the $\mathrm{Spin}_{7}$-structure is parallel.\\

{\it Key words:} Conformal-Killing equation, fundamental forms,
$G_{2}$ and $\mathrm{Spin}_{7}$-structures.

\section{Introduction}

A vector field $X$ on a Riemannian manifold $(M^{m}, g)$ is
Killing if its covariant derivative $\nabla X$ with respect to the
Levi-Civita connection $\nabla$ is totally skew-symmetric, or
$\nabla X =\frac{1}{2}dX$ (here and everywhere in this note we
identify vector fields with $1$-forms using the Riemannian
duality). More generally, a $p$-form $\psi\in\Omega^{p}(M)$ is
Killing if $\nabla \psi = \frac{1}{p+1}d\psi .$ In the same way as
Killing forms generalize Killing vector fields, conformal-Killing
forms generalize conformal-Killing vector fields.  In order to
define conformal-Killing $p$-forms ($1\leq p\leq m$) on $(M,g)$,
consider the decomposition of $T^{*}M\otimes \Lambda^{p}(M)$ into
irreducible $O(m)$-sub-bundles:
\begin{equation}\label{dec-tangent}
T^{*}M\otimes \Lambda^{p}(M)\cong \Lambda^{p+1}(M)\oplus
\Lambda^{p-1}(M)\oplus {\mathcal T}^{p}(M),
\end{equation}
where ${\mathcal T}^{p}(M)$ is the intersection of the kernels of
skew-symmetrization and natural contraction maps. The covariant
derivative $\nabla\psi$ of a $p$-form $\psi \in\Omega^{p}(M)$ is a
section of $T^{*}M\otimes \Lambda^{p}(M)$ and its projections onto
$\Lambda^{p+1}(M)$ and $\Lambda^{p-1}(M)$ according to
decomposition (\ref{dec-tangent}), are essentially given by the
exterior derivative $d\psi$ and the codifferential $\delta \psi$
respectively. The projection of $\nabla \psi$ onto the remaining
component ${\mathcal T}^{p}(M)$ defines the conformal-Killing
operator. A $p$-form $\psi$ is conformal-Killing if it belongs to
the kernel of the conformal-Killing operator, i.e. $\nabla\psi$ is
a section of the direct sum bundle
$\Lambda^{p+1}(M)\oplus\Lambda^{p-1}(M)$, or, equivalently, the
conformal-Killing equation
\begin{equation}
\nabla_{X}\psi = \frac{1}{p+1}i_{X}d\psi -\frac{1}{m-p+1}X\wedge
\delta \psi ,\quad\forall X\in TM.
\end{equation}
is satisfied.

Conformal-Killing forms exist on spaces of constant curvature, on
Sasaki manifolds and on some classes of K\"{a}hler manifolds (like
Bochner-flat or conformally Einstein) where they are closely
related to the so called Hamiltonian $2$-forms \cite{acg}.
Conformal-Killings forms exist also on Riemannian manifolds
admitting twistor spinors \cite{sem}. The space of
conformal-Killing forms on a Riemannian manifold is always finite
dimensional (even in the non-compact case) and an upper bound for
the dimension is realized on the standard sphere, where any
conformal-Killing form is a sum of two eigenforms of the Laplace
operator, with eigenvalues depending on the dimension and the
degree of the form \cite{sem}.

There are two results in the literature which motivate this note.
The first was proved in \cite{sem} and states that an almost
Hermitian manifold whose K\"{a}hler form is conformal-Killing, is
necessarily nearly K\"{a}hler; the second motivating result was
proved in \cite{liana} and states that an almost
quaternionic-Hermitian manifold whose fundamental $4$-form is
conformal-Killing, is necessarily quaternionic-K\"{a}hler. In this
note we prove the analogous statements for the fundamental form of
$G_{2}$ and $\mathrm{Spin}_{7}$-structures. More precisely, we
prove the following result:

\begin{thm}\label{main}
i) Let $M$ be a $7$-manifold with a $G_{2}$-structure defined by
$\phi\in \Omega^{3}_{+}(M)$ and let $g_{\phi}$ be the associated
Riemannian metric on $M$. Then $\phi$ is conformal-Killing with
respect to $g_{\phi}$ if and only if the $G_{2}$-structure is
nearly parallel.

ii)  Let $M$ be an $8$-manifold with a
$\mathrm{Spin}_{7}$-structure defined by
$\psi\in\Omega^{4}_{+}(M)$ and let $g_{\psi}$ be the associated
Riemannian metric on $M$. Then $\psi$ is conformal-Killing with
respect to $g_{\psi}$ if and only if the
$\mathrm{Spin}_{7}$-structure is parallel.

\end{thm}

The plan of this note is the following. In Section \ref{g2} we
recall basic facts on $G_{2}$-structures and we prove the
statement for $G_{2}$. The statement for $\mathrm{Spin}_{7}$
is proved in Section \ref{spin7}.\\

{\bf Acknowledgements.} I am grateful to Uwe Semmelmann for useful
discussions and interest in this work. Useful discussions with
Andrei Moroianu are also acknowledged. This work is partially
supported by a CNCSIS grant IDEI "Structuri geometrice pe
varietati diferentiabile" (code no. 1187/2008).

\section{The statement for $G_{2}$}\label{g2}

\subsection{Basic facts about $G_{2}$-structures}\label{g2-basic}
Let $\{ e_{1}, \cdots , e_{7}\}$ be the standard basis of $V=
\mathbb{R}^{7}$ and $\{ e^{1}, \cdots , e^{7}\}$ the dual basis.
We shall use the notation $e^{i_{1}\cdots i_{k}}$ for the wedge
product $e^{i_{1}}\wedge\cdots \wedge
e^{i_{k}}\in\Lambda^{k}(V^{*})$. Recall that $G_{2}< GL(V)$ is a
$14$-dimensional compact, connected, simple Lie group defined as
the stabilizer of the $3$-form
\begin{equation}\label{phi}
\phi_{0} := e^{123} +e^{145} + e^{167}+ e^{246} - e^{257} -e^{347}
- e^{356}.
\end{equation}
It can be shown any $g\in G_{2}$ preserves the standard metric
$\langle\cdot , \cdot\rangle_{7}$ and the orientation of $V$ for
which the basis $\{ e^{1},\cdots , e^{7}\} $ is orthonormal and
positive oriented. Let $*_{7}$ be the associated Hodge star
operator. It follows that any $g\in G_{2}$ stabilizes also the
$4$-form
\begin{equation}\label{starphi}
*_{7}\phi_{0}= e^{4567} + e^{2367} + e^{2345} + e^{1357} -
e^{1346} - e^{1256} - e^{1247}.
\end{equation}

The standard representation of $G_{2}$ on $V\cong V^{*}$ is
irreducible, but the representation of $G_{2}$ on higher degree
tensors is reducible, in general. For our purpose we need to know
the irreducible decompositions of the $G_{2}$-modules
$\Lambda^{2}(V^{*})$ and $S^{2}(V^{*})$, only. It is known that
$\Lambda^{2}(V^{*})$ decomposes into two $G_{2}$-irreducible
sub-representations
\begin{equation}\label{d1}
\Lambda^{2}(V^{*}) =\Lambda^{2}_{7}(V^{*})\oplus
\Lambda^{2}_{14}(V^{*})
\end{equation}
where
$$
\Lambda^{2}_{7}(V^{*}) := \{ *_{7}(\alpha\wedge *_{7}\phi_{0} ),\
\alpha\in V^{*}\} = \{ \beta\in\Lambda^{2}(V^{*}),\ 2*_{7}\beta =
\beta\wedge \phi_{0} \}
$$
is of dimension $7$ and
$$
\Lambda^{2}_{14}(V^{*}):= \{\beta\in\Lambda^{2}(V^{*}),\
*_{7}\beta =-\beta\wedge \phi_{0}\}
$$
is isomorphic to the adjoint representation and has dimension
$14.$ Similarly, $S^{2}(V^{*})$ decomposes into two
$G_{2}$-irreducible sub-representations
\begin{equation}\label{d2}
S^{2}(V^{*}) = S^{2}_{0}(V^{*})\oplus \mathbb{R}\langle\cdot ,
\cdot\rangle_{7}
\end{equation}
where $S^{2}_{0}(V^{*})$ denotes trace-less symmetric
$(2,0)$-tensors and $\mathbb{R}\langle\cdot , \cdot\rangle_{7}$ is
the $1$-dimensional representation generated by $\langle\cdot ,
\cdot\rangle_{7}.$ For proofs of these facts and more about the
representation theory of $G_{2}$, see e.g. \cite{bryant2},
\cite{andrei}.\\

Consider now a $7$-dimensional manifold $M$ with a
$G_{2}$-structure, defined by $\phi\in\Omega^{3}_{+}(M) .$ This
means that $\phi$ is a smooth $3$-form, linearly equivalent to
$\phi_{0}$ (i.e. at any point $p\in M$, there is a linear
isomorphism $f_{p}: T_{p}M\rightarrow V$ such that
$f_{p}^{*}(\phi_{0} ) =\phi_{p}$). The stabilizer $G_{2}\subset
GL(V)$ of $\phi_{0}$ being included in $SO(V)$, the standard
metric and orientation of $V$ induce, by means of the isomorphisms
$f_{p}$, a well-defined metric $g_{\phi}$ and an orientation on
$M$, such that $f_{p}$ is an orientation preserving isometry. We
denote by $*_{\phi}$ the associated Hodge star operator and we
freely identify vectors and covectors on $M$ using $g_{\phi}.$ Let
$\nabla$ be the Levi-Civita connection of $g_{\phi}.$ According to
\cite{sal}, \cite{gray} the covariant derivative $\nabla\phi$,
which is a section of $T^{*}M\otimes \Lambda^{3}(M)$, is actually
a section of $T^{*}M\otimes \Lambda^{3}_{7}(M)$, where
$\Lambda^{3}_{7}(M)$ is a sub-bundle of $\Lambda^{3}(M)$, defined
as
$$
\Lambda^{3}_{7}(M)= \{ *_{\phi} (\alpha\wedge \phi ),\ \alpha\in
T^{*}M\} .
$$
We shall identify $\Lambda^{3}_{7}(M)$ with $T^{*}M$ by means of
the isomorphism
\begin{equation}\label{identificare}
\Lambda^{3}_{7}(M)\ni \beta \rightarrow *_{\phi} (\beta\wedge \phi
)\in T^{*}M,
\end{equation}
which is the inverse (up to a multiplicative constant) of the
isomorphism
$$
T^{*}M\ni \alpha\rightarrow *_{\phi} (\alpha\wedge \phi )\in
\Lambda^{3}_{7}(M).
$$
More precisely, the following general identity holds:
\begin{equation}\label{general}
*_{\phi}( *_{\phi}(\alpha \wedge\phi )\wedge \phi ) =
-4\alpha,\quad\forall \alpha\in T^{*}M.
\end{equation}
Finally, from the isomorphism (\ref{identificare}) and the
decompositions (\ref{d1}) and (\ref{d2}), we get the following
decomposition of $T^{*}M\otimes \Lambda^{3}_{7}(M)$ into
irreducible sub-bundles:
\begin{equation}\label{var1}
T^{*}M\otimes \Lambda^{3}_{7}(M) \cong T^{*}M\otimes T^{*}M\cong
\Lambda^{2}_{7}(M)\oplus \Lambda^{2}_{14}(M)\oplus
S^{2}_{0}(M)\oplus\mathbb{R}_{\phi}(M),
\end{equation}
where
\begin{align*}
\Lambda^{2}_{7}(M) &= \{ \alpha\in \Lambda^{2}(M):\
2*_{\phi}\alpha =\alpha\wedge \phi \}\\
\Lambda^{2}_{14}(M) &= \{ \alpha\in \Lambda^{2}(M):\
*_{\phi}\alpha =- \alpha\wedge \phi \},
\end{align*}
$S^{2}_{0}(M)$ denotes the bundle of symmetric $(2,0)$-traceless
tensors and $\mathbb{R}_{\phi}(M)= \langle g_{\phi}\rangle$ is the
trivial rank one bundle generated by $g_{\phi }.$

From (\ref{var1}), there are 16 classes of $G_{2}$-manifolds in
the Gray-Hervella classification \cite{gray}. The
$G_{2}$-structure defined by $\phi$ is called {\it nearly
parallel} if $\nabla\phi$ is a section of the line bundle
$\mathbb{R}_{\phi}(M).$ When viewed as a sub-bundle of
$T^{*}M\otimes \Lambda^{3}_{7}(M)$, $\mathbb{R}_{\phi }(M)$ is
generated by $\sum_{k=1}^{7}e^{k}\otimes *_{\phi}(e^{k}\wedge\phi
)$, where $\{ e^{1}, \cdots , e^{7}\}$ is a local orthonormal
basis of $T^{*}M.$ Thus $\nabla\phi$ is a section of
$\mathbb{R}_{\phi}(M)$ if and only if $\nabla\phi$ is a multiple
of $*_{\phi}\phi .$

\subsection{Proof for the $G_{2}$ statement}

In this Section we prove Theorem \ref{main} {\it i)}. We use a
representation theoretic argument. Similar arguments already
appear in the literature \cite{liana}, \cite{swann2},
\cite{swann3}. Let $M$ be a $7$-manifold with a $G_{2}$-structure
defined by $\phi\in\Omega^{3}_{+}(M).$ With the notations from the
previous Section,  we aim to prove the following Proposition.

\begin{prop}\label{proof-g2} The $3$-form $\phi$ is conformal-Killing
with respect to $g_{\phi}$ if and only if the $G_{2}$-structure
defined by $\phi$ is nearly parallel.
\end{prop}

In order to prove Proposition \ref{proof-g2}, define the algebraic
conformal-Killing operator
$$
{\mathcal T}_{3}: T^{*}M\otimes \Lambda^{3}_{7}(M)\rightarrow
T^{*}M\otimes \Lambda^{3}(M)
$$
given on decomposable tensors by
\begin{equation}\label{p-1}
{\mathcal T}_{3}(\gamma\otimes\beta )(X) =\frac{3}{4}\gamma (X)
\beta +\frac{1}{4}\gamma\wedge i_{X}\beta -\frac{1}{5} X\wedge
i_{\gamma}\beta
\end{equation}
where $\gamma\in T^{*}M$, $\beta\in \Lambda^{3}_{7}(M)$, $X\in TM$
is identified with the dual $1$-form and $i_{\gamma}\beta :=\beta
(\gamma^{\flat}, \cdot )$ is the interior product of $\beta$ with
the vector field $\gamma^{\flat}$ dual to $\gamma$. (The operator
${\mathcal T}_{3}$ usually acts on the entire $T^{*}M\otimes
\Lambda^{3}(M)$, but we consider its restriction to $T^{*}M\otimes
\Lambda^{3}_{7}(M)$ only, because the covariant derivative
$\nabla\phi$ with respect to the Levi-Civita connection $\nabla$
of $g_{\phi}$ is a section of this bundle). Notice that
$$
{\mathcal T}_{3}(\nabla\phi )(X)= \nabla_{X}\phi
-\frac{1}{4}i_{X}d\phi +\frac{1}{5}X\wedge\delta \phi
,\quad\forall X\in TM.
$$
Thus $\phi$ is conformal-Killing if and only if
\begin{equation}\label{equa}
{\mathcal T}_{3}(\nabla \phi )=0.
\end{equation}
Recall now the decomposition of $T^{*}M\otimes \Lambda^{3}_{7}(M)$
into irreducible sub-bundles:
\begin{equation}\label{prime}
T^{*}M\otimes \Lambda^{3}_{7}(M)\cong \Lambda^{2}_{7}(M)\oplus
\Lambda^{2}_{14}(M)\oplus S^{2}_{0}(M)\oplus\mathbb{R}_{\phi}(M).
\end{equation}
From (\ref{equa}), Proposition \ref{proof-g2} is a consequence of
the following general result.

\begin{prop}\label{prime-prop} The restriction of ${\mathcal T}_{3}$ to $\Lambda^{2}(M)\oplus
S^{2}_{0}(M)$ is injective and the restriction of ${\mathcal
T}_{3}$ to $\mathbb{R}_{\phi}(M)$ is identically zero.
\end{prop}

We divide the proof of Proposition \ref{prime-prop} into two
Lemmas. First we need to introduce some notations. Define a map
\begin{equation}\label{extinsa}
T^{*}M\otimes \Lambda^{3}(M)\rightarrow T^{*}M\otimes T^{*}M,\quad
\alpha\otimes\beta \rightarrow \alpha\otimes
*_{\phi}(\beta\wedge\phi ).
\end{equation}
For any component $W$ in the decomposition (\ref{var1}) of
$T^{*}M\otimes T^{*}M$, let
$$
\mathrm{pr}_{W} :T^{*}M\otimes \Lambda^{3}(M)\rightarrow W
$$
be the composition of the map (\ref{extinsa}) with the projection
from $T^{*}M\otimes T^{*}M$ to $W$, according to the decomposition
(\ref{var1}). Let $\{ e^{1}, \cdots, e^{7}\}$ be a local
orthonormal positive oriented frame of $T^{*}M$, so that $\phi$ is
of the form (\ref{phi}), and define
$$
\eta := e^{1}\otimes *_{\phi}(e^{2}\wedge\phi ).
$$
From its very definition, $\eta$ is a section of $T^{*}M\otimes
\Lambda^{3}_{7}(M)$.

\begin{lem} With the notations above,
\begin{equation}\label{la-1}
(\mathrm{pr}_{\Lambda^{2}(M)}\circ{\mathcal T}_{3})(\eta ) =
-\frac{9}{10}(3e^{12} + e^{47} + e^{56})
\end{equation}
and
\begin{equation}\label{la-2}
(\mathrm{pr}_{S^{2}_{0}(M)}\circ{\mathcal T}_{3})(\eta )=
-\frac{7}{4} (e^{1}\otimes e^{2}+ e^{2}\otimes e^{1}).
\end{equation}
In particular $(\mathrm{pr}_{\Lambda^{2}_{7}(M)}\circ{\mathcal
T}_{3})(\eta )$, $(\mathrm{pr}_{\Lambda^{2}_{14}(M)}\circ{\mathcal
T}_{3})(\eta )$ and $(\mathrm{pr}_{S^{2}_{0}(M)}\circ{\mathcal
T}_{3})(\eta )$ are non-zero and ${\mathcal
T}_{3}\vert_{\Lambda^{2}(M)\oplus S^{2}_{0}(M)}$ is injective.
\end{lem}

\begin{proof} From (\ref{p1}),
$$
{\mathcal T}_{3}(\eta ) =\frac{3}{4}e^{1}\otimes
*_{\phi}(e^{2}\wedge\phi ) +\frac{1}{4} e^{k}\otimes
 e^{1}\wedge *_{\phi}(e^{k}\wedge e^{2}\wedge\phi ) -\frac{1}{5} e^{k}\otimes
e^{k}\wedge *_{\phi}(e^{1}\wedge e^{2}\wedge \phi )
$$
where, in order to simplify notations, we omitted the summation
sign over $1\leq k\leq 7$. Therefore, from the identification
(\ref{identificare}),
\begin{align*}
(\mathrm{pr}_{\Lambda^{2}(M)}\circ{\mathcal T}_{3})(\eta )&=
\frac{3}{4} e^{1}\wedge *_{\phi}(*_{\phi}(e^{2}\wedge \phi )
\wedge \phi )+\frac{1}{4} e^{k}\wedge *_{\phi} (e^{1}\wedge
*_{\phi}(e^{k}\wedge e^{2}\wedge\phi ) \wedge\phi )\\
&-\frac{1}{5} e^{k}\wedge *_{\phi} (e^{k}\wedge *_{\phi}
(e^{1}\wedge e^{2}\wedge\phi )\wedge \phi ).
\end{align*}
The first term from the right hand side of this equality is equal
to $-3e^{1}\wedge e^{2}$, using the identity (\ref{general}). For
the second term, we compute
$$
*_{\phi}(e^{2}\wedge \phi )= - (e^{367} + e^{345} + e^{156}
+e^{147})
$$
and, for any $k\in \{ 1,2,\cdots , 7\}$,
\begin{align*}
*_{\phi}( e^{k}\wedge e^{2}\wedge \phi ) &= -
\delta_{k1}(e^{47} + e^{56}) - \delta_{k3} (e^{45} + e^{67})\\
&+\delta_{k4} (e^{17} + e^{35}) + \delta_{k5} ( e^{16}- e^{34})\\
&+\delta_{k6} (e^{37}-e^{15}) - \delta_{k7} (e^{14}+ e^{36}).
\end{align*}
It follows that
$$
e^{1}\wedge \phi \wedge *_{\phi} (e^{k}\wedge e^{2}\wedge\phi )=
2\delta_{k1} e^{134567} -\delta_{k4}e^{123456} +\delta_{k5}
e^{123457} +\delta_{k6} e^{123467} -\delta_{k7} e^{123567}
$$
and
$$
*_{\phi}(e^{1}\wedge \phi \wedge *_{\phi} (e^{k}\wedge
e^{2}\wedge\phi ))=- 2\delta_{k1} e^{2} -\delta_{k4}e^{7}
-\delta_{k5} e^{6} +\delta_{k6} e^{5} +\delta_{k7} e^{4}.
$$
Therefore,
$$
e^{k}\wedge *_{\phi}(e^{1}\wedge \phi \wedge *_{\phi} (e^{k}\wedge
e^{2}\wedge\phi ))= -2(e^{12} +e^{47} +e^{56}),
$$
which provides the second term in the expression of
$(\mathrm{pr}_{\Lambda^{2}(M)}\circ{\mathcal T}_{3})(\eta ).$ A
similar calculation finally shows that
$$
e^{k}\wedge *_{\phi} ( e^{k}\wedge *_{\phi}(e^{1}\wedge
e^{2}\wedge \phi ) \wedge \phi )= 2(- 2e^{12}+e^{47} + e^{56}).
$$
Putting together the results of these computations, we obtain
(\ref{la-1}). Relation (\ref{la-2}) follows from a similar
computation. Using (\ref{la-1}) it can be checked that
\begin{equation}\label{rel1}
(\mathrm{pr}_{\Lambda^{2}(M)}\circ{\mathcal T}_{3})(\eta )\wedge
\phi = \frac{9}{5}( e^{12347} + e^{12356} + e^{34567})
\end{equation}
and
\begin{equation}\label{rel2}
*_{\phi}(\mathrm{pr}_{\Lambda^{2}(M)}\circ {\mathcal T}_{3})(\eta
) = -\frac{9}{10}( e^{12347} + e^{12356} + 3e^{34567}).
\end{equation}
Relations (\ref{rel1}) and (\ref{rel2}) imply that
$(\mathrm{pr}_{\Lambda^{2}_{7}(M)}\circ{\mathcal T}_{3})(\eta )$
and $(\mathrm{pr}_{\Lambda^{2}_{14}(M)}\circ{\mathcal T}_{3})(\eta
)$ are both non-zero. From the decomposition (\ref{prime}) and an
easy argument which uses the Schur's lemma,
$\mathrm{pr}_{\Lambda^{2}_{7}(M)}\circ{\mathcal T}_{3}$,
$\mathrm{pr}_{\Lambda^{2}_{14}(M)}\circ{\mathcal T}_{3}$ and
$\mathrm{pr}_{S^{2}_{0}(M)}\circ{\mathcal T}_{3}$ non-trivial
imply that ${\mathcal T}\vert_{\Lambda^{2}(M)\oplus S^{2}_{0}(M)}$
is injective, as required.
\end{proof}

In order to conclude the proof of Proposition \ref{prime-prop} we
still need to show that $\mathcal T$ is identically zero on
$\mathbb{R}_{\phi}(M)$. This is done in the following Lemma.

\begin{lem} The restriction ${\mathcal T}_{3}\vert_{\mathbb{R}_{\phi}(M)}$
is identically zero.
\end{lem}

\begin{proof} Recall that $T^{*}M\otimes \Lambda^{3}(M)$
contains $\mathbb{R}_{\phi }(M)$ with multiplicity one and the
projection $\mathrm{pr}_{\mathbb{R}_{\phi}(M)}$ onto
$\mathbb{R}_{\phi}(M)$ has the following expression:
\begin{equation}\label{p1}
T^{*}M\otimes\Lambda^{3}(M)\ni\gamma\otimes\beta \rightarrow
*_{\phi}(\gamma\wedge \beta\wedge \phi
)g_{\phi}\in\mathbb{R}_{\phi}(M).
\end{equation}
Schur's Lemma again implies that ${\mathcal
T}_{3}\vert_{\mathbb{R}_{\phi}(M)}$ is identically zero if and
only if $\mathrm{pr}_{\mathbb{R}_{\phi}(M)}\circ{\mathcal T}_{3}$
is identically zero. On the other hand, from definition
(\ref{p-1}) of ${\mathcal T}_{3}$ and (\ref{p1}),
\begin{align*}
(\mathrm{pr}_{\mathbb{R}_{\phi}(M)}\circ {\mathcal T}_{3})(\gamma
\otimes\beta )&= \mathrm{pr}_{\mathbb{R}_{\phi}(M)}\left(
e^{k}\otimes {\mathcal T}_{3}(\gamma\otimes\beta )(e_{k})\right)\\
&= \mathrm{pr}_{\mathbb{R}_{\phi}(M)} \left( \frac{3}{4}
\gamma\otimes\beta +\frac{1}{4} e^{k}\otimes \gamma\wedge
i_{e_{k}}\beta -\frac{1}{5} e^{k}\otimes e^{k}\wedge
i_{\gamma}\beta \right)\\
&= *_{\phi}\left(\left( \frac{3}{4}\gamma\wedge \beta +\frac{1}{4}
e^{k}\wedge \gamma\wedge i_{e_{k}}\beta \right)\wedge\phi\right) g_{\phi}\\
\end{align*}
which is zero, since
$$
3\gamma\wedge\beta +e^{k}\wedge \gamma\wedge i_{e_{k}}\beta
=0,\quad\forall\gamma\in T^{*}M,\quad\forall\beta\in
\Lambda^{3}(M).
$$
Our claim follows.
\end{proof}

The proof is of Proposition \ref{prime-prop} is now completed.
Thus Proposition \ref{proof-g2} and Theorem \ref{main} {\it ii)}
follow.

\section{The statement for $\mathrm{Spin}_{7}$}\label{spin7}

\subsection{Basic facts about
$\mathrm{Spin}_{7}$-structures}\label{spin7-basic}

Extend the vector space $V =\mathbb{R}^{7}$ of Section
\ref{g2-basic} to $V_{+}: = \mathbb{R}e_{0}\oplus V$. The group
$\mathrm{Spin}_{7}< GL(V_{+})$ is a $21$-dimensional compact,
connected, simply connected Lie group defined as the stabilizer of
the $4$-form
\begin{equation}\label{psi0}
\psi_{0} = e^{0}\wedge\phi_{0} + *_{7}\phi_{0} ,
\end{equation}
where we used the isomorphism $\Lambda^{4}(V_{+}^{*}) \cong
\Lambda^{3}(V^{*})\oplus \Lambda^{4}(V^{*})$, $\phi_{0}$ and
$*_{7}$ were defined in Section \ref{g2-basic} and $e^{0}\in
(V_{+})^{*}$ takes value one on $e_{0}$ and annihilates $V.$ In
terms of the standard basis $\{ e_{0}, \cdots , e_{7}\}$ of
$V_{+}$ and the dual basis $\{ e^{0}, \cdots , e^{7}\}$,
\begin{align*}
\psi_{0} =e^{0123}+e^{0145}+e^{0167}+e^{0246}-e^{0257}-e^{0347}-e^{0356}\\
+e^{4567} +e^{2367}+e^{2345}+e^{1357}-e^{1346}-e^{1256}-e^{1247}.
\end{align*}
From (\ref{psi0}), the stabilizer of $e_{0}$ in
$\mathrm{Spin}_{7}$ is isomorphic to $G_{2}.$ The vector space
$V_{+}$ has a standard metric $\langle\cdot , \cdot\rangle_{8}$
and orientation, for which $\{ e^{0}, \cdots , e^{7}\}$ is
orthonormal and positive oriented and we shall denote by $*_{8}$
the associated Hodge star operator. It can be shown that
$\mathrm{Spin}_{7}$ preserves this metric and orientation and the
map $\mathrm{Spin}_{7}\rightarrow S^{7}$ defined by $g\rightarrow
g(e_{0})$ is a $G_{2}$-fibration.

The group $\mathrm{Spin}_{7}$ acts irreducibly on $V_{+}\cong
V_{+}^{*}$, but its action on higher degree forms is reducible in
general. For our purpose we need to recall the irreducible
decomposition of $\Lambda^{3}(V_{+}^{*})$ only. As shown in
\cite{bryant1}, $\Lambda^{3}(V_{+}^{*})$ decomposes into
irreducible $\mathrm{Spin}_{7}$-modules as
\begin{equation}\label{lambda3}
\Lambda^{3}(V_{+}^{*})=\Lambda^{3}_{8}(V_{+}^{*}) \oplus
\Lambda^{3}_{48}(V_{+}^{*}),
\end{equation}
where $\Lambda^{3}_{8}(V_{+}^{*})$ and
$\Lambda^{3}_{48}(V_{+}^{*})$, of dimension $8$ and $48$
respectively, are defined by
\begin{align*}
\Lambda^{3}_{8}(V_{+}^{*}) &:= \{ *_{8} (\psi_{0}\wedge \alpha ),
\quad\alpha\in V_{+}^{*}\}\\
\Lambda^{3}_{48}(V_{+}^{*}) &:= \{ \beta\in\Lambda^{3}(V_{+}^{*}),
\quad \beta\wedge \psi_{0} =0\} .
\end{align*}

Consider now a $\mathrm{Spin}_{7}$-structure on an $8$-manifold
$M$, defined by a smooth $4$-form $\psi\in \Omega^{4}_{+}(M)$
linearly equivalent to $\psi_{0}$ at any point of $M$. Since
$\mathrm{Spin}_{7}< SO(V_{+})$, $\psi$ determines a canonical
metric $g_{\psi}$ and an orientation on $M$, for which any linear
isomorphism $f_{p}: T_{p}M\rightarrow V_{+}$ with
$f_{p}^{*}(\psi_{0})= \psi_{p}$ is an orientation preserving
isometry. We denote by $*_{\psi}$ the Hodge star operator
associated to $g_{\psi}$ and this orientation. We shall freely
identify vectors and covectors on $M$ using $g_{\psi }.$ Let
$\nabla$ be the Levi-Civita connection of $g_{\psi}.$  As shown in
\cite{sal} and \cite{gray} (see also \cite{bryant2}) the covariant
derivative $\nabla\psi$ is a section of the tensor product
$T^{*}M\otimes \Lambda^{4}_{7}(M)$, where $\Lambda^{4}_{7}(M)$ is
a rank $7$-bundle generated by $4$-forms $X\wedge i_{Y}\psi -
Y\wedge i_{X}\psi$, for any $X, Y\in TM$. Moreover,  the inner
contraction map
\begin{equation}\label{inner}
T^{*}M\otimes \Lambda^{4}_{7}(M)\rightarrow \Lambda^{3}(M),
X\otimes \alpha \rightarrow i_{X}\alpha
\end{equation}
is an isomorphism \cite{fernandez}. From (\ref{lambda3}) and
(\ref{inner}), the isomorphic bundles $T^{*}M\otimes
\Lambda^{4}_{7}(M)$ and $\Lambda^{3}(M)$ decompose into
irreducible sub-bundles as
\begin{equation}\label{ire}
T^{*}M\otimes \Lambda^{4}_{7}(M) \cong \Lambda^{3}(M)\cong
\Lambda^{3}_{8}(M)\oplus \Lambda^{3}_{48}(M),
\end{equation}
where
$$
\Lambda^{3}_{8}(M) := \{ *_{\psi} (\psi \wedge \alpha ), \quad
\alpha\in T^{*}M\}
$$
and
$$
\Lambda^{3}_{48}(M) := \{ \beta\in\Lambda^{3}(M), \quad
\beta\wedge \psi =0\} .
$$
It follows that there are four classes of
$\mathrm{Spin}_{7}$-manifolds \cite{fernandez}. The
$\mathrm{Spin}_{7}$-structure is called {\it parallel} if
$\nabla\psi =0.$

\subsection{Proof for the $\mathrm{Spin}_{7}$ statement}

In this Section we prove Theorem \ref{main} {\it ii)}. We use a
similar method like for Theorem \ref{main} {\it i)}, but the
computations are more involved. Let $M$ be an $8$-manifold with a
$\mathrm{Spin}_{7}$-structure defined by
$\psi\in\Omega^{3}_{+}(M).$ With the notations from the previous
Section, we aim to prove the following Proposition.

\begin{prop}\label{p-spin} The $4$-form $\psi$ is conformal-Killing
with respect to $g_{\psi}$ if and only if the
$\mathrm{Spin}_{7}$-structure defined by $\psi$ is parallel.
\end{prop}

Like before, consider the algebraic conformal-Killing operator
\begin{equation}\label{o1}
{\mathcal T}_{4} :T^{*}M\otimes \Lambda^{4}_{7}(M) \rightarrow
T^{*}M\otimes \Lambda^{4}(M)
\end{equation}
given on decomposable tensors by
\begin{equation}\label{o2}
{\mathcal T}_{4}(\gamma\otimes \alpha ) (X)= \frac{4}{5} \gamma
(X)\alpha +\frac{1}{5} \gamma \wedge i_{X}\alpha -\frac{1}{5}
X\wedge i_{\gamma}\alpha
\end{equation}
where $\alpha\in \Lambda^{4}_{7}(M)$, $X\in TM$ is identified with
a covector using $g_{\psi}$ and $i_{\gamma}\alpha := \alpha
(\gamma^{\flat},\cdot )$ denotes the inner product of $\alpha$ and
the dual vector field $\gamma^{\flat}$ corresponding to $\gamma
\in T^{*}M$. (${\mathcal T}_{4}$ usually acts on the entire
$T^{*}M\otimes \Lambda^{4}(M)$ but we consider its restriction to
$T^{*}M\otimes \Lambda^{4}_{7}(M)$ only, because the covariant
derivative $\nabla\psi$ with respect to the Levi-Civita connection
$\nabla$ of $g_{\psi}$ is a section of this bundle). Note that
$$
{\mathcal T}_{4}(\nabla \psi )(X) =\nabla_{X}\psi -
\frac{1}{5}i_{X}d\psi +\frac{1}{5} X\wedge \delta \psi
,\quad\forall X\in TM
$$
and thus $\psi$ is conformal-Killing if and only if
\begin{equation}\label{spin-conf}
{\mathcal T}_{4}(\nabla \psi )=0.
\end{equation}

From (\ref{spin-conf}), Proposition \ref{p-spin} is a consequence
of the following general result.

\begin{prop}\label{a-spin} The algebraic conformal-Killing operator
${\mathcal T}_{4}$ defined by (\ref{o1}) and (\ref{o2}) is
injective.\end{prop}

In order to prove Proposition \ref{a-spin}, we will find maps
$$
P_{8}: T^{*}M\otimes \Lambda^{4}(M) \rightarrow \Lambda^{3}_{8}(M)
$$
and
$$
P_{48}: T^{*}M\otimes \Lambda^{4}(M) \rightarrow
\Lambda^{3}_{48}(M)
$$
such that the compositions $P_{8}\circ {\mathcal T}_{4}$ and
$P_{48}\circ {\mathcal T}_{4}$ are non-trivial. (The existence of
such maps would readily imply, from Schur's lemma and the
decomposition (\ref{ire}) of $T^{*}M\otimes \Lambda^{4}_{7}(M)$,
that ${\mathcal T}_{4}$ is injective, as required in Proposition
\ref{a-spin}).\\

In order to define the maps $P_{8}$ and $P_{48}$ we need to
introduce more notations, as follows. First, define
$$
p:\Lambda^{4}(M) \rightarrow \Lambda^{2}(M)
$$
by
\begin{equation}\label{def-p}
p(\beta ) (X, Y) = \langle \psi , i_{X}\beta\wedge Y - i_{Y}\beta
\wedge X\rangle ,\quad \beta\in\Lambda^{4}(M),\quad X ,Y\in TM,
\end{equation}
where $\langle\cdot , \cdot\rangle$ denotes the inner product
which on decomposable skew-symmetric multi-vectors (or forms) is
defined by
$$
\langle X_{1}\wedge \cdots X_{p}, Y_{1}\wedge \cdots Y_{p}\rangle
=\mathrm{det}g_{\psi}(X_{i}, Y_{j}).
$$
Next, by tensoring the map $p$ with the identity map on $T^{*}M$
we get a map
$$
\mathrm{id}\otimes p:T^{*}M\otimes \Lambda^{4}(M)\rightarrow
T^{*}M\otimes \Lambda^{2}(M),
$$
which, composed at the right with skew symmetrization, gives a map
$$
P: T^{*}M\otimes \Lambda^{4}(M) \rightarrow \Lambda^{3}(M).
$$
Finally, composing further $P$ with the projections
$\Lambda^{3}(M)\rightarrow \Lambda^{3}_{8}(M)$ and
$\Lambda^{3}(M)\rightarrow \Lambda^{3}_{48}(M)$ according to
(\ref{ire}), we get the two maps $P_{8}$ and $P_{48}$ we were
looking for.

In order to show that the compositions $P_{8}\circ {\mathcal
T}_{4}$ and $P_{48}\circ {\mathcal T}_{4}$ are non-trivial, we
will find a particular $\eta\in T^{*}M\otimes\Lambda^{4}(M)$ such
that both $P_{8}\circ {\mathcal T}_{4}$ and $P_{48}\circ {\mathcal
T}_{4}$ take non-zero value on $\eta .$ To define $\eta$, consider
a local positive oriented orthonormal frame $\{ e_{0}, \cdots ,
e_{7}\}$ of $TM$ and its dual frame $\{ e^{0}, \cdots , e^{7}\}$,
such that $\psi$ has the form
\begin{align*}
\psi =
e^{0123}+e^{0145}+e^{0167}+e^{0246}-e^{0257}-e^{0347}-e^{0356}\\
+e^{4567} +e^{2367}+e^{2345}+e^{1357}-e^{1346}-e^{1256}-e^{1247}.
\end{align*}
Define
$$
\alpha_{0}:= i_{e_{0}}\psi \wedge e^{1}- i_{e_{1}}\psi \wedge
e^{0}
$$
and
$$
\eta := e^{0}\otimes \alpha_{0}.
$$
From its very definition, $\alpha_{0}$ is a section of
$\Lambda^{4}_{7}(M)$ and $\eta$ is a section of $T^{*}M\otimes
\Lambda^{4}_{7}(M).$ In the following Lemmas we will compute
$(P\circ {\mathcal T}_{4})(\eta )$. Since
\begin{equation}\label{pc}
(P\circ {\mathcal T}_{4})(\gamma\otimes \alpha ) =
\frac{4}{5}\gamma\wedge p(\alpha ) +\frac{1}{5} e^{k}\wedge p(
\gamma\wedge i_{e_{k}}\alpha ) - \frac{1}{5}e^{k}\wedge
p(e^{k}\wedge i_{\gamma}\alpha ) ,
\end{equation}
for any $\gamma \in T^{*}M$ and $\alpha\in \Lambda^{4}(M)$,  we
need to compute $p(\alpha_{0})$, $e^{k}\wedge p(e^{k}\wedge
i_{e_{0}}\alpha_{0})$ and $e^{k}\wedge p( e^{0}\wedge
i_{e_{k}}\alpha_{0})$. (As usual, we omit the summation sum over
$0\leq k\leq 7$).\\

First, we compute $p(\alpha_{0}).$

\begin{lem}\label{l1}
The $3$-form $p(\alpha_{0})$ has the following expression:
\begin{equation}\label{e1}
p(\alpha_{0}) = -8(e^{01}+e^{23}+e^{45}+e^{67}).
\end{equation}
\end{lem}

\begin{proof} From the definition (\ref{def-p}) of the map $p$,
$$
p(\alpha_{0})(X,Y)= \langle \psi , i_{X}\alpha_{0}\wedge Y -
i_{Y}\alpha_{0}\wedge X\rangle ,\quad\forall X, Y\in TM.
$$
Define a $1$-form $\psi (i_{X}\alpha_{0}, \cdot )$ by
$$
\psi (i_{X}\alpha_{0}, Y):= \langle \psi , i_{X}\alpha_{0}\wedge
Y\rangle ,\quad \forall Y\in TM.
$$
With this notation,
\begin{equation}\label{alpha-0}
p(\alpha_{0})(X,Y) =\psi (i_{X}\alpha_{0}, Y) -\psi
(i_{Y}\alpha_{0}, X).
\end{equation}
In terms of the local frame $\{ e^{0}, \cdots ,e^{7}\}$ chosen
above,
\begin{equation}\label{detaliat}
\alpha_{0}= -e^{1246} +e^{1257} +e^{1347} +e^{1356} + e^{0357}
-e^{0346}- e^{0256} -e^{0247}
\end{equation}
and, from a straightforward computation,
\begin{align*}
i_{X}\alpha_{0} &= e^{0}(X) (e^{357} -e^{346} - e^{256}
-e^{247}) +e^{1}(X)( -e^{246} +e^{257}+e^{347}+e^{356})\\
& +e^{2}(X) (e^{146}-e^{157}+e^{056}+e^{047})
+e^{3}(X) (-e^{147}-e^{156}-e^{057}+e^{046})\\
&+e^{4}(X)(-e^{126}+e^{137}-e^{036}-e^{027} )
+e^{5}(X)(e^{127}+e^{136}+e^{037}-e^{026})\\
&+e^{6}(X) (e^{124}-e^{135}+e^{034}+e^{025})
+e^{7}(X)(-e^{125}-e^{134}-e^{035}+e^{024}).
\end{align*}
Using this computation and the expression of $\psi$ in the frame
$\{ e_{0}, \cdots , e_{7}\}$ we get
\begin{align*}
\psi (i_{X}\alpha_{0}, \cdot )&= -4e^{0}(X) e^{1} +4e^{1}(X) e^{0}
-4e^{2}(X) e^{3}+4
e^{3}(X) e^{2}\\
& -4e^{4}(X)e^{5} +4e^{5}(X)e^{4} -4e^{6}(X) e^{7}
+4e^{7}(X)e^{6}.
\end{align*}
Applying this relation to $Y$, skew-symmetrizing the result in $X$
and $Y$ and using (\ref{alpha-0}) we get (\ref{e1}), as required.
\end{proof}

Next, we compute $e^{k}\wedge p(e^{k}\wedge i_{e_{0}}\alpha_{0})$.

\begin{lem}\label{l2} The $3$-form $e^{k}\wedge p(e^{k}\wedge
i_{e_{0}}\alpha_{0})$ has the following expression:
\begin{equation}\label{comut1}
e^{k}\wedge p(e^{k}\wedge i_{e_{0}}\alpha_{0})= -8(e^{045}
 +e^{023} +e^{067}) -6(e^{256} +e^{247}+e^{346}-e^{357}).
\end{equation}
\end{lem}

\begin{proof}
From the definition (\ref{def-p}) of the map $p$, for any $k\in \{
0,\cdots , 7\}$ and $X, Y\in TM$,
\begin{align*}
p(e^{k}\wedge i_{e_{0}}\alpha_{0})(X,Y) &= e^{k}(X)\langle \psi
,i_{e_{0}}\alpha_{0}\wedge Y\rangle - e^{k}(Y) \langle \psi ,
i_{e_{0}}\alpha_{0}\wedge X\rangle\\
& + \langle\psi , i_{X}(i_{e_{0}}\alpha_{0})\wedge Y\wedge
e_{k}\rangle -\langle\psi , i_{Y}(i_{e_{0}}\alpha_{0})\wedge X
\wedge e_{k}\rangle .
\end{align*}
From
\begin{equation}\label{i}
i_{e_{0}}\alpha_{0}= e^{357}-e^{346}-e^{256}-e^{247}
\end{equation}
and the expression of $\psi$ in the frame $\{ e_{0}, \cdots ,
e_{7}\}$, we get
\begin{equation}\label{E1}
\langle \psi , i_{e_{0}}\alpha_{0}\wedge X\rangle  =
-4e^{1}(X),\quad\forall X\in TM
\end{equation}
and thus
\begin{equation}\label{1k}
e^{k}(X)\langle \psi ,i_{e_{0}}\alpha_{0}\wedge Y\rangle -
e^{k}(Y) \langle \psi , i_{e_{0}}\alpha_{0}\wedge X\rangle =
4e^{1k}(X, Y).
\end{equation}
It remains to compute the terms of the form $\langle \psi ,
i_{X}(i_{e_{0}}\alpha_{0})\wedge Y\wedge e_{k}\rangle .$ For this,
define a $2$-form $\psi (i_{X}(i_{e_{0}}\alpha_{0}),\cdot )$ whose
value on a pair of vectors $(Y, Z)$ is equal to $\langle \psi ,
i_{X}(i_{e_{0}}\alpha_{0} )\wedge Y\wedge Z\rangle$. Taking the
inner product of $i_{e_{0}}\alpha_{0}$ given by (\ref{i}) with
$X\in TM$ and contracting the resulting expression with $\psi$ we
get
\begin{align*}
\psi (i_{X}(i_{e_{0}}\alpha ), \cdot )&= e^{5}(X)
(-2e^{04} -e^{37} +e^{26} +2e^{15})+e^{6}(X)(2e^{07} -e^{34} -e^{25} +2e^{16})\\
&+e^{4}(X)(2e^{05} +e^{36} +e^{27} +2e^{14})+e^{3}(X)(-2e^{02} -e^{46}+2e^{13} +e^{57})\\
&+e^{2}(X)(2e^{03}+2e^{12} -e^{47} -e^{56}) +e^{7}(X) (
-2e^{06}+e^{35} +2e^{17} -e^{24}).
\end{align*}
Applying $\psi (i_{X}(i_{e_{0}}\alpha ),\cdot )$  to $(Y,Z)$ and
skew-symmetrizing the result in $X$ and $Y$ we obtain
\begin{align*}
\langle \psi , i_{X}(i_{e_{0}}\alpha_{0}) \wedge Y\wedge Z -
i_{Y}(i_{e_{0}}\alpha_{0})\wedge X\wedge Z\rangle =
-4(e^{23}+e^{45} +e^{67})(X,Y)e^{0}(Z)\\
+2(e^{05}+e^{36}-e^{14}+e^{27})(X,Y) e^{4}(Z)
+2(e^{35}-e^{06}-e^{24}-e^{17})(X,Y)e^{7}(Z)\\
+2(e^{57}-e^{46}-e^{13}-e^{02})(X,Y)e^{3}(Z)
+2(e^{52}+e^{61}+e^{43}+e^{07})(X,Y)e^{6}(Z)\\
+2(e^{65}+e^{74}+e^{03}+e^{21})(X,Y)e^{2}(Z)
+2(e^{51}+e^{26}+e^{40}+e^{73})(X,Y)e^{5}(Z),\\
\end{align*}
for any $X, Y, Z\in TM.$ Letting in this expression $Z:= e^{k}$
and using the definition of $p$ together with (\ref{1k}) we get
\begin{align*}
p(e^{k}\wedge i_{e_{0}}\alpha_{0}) &= 4e^{1k}
-4\delta_{k0}(e^{23}+e^{45}+e^{67})+2\delta_{k4}(e^{05} +e^{36}
-e^{14} +e^{27})\\
&+2\delta_{k7}(e^{35} -e^{06} -e^{24} -e^{17})+2\delta_{k3}(
e^{57}-e^{46} -e^{13} -e^{02})\\
&+2\delta_{k6}(e^{52}+ e^{61}+ e^{43}+
e^{07})+2\delta_{k2}(e^{65}+e^{74}+e^{03} +e^{21})\\
& +2\delta_{k5} (e^{51}+e^{26} +e^{40} +e^{73}),
\end{align*}
for any fixed $k$. Relation (\ref{comut1}) follows now easily.
\end{proof}

Finally, it remains to compute $e^{k}\wedge p(e^{0}\wedge
i_{e_{k}}\alpha_{0}).$

\begin{lem}\label{l3} The $3$-form $e^{k}\wedge p(e^{0}\wedge
i_{e_{k}}\alpha_{0})$ has the following expression:
$$
e^{k}\wedge p(e^{0}\wedge i_{e_{k}}\alpha_{0}) =
6(e^{247}-e^{357}+e^{256}+e^{346}).
$$
\end{lem}

\begin{proof}
From the definition of the map $p$, one can check that
\begin{align*}
p(e^{0}\wedge i_{e_{k}}\alpha_{0} )(X,Y)&= 4 (-\delta_{k0}e^{01}
-\delta_{k2}e^{03}+\delta_{k3}e^{02}-\delta_{k4}e^{05})(X,Y)\\
& +4(\delta_{k5}e^{04}-\delta_{k6}e^{07}
+\delta_{k7}e^{06})(X,Y)\\
&-\langle \phi , i_{X}(i_{e_{k}}\alpha_{0})\wedge Y -
i_{Y}(i_{e_{k}}\alpha_{0})\wedge X\rangle ,
\end{align*}
where
$$
\phi := e^{123} +e^{145} + e^{167}+ e^{246} - e^{257} -e^{347} -
e^{356}
$$
and $k\in \{ 0,\cdots , 7\}$ is fixed. Note that
\begin{align*}
&e^{k}\wedge ( -\delta_{k0}e^{01}
-\delta_{k2}e^{03}+\delta_{k3}e^{02}-\delta_{k4}e^{05}
+\delta_{k5}e^{04}-\delta_{k6}e^{07} +\delta_{k7}e^{06}))\\
&=2(e^{023}+ e^{045} +e^{067}).
\end{align*}
For any $1\leq k\leq 7$ define a $2$-form by
$$
\beta_{k}(X, Y) = \langle \phi , i_{X}(i_{e_{k}}\alpha_{0})\wedge
Y - i_{Y}(i_{e_{k}}\alpha_{0})\wedge X\rangle .
$$
From a long but straightforward computation which uses the
expressions of $\alpha_{0}$ and $\phi$ in the frame $\{ e_{0},
\cdots , e_{7}\}$,
\begin{equation}\label{betak}
e^{k}\wedge\beta_{k}= 6(-e^{247} +e^{357}-e^{256} -e^{346})
+8(e^{023} +e^{045}+ e^{067}).
\end{equation}
Combining (\ref{betak}) with the expression of $p(e^{0}\wedge
i_{e_{k}}\alpha )$ we get our claim.
\end{proof}

The following Lemma concludes the proof of Proposition
\ref{a-spin}.

\begin{lem}\label{FINAL} The value of $P\circ {\mathcal T}_{4}$ on $\eta_{0}$
has the following expression:
\begin{equation}\label{fin}
5(P\circ {\mathcal T}_{4})(\eta_{0}) =
-24(e^{023}+e^{045}+e^{067}) +12(e^{247}-e^{357}+e^{256}+e^{346}).
\end{equation}
In particular, $(P_{8}\circ {\mathcal T}_{4})(\eta_{0})$ and
$(P_{48}\circ {\mathcal T}_{4})(\eta_{0})$ are non-zero.
\end{lem}

\begin{proof} Relation (\ref{fin}) follows from relation (\ref{pc}) and the previous Lemmas.
A direct check shows that (\ref{fin}) is not of the form
$i_{X}\psi$, for $X\in TM$  and thus $(P_{8}\circ {\mathcal
T}_{48}) (\eta_{0})$ is non-zero. Moreover it can be checked that
$$
5(P\circ {\mathcal T}_{4})(\eta_{0})\wedge \psi = -24 e^{0234567}.
$$
In particular,  $(P_{8}\circ {\mathcal T})(\eta_{0})$ is also
non-zero.
\end{proof}

The proof of Proposition \ref{a-spin} is now completed.
Proposition \ref{p-spin} and Theorem \ref{main} {\it ii)} follow.


TITLE: Brightness and the incorrect direction of current
QUESTION [0 upvotes]: I was thinking of a circuit similar to this:
$(+)$ $\rightarrow  R_1 \rightarrow  R_2 \rightarrow $ $(-)$
where $R_1$ and $R_2$ are lightbulbs. I have read that as the voltage drops at $R_1$, we then expect a lower voltage through $R_2$, thus, less brightness at the light bulb.
However, the conventional current direction is 'incorrect' as it is the negative charges that flow from the negative end to the positive end. Considering that 'direction', wouldn't we expect $R_2$ to be brighter than $R_1$, since true current passes through $R_1$ first?
How can I make sense of the true current direction and the voltage drops across the resistors?
Thank you guys for your help.

REPLY [1 votes]: . . . . we then expect a lower voltage through ????? R2 . . . .
This statement may be the reason for your question.
The (conventional = equivalent to the flow of positive charges) current through both resistors is the same as both resistors are connected in series.
The potential difference (voltage) is measured across each of the resistors.
The current flows from a point of higher potential to a point of lower potential.
If both resistors have the same resistance then the potential difference across each of the resistors is the same.
You should be under the misapprehension that a charged particle leaves the terminal of a battery with lost of energy, goes around an electrical circuit losing energy and then arriving at the other terminal of the battery with less energy.
What happens is that a charged particle leaves the terminal of a battery and is accelerated by an electric field thus gaining kinetic energy.
The charged particle then collides with an ion (charged atom of the conductor) and loses kinetic energy in the process which then means that the ion vibrates more.
The process then repeats itself so that the moving charge repeated gains and loses energy so in effect has the same kinetic energy when it leaves one terminal of the battery and arrives at the other terminal of the battery.


text:
\begin{document}
\date{}
\maketitle

\begin{abstract}
We prove that finiteness of the
index of the intersection of a finite set of finite index subalgebras
in a von Neumann algebra (with small centre) is equivalent to the
finite dimensionality of the algebra generated by
the conditional expectations onto the subalgebras. \par
2000 Mathematics Subject Classification. 46S99, 81R10.
Key words: Subfactors, Conformal Field Theories.
\end{abstract}

\newpage

\section{Introduction}
Subfactor theory provides an entry point into a world of mathematics and
physics containing large parts of conformal field theory, 2-dimensional statistical mechanical
models, quantum groups, finitely presented groups and low dimensional topology.
We cite two facts attesting to the effectiveness of the subfactor point of view.
The first is the discovery of a polynomial invariant of links in \cite{J3}
and the second is the use of subfactors by Wassermann in \cite{W} to give a
rigorous and entirely unitary definition of fusion of loop group representations.

But the very breadth of areas in this world means that one would not expect
subfactors to yield the most effective technique for
any of the particular topics mentioned above. As things stand for instance it
would be foolish to expect subfactor ideas to have a big impact on
finite group theory. It is more reasonable to expect subfactors to provide the
machinery for theories close to the abstract structure of a subfactor itself.
The most obvious such structure is an intermediate subfactor and the
first author and Bisch in \cite{BJ} led to the discovery of an entirely new
algebra, in the spirit of the type A Hecke algebra,  which they called the
"Fuss-Catalan" algebra. This success is a compelling argument for further work
on the lattice of intermediate subfactors of a factor, which had been suggested in
papers by Watatani (\cite{Wat})  and Watatani and Sano (\cite{SW}) . The main contribution of
\cite{SW} was to introduce the angle operator between two subfactors as an invariant
of a pair of subfactors and calculate several examples. The following question
is hinted at but never mentioned explicitly in \cite{SW}:

$\cal Q$:{\it If the spectrum of the angle operator between two finite index subfactors is finite,
is the intersection of finite index? }

(Note that it follows from the elementary theory of the index that the angle condition
is necessary for the intersection to have finite index.) The simplest non-trivial example
of a subfactor is given by a pair $\Gamma_0 \subseteq \Gamma$ of discrete groups each having
infinite non-identity conjugacy  classes. The group von Neumann algebras give the factor
and subfactor $M$ and $N$ and $[M:N]=[\Gamma:\Gamma_0]$. Of course the intersection of
two finite index subgroups has finite index so this case is rather special. Indeed what
is special is that the orthogonal projections (=conditional expectations) onto these subgroup
subfactors all commute so that angles are all either $0$ or $\pi/2$ so the angle operator has
only two elements in its spectrum. Another simple construction of subfactors is as fixed point
algebras. If $G$ and $H$ are two finite groups of automorphisms of a II$_1$ factor $M$ then
the fixed point algebras $M^G$ and $M^H$ are of finite index and the conditional expectations
will not commute so that $\cal Q$ becomes quite relevant. The simplest example is when
$G$ and $H$ are are $\mathbb Z /2\mathbb Z$ so that the group generated by them is dihedral.
It is easy to check that the answer to $\cal Q$ is affirmative in this case. This example,
and question $\cal Q$, make perfect sense in Galois theory: if $E$ and $F$ are two subfields
of a field $K$ with $[K:E]$ and $[K:F]$ finite there are trace maps $Tr_E$ and $Tr_F$ which
play the role of the conditional expectations and one may ask if  finite rank
of the $\mathbb Z$-algbera generated by $Tr_E$ and $Tr_F$ guarantees that
$[K:E\cap F]<\infty$.
George Bergmann (\cite{Berg}) answered this question in the affirmative in zero characteristic,
thus reinforcing $\cal Q$. He further extended it to a finite collection of subfields rather
than just two. Similarly we can extend $\cal Q$ to $\cal Q$':

$\cal Q$' {\it If the conditional expectations onto a finite family of finite index subfactors
generate a finite dimensional algebra, is the intersection of the subfactors of finite index?}

Of course there is no reason for the intersection of subfactors to be a factor so
we must use some alternative definition of finite index to properly formulate
$\cal Q$' in detail. Several equivalent definitions are available, the most general
being the probabilistic one of Pimsner and Popa in \cite{PP}. In the case of finite von Neumann
algebras one may use the property that the large algebra is a finitely generated
left (or right) module over the small one. For properly infinite  algebras we will use
the endomorphism theory pioneered by Longo (see \cite{L1}). In this paper  we will answer
$\cal Q$' in the affirmative. Our proof in the properly infinite case
uses the properties of type III factors and has applications to conformal
field theories where type III factors appear naturally.\par

The paper is organized as follows: After introducing some
basics of index theory and setting up notations in \S2, in \S3 we prove
$\cal Q$' in   Th. \ref{finitecase}, and we give two
applications in Cor. \ref{3.2} and Th. \ref{3.3}. In \S4 we first prove
Th. \ref{key} using two lemmas. An averaging technique in \cite{ILP}
plays a key role in the proof of Lemma \ref{s=1}. Th. \ref{key} implies
Cor. \ref{general} which proves the extension of $\cal Q$'. In \S4.3,
we describe the setting of conformal nets where the assumptions of
Cor. \ref{general} are naturally satisfied (cf. Lemma \ref{irred1},
Cor. \ref{subnet1} and Lemma \ref{irred2}), and apply Cor. \ref{general} to
a large class of conformal nets in Cor. \ref{subnet2}.

The first author would like to thank D. Bisch for useful discussions and motivation
for the problem.

\section{Preliminaries}
Index theory of subfactors was initiated in \cite{J1} in
the setting of finite factors. In this paper we will use the following
more general definition from \cite{PP} and \cite{P}:
\begin{definition}\label{ind}
Let $N\subset M$ be an inclusion of von Neumann algebras with a
conditional expectation $E:M\rightarrow N$. The (probabilistic) index
of $E$, denoted simply by $\Ind E$, is defined by
$$
\Ind E= (sup \{c\geq 0| E(m)\geq cm, \forall m\in M_+\})^{-1}.
$$
The inequality in the above definition will be referred to as
Pimsner-Popa inequality.
The inclusion  $N\subset M$ has finite index if there exists a
conditional expectation $E:M\rightarrow N$ such that
$\Ind E<\infty$.
\end{definition}
Let $N\subset M$ be an inclusion of von Neumann algebras and $E:
M\rightarrow N$ a normal faithful conditional expectation. Let $\phi$
be a normal faithful state on $M$ such that $\phi=\phi\cdot E$. Let
$\H$ be the Hilbert space of the GNS representation associated to
$\phi$, and let $\Omega\in \H$ be the vector such that
$\phi(m)=\langle m\Omega, \Omega
\rangle, \forall m\in M.$  Let $e$ be the Jones projection from
$\H$ to $\overline{N\Omega}$.
\begin{definition}\label{basis}
A family of elements $\{ m_j\} \subset M$ satisfying the conditions:\par
(1) $E(m_i^*m_j)= \delta_{ij} f_j$ where $f_j$ is a projection in $N,
\forall i,j$;\par
(2) $\overline{\sum_j m_j(e\H)}=\H$ \par
is called an orthonormal basis of $M$ over $N$ via $E$.
\end{definition}

\begin{lemma}\label{basic}
Let $M\subset B(\H)$ be a von Neumann algebra represented standardly
on a Hilbert space $\H$ with a cyclic separating vector $\Omega$ and
denote by $\phi$ the vector state on $M$ with
$\phi(m)=\langle m\Omega, \Omega
\rangle.$
Let $N\subset M$ be a von Neumann algebra and assume that $E:M\rightarrow N$
is a faithful conditional expectation with $\phi(E(m))= \phi(m),
\forall m\in M.$ Let $e$ be the Jones projection from $\H$ to
$\overline{N\Omega}$, and $J$ the canonical conjugation of $M$
with respect to $\Omega$. Denote by $\langle M, e\rangle$ the von Neumann
algebra generated by $M, e$.
Then: \par
(1) $MeM$ is weakly dense in $\langle M, e\rangle$; \par
(2) $\langle M, e\rangle= JN'J$; \par
(3) $N\subset M$ has finite index if and only if $M\subset \langle M, e\rangle$
has finite index; \par
(4) $N\subset M$ has finite index if and only if
$N\otimes B \subset M\otimes B$ has finite index where $B$ is a factor.
\end{lemma}
\proof
(1), (2) is contained in \cite{J1}. (3) follows by 1.2 of Page 9 of
\cite{P}. As for (4), note that if $E: M\otimes B\rightarrow N\otimes B$ is
a conditional expectation which verifies Pimsner-Popa inequality for
some constant $c>0$, then $E(M\otimes 1)= (N\otimes1)$, hence by
restriction $N\subset M$ has finite index. On the other hand if
$N\subset M$ has finite index and assume that
$E: M\rightarrow N$ is
a conditional expectation which verifies Pimsner-Popa inequality for
some constant $c'>0$, by Th. 1.1.6 of \cite{P} there is an orthonormal
basis $\{ m_i\}$ of $M$ over $N$ via $E$, such that
$\sum_jm_jm_j^*$ is bounded.  By definition
$\{ m_i\otimes 1\}$  is
also an orthonormal basis of  $M\otimes B$ over $N\otimes B$ via
$E\otimes id$, and hence   by Th. 1.1.6 of \cite{P} again
$N\otimes B \subset M\otimes B$ has finite index.
\endproof

\section{Finite type von Neumann algebra case}\label{finite}
Let $M$ be a direct sum of finitely many finite
factors with faithful trace $tr$.  Let $\cal P$ be a finite set of (unital)
finite index subalgebras of $M$ and for each $P\in \cal P$
let $e_P$ be the projection from $L^2(M,tr)$ onto $L^2(P,tr)$.
Let ${\cal F}= \{e_P : P\in {\cal P}\}$.

\begin{theorem} \label{finitecase}
If $N=\bigcap_{P\in {\cal P}} P$ then

$$[M:N]<\infty \iff \dim \{{\cal F}''\} <\infty.$$
\end{theorem}

\proof (only if) If $[M:N]<\infty$ then $\dim \{ N'\cap \langle M,e_N \rangle\}<\infty$
 by reduction to the factor case. But ${\cal F}'' \subseteq  N'\cap \langle M,e_N \rangle$.
\vskip 4pt
(if)  The key thing to establish is that $Me_N M$ is contained in a finitely
generated
left $M$-module contained in $\langle M,e_N \rangle$.

Since ${\cal F}''$ is finite dimensional and $e_N =\inf_{f\in {\cal F}} f$,
$e_N$ is a polynomial in the $\{f\in \cal F\}$. Thus
$Me_NM \subseteq \sum_{w\in {\cal W}} MwM$ where $\cal W$ is a finite set of
words on $\{f\in \cal F\}$. Now let $w \in {\cal W}$ be $f_1f_2...f_k$.
Then for each $i$, $Mf_iM$ (which is contained in $\langle M,e_N \rangle$)
is a basic construction for a finite index
subalgebra so by [PP],[GHJ] it is a finitely generated left $M$-module.
Hence $Mf_1Mf_2Mf_3...f_kM$ is also a finitely generated left $M$-module,
being the image under the multiplication map of
$(Mf_1M) \otimes_M (Mf_2M)\otimes_M (Mf_3M)\otimes_M... \otimes_M (Mf_kM)$.
Thus since $MwM\subseteq Mf_1Mf_2Mf_3...f_kM$, $Me_NM$ is contained in
a finitely generated left $M$-module inside $\langle M,e_N \rangle$, say $$Me_NM\subseteq
\sum
_{i=1}^n Mr_i$$
for $r_i \in \langle M,e_N \rangle$.

Now consider $\langle M,e_N \rangle$. It has a faithful normal trace $Tr$
so we may consider its action on $L^2(\langle M,e_N \rangle, Tr)$. By contradiction
we suppose $\langle M,e_N \rangle$ is not a finite von Neumann algebra.

First suppose $M_*$ is separable,
then so is $L^2(\langle M,e_N \rangle, Tr)$ so by proposition 3.14 of \cite{Tak}
there is a cyclic vector for  $Me_NM$ in $L^2(\langle M,e_N \rangle, Tr)$, call it $\Omega$.
Then $\{r_i \Omega\}$ is a finite set which is cyclic for $M$.
By the coupling constant, $M'$ is finite (on $L^2(\langle M,e_N \rangle, Tr)$)
and $\langle M,e_N \rangle\subseteq JM'J$.

The nonseparable case requires a little more care. We know that $\langle M,e_N \rangle$ is
a semifinite von Neumann algebra whose center, being the same as the center
of $N$, has only countably many mutually orthogonal projections. Thus we may,
by   proposition 1.40 of \cite{Tak}, find an infinite, $\sigma$-finite
projection
$p\in \langle M,e_N \rangle$ of central support $1$ with $p=\sum_n p_n$ where $p_n$ are
mutually orthogonal projections with $Tr(p_n)<\infty$. Choose positive numbers
$a_n$ with $\sum_n a_n^2 Tr(p_n) <\infty$.
Then $\Omega=\sum_n a_n p_n$ is a vector in $L^2(\langle M,e_N \rangle,Tr)$
and it is a separating vector for $p\langle M,e_N \rangle p$ on $pL^2(\langle M,e_N \rangle,Tr)$.
Since the central support of $p$ is $1$, the action of $\langle M,e_N \rangle$ on
$L^2(\langle M,e_N \rangle,Tr)p$
is faithful so as in the previous paragraph  $\{r_i \Omega\}$ is a finite
set which is cyclic for $M$.
Hence by the coupling constant, $M'$ is finite (on $L^2(\langle M,e_N \rangle,
 Tr)p$)
and it contains the infinite von Neumann algebra
$Jp\langle M,e_N \rangle pJ\quad -\quad$a contradiction. (Note that
$J$ is the canonical involution on $L^2(\langle M,e_N \rangle, Tr)$.)
\endproof

If we know that $N$ and $M$ are II$_1$ factors with $[M:N] <\infty$, the
proof of \ref{finitecase}
gives a bound on $[M:N]$ in terms of $\dim A$
(where we set $A={\cal F}''$) and
the individual
$[M:P]$ for $P\in \cal P$. To give an explicit (but rather crude) bound, let
$\displaystyle L=\max_{P\in {\cal P}} [M:P]$ and $\ell$ be the length of
the longest
word $w$ in some basis of $A$ consisting of words on $\cal F$.

\begin{corollary}\label{3.2}
With hypotheses as above, $$[M:N] \leq L^\ell \hspace{3pt} \dim A.$$

\end{corollary}

\proof Since $\langle M,e_N \rangle \quad \subseteq \quad \sum_w MwM$ it suffices to bound
$\dim_M(\overline{MwM})$ for each $w$. But since all indices are finite one
may
take the Connes
tensor product of the $L^2(MfM)$ instead of the algebraic one and
 $\dim_M(\overline{MwM}) \leq L^{length(w)}$ follows from the
multiplicativity of the
 $M$-dimension under tensor product and the fact that $MfM$ is just a
basic construction for
 a subactor in $\cal P$,so its $M$-dimension is bounded by $L$.
\endproof
Using the formalism of planar algebras (cf.\cite{J2}) the first author has
found the slightly better bound
(when $|{\cal P}|=2$),
$$[M:N] \leq \frac{L^\ell \hspace{3pt} \dim A}{4}.$$
In practice the bound seems to be
a lot sharper. \par
Given a subfactor it is always possible to perturb it
(for instance conjugating it
by a unitary) to obtain a pair of subfactors to which \ref{finitecase} can be
applied. We have only carried this analysis out in one case,
namely the locally
trivial subfactors coming from automorphisms which can be thought of as
perturbations
of the tensor product subfactor. We obtain the following result which does not
seem easy to prove by other means.

\begin{theorem} \label{3.3}
Let $\cal A$ be a finite set of automorphisms of the finite
factor
$M$. Then $N^{\cal A} = \{x\in N | \alpha(x)=x \hspace{3pt}\forall \alpha
\in {\cal A}\}$ is of
finite index in $N$ iff the spectrum of the operator
 $$T=\sum_{\alpha, \beta \in {\cal A}}\alpha \beta^{-1}$$
is finite.
\end{theorem}
\proof
Let $M$ be the finite direct sum of factors
$\oplus_{\alpha \in {\cal A}} N$
 where we take a copy of $N$ for each $\alpha$, and
let $P$ and $Q$ be the
 two subfactors $P=\{\oplus_\alpha x |x\in N\}$ and
$Q=\{\oplus_\alpha \alpha(x)
|x\in N\}$.
 Then $P\cap Q =\{\oplus_\alpha x |x\in N^{\cal A}\}$ so that
the index of $N^{\cal A}$ in
 $N$ is finite iff the index of $P\cap Q$ in $N$ is finite.
Use the obvious trace on
 $M$ to form conditional expectations. Identify $P$ with $N$ in the
obvious way. We have
$\displaystyle E_P(\oplus_\alpha x_\alpha)=
\frac{\sum_\alpha x_\alpha}{|{\cal A
}|}$
and
$$ E_Q(\oplus_\alpha x_\alpha)=
\oplus_{\beta \in {\cal A}} \quad
\beta(\frac{\sum_\alpha \alpha^{-1}(x_\alpha)}{|{\cal A}|}).$$
So we see that $E_PE_QE_P$ is, up to a multiple, our operator $T$. Now
the algebra generated by two idempotents $p$ and $q$ is finite dimensional iff
the algebra generated by $pqp$ is finite dimensional, which in this context
is the same as the finiteness of the spectrum of $pqp$. So by \ref{finitecase}
we are done.
\endproof

The result appears to be non-trivial especially if the
automorphisms $\alpha$ are inner,
in which case it states that the algebra generated by
$Ad (u)$ for some finite set
of unitaries $u$ is finite dimensional iff
$\sum_{u,v} Ad (uv^*)$ has finite spectrum.
Note that it is NOT true that the algebra generated by $u$ is
finite dimensional iff $\sum_{u,v}  uv^*$ has finite spectrum.

\section{General factor case (with separable predual)}
\subsection{Preliminaries on Sectors}
Let $M$ be a properly infinite factor.  The {\it sectors of $M$}  are given by
$$\text{Sect}(M) = \text{End}(M)/\text{Inn}(M),$$
namely $\text{Sect}(M)$ is the quotient of the semigroup of the
endomorphisms of $M$ modulo the equivalence relation: $\rho,\rho'\in
\text{End}(M),\, \rho\thicksim\rho'$ iff there is a unitary $u\in M$
such that $\rho'(x)=u\rho(x)u^*$ for all $x\in M$.

$\text{Sect}(M)$ is a $^*$-semiring (there is an addition, a product and
an involution) equivalent to the Connes correspondences (bimodules) on
$M$ up to unitary equivalence. If $\r$ is
an element of $\text{End}(M)$ we shall denote by $[\r]$
its class in $\text{Sect}(M)$. The operations are:

{\it Addition} (direct sum): Let $\rho_1,\rho_2,\dots\r_n\in\text{End}(M)$.
Choose a non-degenerate $n$-dimensional Hilbert $H$ space of isometries in $M$
and a basis $v_1,\dots v_n$ for $H$. Here $H$ is non-degenerate means that
the isometries $v_1,\dots v_n$ verifies $\sum_i v_iv_i^*= 1$.

Then
\[
\r(x)\equiv\sum_{i=1}^n v_i\r_i(x)v^*_i,\quad x\in M,
\]
is an endomorphism of $M$. The definition of the direct sum
endomorphism $\r$ does not depend on the choice of $H$ or on the
basis, up to inner automorphism of $M$,
namely $\r$ is a well-defined sector of $M$.

{\it Composition} (monoidal product). The usual composition of maps
$$\rho_1\cdot\rho_2(x) = \rho_1(\rho_2(x)), \qquad x\in M,$$ defined on
$\text{End}(M)$ passes to the quotient $\text{Sect}(M)$.
Let $\rho \in \text{\rm End}(M)$ and $\e$ be a normal
faithful conditional expectation
$\e:
M\rightarrow \rho(M)$.  We define a number $d_\e\geq 1$ (possibly
$\infty$) by:
$$
d_\e^{-2} :=\text{\rm Max} \{t\in [0, +\infty)|
\e (m_+) \geq t m_+, \forall m_+ \in M_+
\}$$ (Pimsner-Popa inequality in \cite{PP}).\par
 We define
$$
d(\rho) = \text{\rm Min}_\e \{ d_\e \},
$$
where the minimum is taken over $\e$ with $d_\e < \infty$ (otherwise
we put $d(\r) =\infty$).  $d(\rho)$ is called the  dimension of  $\rho$.
We say that $\rho$ has finite index if $d(\rho)< \infty.$
It is
clear from the definition that  the dimension  of  $\rho$
depends only the sector $[\rho]$.
The following properties of the dimension can be found in
\cite{L1}.
\begin{lemma}\label{dim}
(1) If $[\rho]=[\rho_1]+[\rho_2],$ then $
d(\rho)= d(\rho_1)+d(\rho_2);$\par
(2) $d(\rho_1\rho_2)= d(\rho_1) d(\rho_2).$
\end{lemma}
For $\lambda $, $\mu \in \text{\rm End}(M)$, we will use
$\text{\rm Hom}(\lambda , \mu )$ denote the vector space of intertwiners from
$\lambda $ to $\mu $, i.e. $a\in \text{\rm Hom}(\lambda , \mu )$ iff
$a\in M, a \lambda (x) = \mu (x) a $ for any $x \in M$.
A sector $\lambda$
is said to be irreducible if the vector space
$\text{\rm Hom}(\lambda , \lambda )$ has
dimension one.\par
\begin{lemma}\label{can}
Let $N\subset M$ be a properly infinite subalgebra and assume that there
exists a normal faithful conditional expectation $E: M\rightarrow N$.
Suppose that M is represented standardly on a Hilbert space $H$ with
a cyclic separating vector $\Omega,$  and the vector state
$\omega (m)=\langle m\Omega,\Omega\rangle$ on $M$ verifies
$\omega (E(m))=\omega (m), \forall m\in M.$ Let $e$ be the Jones projection
from $H$ onto $\overline{N\Omega}$ and $M_1:=\langle M,e\rangle$ be
the von Neumann algebras generated by $M,e$. Then:\par
(1) Let $v\in M_1$ be an isometry with $vv^*=e.$ Then for every
$m_1\in M_1$, there is a unique element, denoted by $\gamma(m_1) \in N,$
such that $vm_1 = \gamma(m_1) v$, and $\gamma \in  \text{\rm End}(M_1)$;\par
(2) We will use the same notation $\gamma$ to denote its restriction to
$M$. Then $N\subset M$ has finite index if and only if
$\gamma(M)\subset M$ has finite index.
\end{lemma}
\proof
The first part is Prop. 2.9 of \cite{LR}, and the second part follows from
(3) of Lemma \ref{basic}.
\endproof
\begin{remark}
The endomorphism $\gamma$ in (2) of Lemma \ref{can} is called
canonical endomorphism for $N\subset M$ in \cite{LR}.
\end{remark}
\subsection{General factor case}
\begin{theorem}\label{key}
Let $M$ be a factor with separable predual, and let
$N\subset M$, $E:M\rightarrow N, \H , \Omega$ and $e$
be as in Lemma \ref{basic}.
If $e=\sum_{1\leq i\leq k} m_i R_i$ with $m_i\in M, R_i\in B(\H)$ such that
$R_i m = \rho_i(m) R_i, \forall m\in M$ and each $\rho_i\in  \text{\rm End}(M)
$ has finite index, $i=1,...,k.$ Then
$N\subset M$ has finite index.
\end{theorem}
\begin{remark}
Note that  each $\rho_i\in  \text{\rm End}(M)
$ has finite index, $i=1,...,k $ is a necessary condition when
$N\subset M$ are properly infinite von Neumann algebras,  since
by Lemma \ref{can} $e= vv^*= \gamma(v^*) v, vm=\gamma(m)v , \forall m\in M$
for
any pair $N\subset M$ as in Lemma  \ref{can}, including the case
when $N\subset M$ has infinite index. Also note that the conditions
on Jones projection $e$ in Theorem \ref{key} are similar to that
of Theorem \ref{finitecase}, but neither theorem implies the other.
\end{remark}
The proof of this theorem is divided into following steps consisting of
two lemmas. \par
First note that replacing
$N,M, E, e, m_i, R_i, \rho_i$ by
$N\otimes B,M\otimes B, E\otimes id, e\otimes 1,
m_i\otimes 1, R_i\otimes 1, \rho_i\otimes id$ respectively if necessary
where $B$ is a type III factor with
separable dual, by (4) of Lemma \ref{basic} it is enough to prove
the theorem for $N\otimes B\subset M\otimes B$. So
we can assume that $M$ is a type III factor, and $N$ is a type III
von Neumann algebra. Let $v, \gamma$ be as in Lemma \ref{basic} such
that $e=vv^*= \gamma(v^*)v, vm=\gamma(m)m, \forall m\in M.$
Note that since each $\rho_i\in  \text{\rm End}(M)
$ has finite index, $i=1,...,k$, $\rho_i$ can be decomposed into
sum of finitely many irreducible sectors. Hence we can assume that
$e=\sum_{1\leq i\leq n} \sum_{1\leq \alpha\leq l(i)}
m_{i\alpha} R_{i\alpha}$ with $m_{i\alpha}\in M,
R_{i\alpha}\in B(\H)$ such that
$R_{i\alpha} m = \rho_i(m) R_{i\alpha}, \forall m\in M$ and each
$\rho_i\in  \text{\rm End}(M)
$ is irreducible , has finite index,
$1\leq \alpha\leq l(i) <\infty$ is a label, and
$[\rho_i]\neq [\rho_j], {\mathrm if} i\neq j, i,j=1,...,n.$ Moreover,
replacing $ \{R_{i\alpha}\}$ by a maximal linearily independent
subset  over ${\mathbb C}$ if necessary,
we can
assume that $ \{R_{i\alpha}\}$ are linearily independent over ${\mathbb C}$.
\begin{definition}\label{V}
Let $V\subset M$ be a vector space consisting of all finite linear
combinations of
elements $s_i$ with the property that
$s_i ^* v = \sum_{1\leq \alpha \leq l(i)} c_\alpha R_{i\alpha},
c_\alpha\in {\mathbb C}, i=1,...,n.$ Denote by
$W$ the vector space spanned by $\{R_{i\alpha}\}$.
\end{definition}
\begin{lemma}\label{Hil}
$V$ is a finite dimensional Hilbert space with natural
inner product $\langle s, t\rangle = t^* s$, and when $V$ is not
zero we can choose an orthonormal basis $\{s_{i\beta}\}$ with the property
$s_{i\beta} \in \Hom(\rho_i,\gamma), s_{i\beta}^* s_{i'\beta'}=
\delta_{ii'}\delta_{\beta\beta'}.$
\end{lemma}
\proof
Let $W$ be the vector space spanned by $\{R_{i\alpha}\}$. Then
$W$ is a vector space with dimension $\prod_{1\leq i\leq n} l(i)$.
Let $F:V\rightarrow W$ be a conjugate linear map defined by
$F(s)= s^*v$. We claim that $F$ is one-to-one: if
$s^*v=0$, then $s^* vv^*= s^*e=0$, and so
$s^* \Omega=0$ which implies that $s=s^*=0$ since $\Omega$ is
separating for $M$. Hence $dim V\leq dim W$.\par
Assume that $s_i\in V, s_i ^* v = \sum_{1\leq \alpha \leq l(i)} c_\alpha R_{i\alpha},
c_\alpha\in {\mathbb C}.$ Then
$$
s_i^* vm = s_i^* \gamma(m) v
= \rho_i (m) s_i^* v
$$
by the intertwining property of $ R_{i\alpha}$, hence
$ s_i^* \gamma(m) vv^*
= \rho_i (m) s_i^* vv^*
, \forall m\in M$
and using the separating property of $\Omega$ for $M$ again we have
$s_i^*\in \Hom(\gamma,\rho_i)$. Since $\rho_i$ are irreducible,
$\Hom(\rho_i, \rho_j)= \delta_{ij} {\mathbb C}, i,j=1,...,n.$ It follows
that  $\langle s, t\rangle = t^* s$ is an inner product, and
the last part of the lemma follows.
\endproof
Let $s=\sum_{i\beta} s_{i\beta} s_{i\beta}^*$ where $s_{i\beta}$ is the
orthonormal basis as in Lemma \ref{Hil}. When $V$ is zero we set
$s=0$. By construction we have $st=t, \forall t\in V$, in fact
$s$ is the left support of $V$ in $M$.
\begin{lemma}\label{s=1}
$s=1.$
\end{lemma}
\proof
Let us compute
\[
\gamma (v^*) (1-s) v  = \gamma(v^*) v- \sum_{i\beta} \gamma(v^*)
s_{i\beta} s_{i\beta}^* v =
\sum_{i\alpha} m_{i\alpha} R_{i\alpha} - \sum_{i\beta} \gamma(v^*)
s_{i\beta}s_{i\beta}^* v
\]
Note that $\gamma(v^*)s_{i\beta} \in M$,  $s_{i\beta}^* v\in W$ by
Definition \ref{V}. So
\begin{equation}\label{1}
\gamma (v^*) (1-s) v= \sum_{i\alpha} m_{i\alpha}' R_{i\alpha}
\end{equation}
for some $m_{i\alpha}' \in M$.
If the left hand side above is non-zero, then there is at least
one $m'_{i\alpha}\neq 0$, and we will derive a contradiction by
an averaging trick as on Page 46 of \cite{ILP}.
Since $M$ is a type III factor, $ m_{i\alpha}' \neq 0$,
as on Page 46 of \cite{ILP} we can find
$a,b\in M$ such that $E_{\rho_i}(a m_{i\alpha}' \rho_i(b))=1$, where
$E_{\rho_i}: M\rightarrow \rho_i(M)$ is a normal faithful conditional
expectation. Let $R$ be an injective simple subfactor in $M$ as on
Page 46 of \cite{ILP} (This  is where the separability of $M_*$ is used),
and let $u$ be an unitary element of $R$.
Multiply both sides of   equation (\ref{1}) by $\rho_i(u^*)a$ on the
left and $bu$ on the right, and use the intertwining properties of
$R_{i\alpha},s$ and $v$ we get:
$$
\rho_i(u^*) a \gamma(v^*) \gamma (b) \gamma (u) (1-s) v
= \rho_i(u^*) a m_{i\alpha}' \rho_i(b)  \rho_i(u) R_{i\alpha}
+ \sum_{(j,\beta) \neq (i,\alpha)}  \rho_i(u^*)
a m_{i\beta}' \rho_j(b)  \rho_j(u) R_{j\beta}.
$$
Averaging by an invariant mean
over the unitary elements of $R$ (which is amenable since
$R$ is injective) as on Page 46 of \cite{ILP} we get
$$
t(1-s) v= c_{i\alpha}  R_{i\alpha} + \sum_{(j,\beta) \neq (i,\alpha)}
c_{j\beta}R_{j\beta},
$$
where $t\in M$, and $c_{j\beta}\in M$ satisfies
$
c_{j\beta} \rho_j(x) = \rho_i(x) c_{j\beta}, \forall x\in R,
$
and $E_{\rho_i} (c_{i\alpha}) = 1$ since $E_{\rho_i}: M\rightarrow
\rho_i(M)$ is normal. Since $\rho_j, j=1,...,n$ has finite index,
by Cor. 2.11 of \cite{ILP} $c_{j\beta} \in \Hom( \rho_j,  \rho_i)
= \delta_{ij} {\mathbb C}$. Hence we have
$c_{i\alpha}=1$ and
$$t(1-s) v= R_{i\alpha} + \sum_{\beta \neq \alpha}
c_{i\beta}R_{i\beta}
$$
where $c_{i\beta} \in {\mathbb C}$. It follows by definition \ref{V}
that $(1-s) t^* \in V$. Since $s$ is the left support of $V$ in $M$,
we have $s(1-s) t^* = (1-s) t^*$, hence
$(1-s) t^*=0$, and
$ R_{i\alpha} + \sum_{\beta \neq \alpha}
c_{i\beta}R_{i\beta} =0 $
which contradicts the linear independence of $\{R_{j\beta}\}$ over
${\mathbb C}$. This shows that the left hand side of equation
(\ref{1}) is zero, i.e.,
$\gamma(v^*) (1-s) v =0.$
Multiply this equation on the
left by $\gamma(m_1)\gamma (v)$ and on the right by $m_2v^*$, and note
that $(1-s) \in\Hom(\gamma,\gamma)$, we  get
$\gamma (m_1 em_2) (1-s) e=0, \forall m_1,m_2\in M.$ By (1) of
Lemma \ref{basic}, $1$ in the weak closure of $MeM$, therefore
$(1-s)e=0$. Using the fact that $\Omega$ is separating for $M$ and
$e\Omega=\Omega$, we conclude that $s=1$.
\endproof
{\bf The end of the proof of Theorem \ref{key}:}\par
By Lemma \ref{s=1}, $s=\sum_{j\beta} s_{j\beta} s_{j\beta}^*=1,$
and each $ s_{j\beta}\in \Hom(\rho_j,\gamma)$ is an isometry.
It follows that
$[\gamma]= \bigoplus_{1\leq j\leq n} l(j)' [\rho_j]$
where $ 0\leq l(j)'< \infty, j=1,...,n.$ By Lemma \ref{can}
$d(\gamma) = \sum_{ 1\leq j \leq n} l(j)' d(\rho_j) < \infty
,$
and by Lemma \ref{can} $N\subset M$ has finite index.
\endproof

\begin{corollary}\label{general}
Let $N_1,...N_n$ be von Neumann subalgebras of  $M$ where $M$ is a factor
with separable predual,
$N=N_1\cap N_2...\cap N_n$,
and each $N_i\subset M$ has finite index, $i=1,...,n.$
Assume that:\par
(1) There is a $\phi$ which is a normal faithful state on M
invariant under normal faithful condition expectations $E_i:M\rightarrow N_i$
and $E:M\rightarrow N.$ Let $e_i, e$ be the corresponding Jones projection in
$B(L^2(M,\phi));$\par

(2) $e=e_1\wedge e_2...\wedge e_n.$\par
Then $N\subset M$ has finite index if and only if $e_1,...,e_n$ generate
a finite dimensional algebra.
\end{corollary}
\proof
The only if part follows from Page 2 of \cite{P} and the assumption that
$M$ is a factor. Let us prove the if part. By replacing $M,N_i, e_i,e,\phi$ by
$M\otimes B, N_i\otimes B, e_i\otimes 1, e\otimes 1, \phi\otimes \phi_1$
respectively
if necessary where $B$ is a type III factor with separable predual and
$\phi_1$ is a normal faithful state on $B$, by (4) of Lemma \ref{basic} it
is enough to prove the if part by assuming that $M$ is a type III
factor, and $N_i,i=1,2,...,n ,N$ are type III von Neumann algebras.\par

Assume that
$e_1,...,e_n$ generate
a finite dimensional algebra. By assumption (2) there exists a
non-commutative polynomial $f$ such that
$e=f(e_1,...,e_n)$. Let $v_i$ be the isometries in $B(L^2(M,\phi))$
as in Lemma \ref{can} such that $v_i v_i^*= e_i, v_im =\gamma_i(m)v_i
,\forall m\in M.$ Note that  for each $i=1,...,n$
$\gamma_i\in \End(M)$ has finite index
by Lemma \ref{basic} since $N_i\subset M$ has finite index by assumption.
Using
$ e_i= v_i v_i^*=\gamma_i(v_i^*)v_i,
\gamma_i(v_i^*) \in M,
$ and the intertwining properties $v_i, i=1,...,n$, it follows that
$$
e= \sum_{ 1\leq j \leq k} m_j R_j
$$
where $ m_j\in M$ and $R_j\in B(L^2(M,\phi)),
R_j (m)= \rho_j(m) R_j, \forall m\in M $ and
$\rho_j\in \End(M)$ is a finite compositions of
$\gamma_i, i=1,...,n$, and hence of finite index by Lemma \ref{dim}.
By Theorem \ref{key} the proof is complete.
\endproof
\begin{remark}
Note that when $M$ is a type $II_1$ factor in Cor. \ref{general},
one can take $\phi$ to be the trace on $M$, then assumption (1) holds
trivially, and assumption (2) holds by \cite{S}. So we obtain
another proof of Th.\ref{finitecase} when $M$ is a type $II_1$ factor
with separable predual, and this in fact partially inspired  simpler
proofs in section \ref{finite} for finite type von Neumann algebra case.
We will see a large class of examples in
the case when $M$ is type III where  assumptions (1) and (2) hold
in Cor. \ref{subnet2}.
\end{remark}
\subsection{Applications to conformal nets}
\subsubsection{Preliminaries on conformal nets}
By an interval of the circle we mean an open connected
non-empty subset $I$ of $S^1$ such that the interior of its
complement $I'$ is not empty.
We denote by $\I$ the family of all intervals of $S^1$.

A {\it net} $\A$ of von Neumann algebras on $S^1$ is a map
\[
I\in\I\to\A(I)\subset B(\H)
\]
from $\I$ to von Neumann algebras on a fixed separable Hilbert space $\H$
that satisfies:
\begin{itemize}
\item[{\bf A.}] {\it Isotony}. If $I_{1}\subset I_{2}$ belong to
$\I$, then
\begin{equation*}
 \A(I_{1})\subset\A(I_{2}).
\end{equation*}
\end{itemize}
If $E\subset S^1$ is any region, we shall put
$\A(E)\equiv\bigvee_{E\supset I\in\I}\A(I)$ with $\A(E)=\mathbb C$
if $E$ has empty interior (the symbol $\vee$ denotes the von Neumann
algebra generated).

The net $\A$ is called {\it local} if it satisfies:
\begin{itemize}
\item[{\bf B.}] {\it Locality}. If $I_{1},I_{2}\in\I$ and $I_1\cap
I_2=\emptyset$ then
\begin{equation*}
 [\A(I_{1}),\A(I_{2})]=\{0\},
 \end{equation*}
where brackets denote the commutator.
\end{itemize}
The net $\A$ is called {\it M\"{o}bius covariant} if in addition
satisfies
the following properties {\bf C,D,E,F}:
\begin{itemize}
\item[{\bf C.}] {\it M\"{o}bius covariance}.
There exists a non-trivial strongly
continuous unitary representation $U$ of the M\"{o}bius group
$\Mob$ (isomorphic to $PSU(1,1)$) on $\H$ such that
\begin{equation*}
 U(g)\A(I) U(g)^*\ =\ \A(gI),\quad g\in \Mob,\ I\in\I.
\end{equation*}
\item[{\bf D.}] {\it Positivity of the energy}.
The generator of the one-parameter
rotation subgroup of $U$ (conformal Hamiltonian), denoted by
$L_0$ in the following,  is positive.
\item[{\bf E.}] {\it Existence of the vacuum}.  There exists a unit
$U$-invariant vector $\Omega\in\H$ (vacuum vector), and $\Omega$ is
cyclic for the von Neumann algebra $\bigvee_{I\in\I}\A(I)$.
\end{itemize}
By the Reeh-Schlieder theorem $\Omega$ is cyclic and separating for
every fixed $\A(I)$. The modular objects associated with
$(\A(I),\Omega)$ have a geometric meaning
\[
\Delta^{it}_I = U(\Lambda_I(2\pi t)),\qquad J_I = U(r_I)\ .
\]
Here $\Lambda_I$ is a canonical one-parameter subgroup of $\Mob$ and $U(r_I)$ is a
antiunitary acting geometrically on $\A$ as a reflection $r_I$ on $S^1$.

This implies {\em Haag duality}:
\[
\A(I)'=\A(I'),\quad I\in\I\ ,
\]
where $I'$ is the interior of $S^1\setminus I$.

\begin{itemize}
\item[{\bf F.}] {\it Irreducibility}. $\bigvee_{I\in\I}\A(I)=B(\H)$.
Indeed $\A$ is irreducible iff
$\Om$ is the unique $U$-invariant vector (up to scalar multiples).
Also  $\A$ is irreducible
iff the local von Neumann
algebras $\A(I)$ are factors. In this case they are III$_1$-factors
with separable predual
in
Connes classification of type III factors.
\end{itemize}
By a {\it conformal net} (or diffeomorphism covariant net)
$\A$ we shall mean a M\"{o}bius covariant net such that the following
holds:
\begin{itemize}
\item[{\bf G.}] {\it Conformal covariance}. There exists a projective
unitary representation $U$ of $\Diff(S^1)$ on $\H$ extending the unitary
representation of $\Mob$ such that for all $I\in\I$ we have
\begin{gather*}
 U(\phi)\A(I) U(\phi)^*\ =\ \A(\phi.I),\quad  \phi\in\Diff(S^1), \\
 U(\phi)xU(\phi)^*\ =\ x,\quad x\in\A(I),\ \phi\in\Diff(I'),
\end{gather*}
\end{itemize}
where $\Diff(S^1)$ denotes the group of smooth, positively oriented
diffeomorphism of $S^1$ and $\Diff(I)$ the subgroup of
diffeomorphisms $g$ such that $\phi(z)=z$ for all $z\in I'$.
\par
Note that if $\phi\in \Diff(I)$, then $U(\phi)\in \A(I)$ by {\bf G}
and Haag duality.\par
Next we  recall some definitions from \cite{KLM} .
Recall that   ${\I}$ denotes the set of intervals of $S^1$.
Let $I_1, I_2\in {\I}$. We say that $I_1, I_2$ are disjoint if
$\bar I_1\cap \bar I_2=\emptyset$, where $\bar I$
is the closure of $I$ in $S^1$.
When $I_1, I_2$ are disjoint, $I_1\cup I_2$
is called a 1-disconnected interval in \cite{Xu3}.
Denote by ${\I}_2$ the set of unions of disjoint 2 elements
in ${\I}$. Let ${\A}$ be an irreducible M\"{o}bius covariant net
. For $E=I_1\cup I_2\in{\I}_2$, let
$I_3\cup I_4$ be the interior of the complement of $I_1\cup I_2$ in
$S^1$ where $I_3, I_4$ are disjoint intervals.
Let
$$
{\A}(E):= A(I_1)\vee A(I_2), \quad
\hat {\A}(E):= (A(I_3)\vee A(I_4))'.
$$ Note that ${\A}(E) \subset \hat {\A}(E)$.
Recall that a net ${\A}$ is {\it split} if ${\A}(I_1)\vee
{\A}(I_2)$ is naturally isomorphic to the tensor product of
von Neumann algebras ${\A}(I_1)\otimes
{\A}(I_2)$ for any disjoint intervals $I_1, I_2\in {\I}$.
${\A}$ is {\it strongly additive} if ${\A}(I_1)\vee
{\A}(I_2)= {\A}(I)$ where $I_1\cup I_2$ is obtained
by removing an interior point from $I$.
\bdefin\label{rational}
\cite{{KLM}, {LX}}
${\A}$ is said to be completely  rational if
${\A}$ is split, and
the index $[\hat {\A}(E): {\A}(E)]$ is finite for some
$E\in {\I}_2$ . The value of the index
$[\hat {\A}(E): {\A}(E)]$ (it is independent of
$E$ by Prop. 5 of \cite{KLM}) is denoted by $\mu_{{\A}}$
and is called the $\mu$-index of ${\A}$. If
the index $[\hat {\A}(E): {\A}(E)]$ is infinity for some
$E\in {\I}_2$, we define the $\mu$-index of ${\A}$ to be
infinity.

\edefin
Note that, by recent results in \cite{LX},
every irreducible, split,
local conformal net with finite $\mu$-index is automatically strongly
additive. Hence we have modified the definition in \cite{KLM} by dropping the
strong additivity requirement in the above definition. Also note that
if $\A$ is completely rational, then $\A$ has only finitely many
irreducible covariant representations by \cite{KLM}.
\par
Let $\A$ be a conformal net. By a {\it conformal subnet} (cf. \cite{L2})
we shall mean a
map
\[
I\in\I\to\B(I)\subset \A(I)
\]
that associates to each interval $I\in \I$ a von Neumann subalgebra $\B(I)$
of $\A(I)$, which is isotonic
\[
\B(I_1)\subset \B(I_2), I_1\subset I_2,
\]
and   M\"{o}bius covariant with respect to the the representation $U$,
namely
\[
U(g) \B(I) U(g)^*= \B(g.I)
\] for all $g\in \Mob$ and $I\in \I$. Note that by Lemma 13
of \cite{L2} for each $I\in \I$ there exists a conditional
expectation $E_I: \A(I)\rightarrow \B(I)$ such that $E$ preserves the
vector state given by the vacuum of $\A$.

\begin{lemma}\label{irred1}
Let $\B\subset \A$ be a conformal subnet and assume that
$U(\Diff(I))\subset \B(I), \forall I\in \I$, and $\B$ is
completely rational.  Then: \par
(1) $\B\subset \A$ is irreducible, i.e.,
$\B(I)'\cap \A(I) = {\mathbb C},  \forall I\in \I$;\par
(2) $\B(I)\subset \A(I)$
has finite index $\forall I\in \I$, and $\A$ is
completely rational.
\end{lemma}
\proof
Let $p\in \B(I)'\cap \A(I)$. Since $\B(I')\subset \A(I')$, we
have $p\in (\B(I)\vee \B(I'))'$. Since $\B$ is completely rational,
by strong additivity $p\in (\vee_{I\in \I} \B(I))'$, and so
$p\in (\vee_{I\in \I} U(\Diff(I)))'$ by assumption. Notice that
$\Diff(I), I\in \I$ generates $\Diff (S^1)$, and so
$p$ commutes with $  U(\Diff (S^1))$, and so $p\Omega$ is an eigenvector
of the conformal Hamiltonian with eigenvalue $0$. By {\bf F} it follows
that $p\Omega = x\Omega, x\in {\mathbb C}$, therefore $p= x$ since
$\Omega$ is separating for $\A(I)$, proving (1). (2) follows from
(1) , Prop. 2.3 of \cite{KL} and Th. 24 of \cite{L2}.
\endproof
\begin{corollary}\label{subnet1}
Let $\A$ be a conformal net, and $\B_1, \B_2,...\B_n$ be conformal
subnets of $\A$. Let $\B$ be the conformal
subnet of $\A$ such that $\B(I)= \B_1(I)\cap  \B_2(I)...\cap \B_n(I),
\forall I\in \I$. Assume that: \par
(1) $U(\Diff(I))\subset \B_i(I), \forall I\in \I, i=1,...,n$; \par
(2) Each $\B_i$ is completely rational,  $i=1,...,n$; \par
(3) $e=e_1\wedge e_2\cdots\wedge e_n$, where
$e$ (resp. $e_i, i=1,...,n$) are Jones projections
onto $\overline{\B(I)\Omega}$
(resp.  $\overline{\B_i(I)\Omega},i=1,...,n $).\par
Then $\B$ is completely rational if and only if
$e_1,...,e_n$ generate a finite dimensional algebra.
\end{corollary}
\proof
Note that by definition and assumption (1) we have
$U(\Diff(I))\subset \B(I), \forall I\in \I$. By Lemma
\ref{irred1} and assumptions (1) and (2) each
$\B_i(I) \subset \A(I)$ has finite index, and $\A$ is
completely rational. By Lemma \ref{irred1} again we have that
 $\B$ is completely rational if and only if $\B(I)\subset \A(I)$
has finite index. We note that $\A(I)$ is a type $III_1$ factor
with separable predual as
stated in {\bf F}. The assumption (1) of Corollary \ref{general}
in this case follows by the remark after the definition of
conformal subnets.
The corollary now follows from
Corollary \ref{general}.
\endproof
A large class of conformal subnets verifying assumptions (1), (2) and
(3) of Cor. \ref{subnet1} come from cosets and orbifolds (cf. \cite{Xu1}
and \cite{Xu2}). Let us recall some definitions.
Let $G$ be a simply connected  compact Lie group. By Th. 3.2
of \cite{FG},
the vacuum positive energy representation of the loop group
$LG$ (cf. \cite{PS}) at level $k$ on a Hilbert space $\H$, denoted by
$\pi^0$,
gives rise to an irreducible conformal net
denoted by {\it ${\A}_{G}$} when $k$ is fixed. We will use
$\Omega$ to denote the vacuum vector. Note that
$ {\A}_{G}(I) = \pi^0(L_IG)''$ where $L_IG$ are these elements of
$LG$ which are equal to identity of $G$ on $I'$.
By Th. 13.4.2 of \cite{PS} (also cf. \cite{GW}) there is a projective
unitary representation $U$ of $\Diff(S^1)$ on $\H$ such that
$U(\phi) \pi^0(f) U(\phi)^*= \pi^0(f\cdot \phi^{-1})$ for any
$f\in LG$.\par
Let $H\subset G$ be a simply connected Lie subgroup. We define
a conformal subnet $\A_{H,G/H}$ of $\A$ by
$\A_{H,G/H}(I):= \pi^0(L_IH)''\vee  (\pi^0(L_IH)'\cap  \pi^0(L_IG)'')$.
Recall from \S3 of \cite{Xu1}  that $H\subset G$ is {\it cofinite}
if $\A_{H,G/H}(I)\subset {\A}_{G}(I)$ has finite index for some $I\in \I$.
Note that Conjecture 2.13 of \cite{Xu1} implies that any such
$H\subset G$ is cofinite. See Cor. 3.4 of \cite{Xu1} for a list of
inclusions which have been proved to be cofinite.\par
When $\Gamma$ is a finite subgroup of $G$, we denote by
$\A^\Gamma$ a conformal subnet of ${\A}_{G}$ such that
$\A^\Gamma (I)= \{ m\in {\A}_{G}(I) | \pi^0(h)m = m \pi^0(h), \forall
h\in \Gamma\}$. This is an example of orbifold construction in
\cite{Xu2}. \par
\begin{lemma}\label{irred2}
Let $H_1,..., H_l$ be simply connected Lie
subgroups of $G$, and let $\Gamma_{l+1},...\Gamma_{n}$ be finite
subgroups of $G$.
Let $\B_i=\A_{H_i,G/H_i}, i=1,...,l$ and
$\B_j= \A^{\Gamma_j}, j=l+1,...,n$ be conformal subnets of $ {\A}_{G}$
as above, and  let $\B$ be the conformal
subnet of $\A$ such that $\B(I)= \B_1(I)\cap  \B_2(I)...\cap \B_n(I),
\forall I\in \I$. Then:\par
(1) $U(\Diff(I))\subset \B_i(I), \forall I\in \I, i=1,...,n$;\par
(2) $e=e_1\wedge e_2\cdots\wedge e_n$, where
$e$ (resp. $e_i, i=1,...,n$) are Jones projections
onto $\overline{\B(I)\Omega}$
(resp.  $\overline{\B_i(I)\Omega},i=1,...,n $).
\end{lemma}
\proof
Ad (1):
When $\B_j= \A^{\Gamma_j}, j=l+1,...,n$ (1) holds trivially since
$U(\Diff (S^1))$ commutes with $\Gamma_j$. Let us assume that
$\B_i=\A_{H_i,G/H_i}$. Let $\phi\in \Diff (I)$. By Th. 13.4.2
of \cite{PS} (also cf. \cite{GW}) we have
$U(\phi) \pi^0(f) U(\phi)^*= \pi^0(f\cdot \phi^{-1})$ for any
$f\in LG$, and by the remark after {\bf G} $U(\phi)\in \A_G(I)$. Apply
the same argument to $LH$ we conclude that there is a unitary
element $\tilde\phi
\in \pi^0(L_IH)''$ such that
$\tilde\phi \pi^0(f) \tilde\phi^*=
\pi^0(f\cdot \phi^{-1})
= U(\phi)\pi^0(f) U(\phi)^*
$ for any
$f\in L_IH$. It follows that
$\tilde\phi^* U(\phi) \in  \pi^0(L_IH)'\cap  \pi^0(L_IG)''$,
hence $ U(\phi)\in \A_{H_i,G/H_i}(I)= \B_i$.\par
Ad (2):
Note by definition
$\overline{B(I)\Omega}\subset \cap_{1\leq i\leq n} \overline{B(I)\Omega}
$, and so it is sufficient to show that for any eigenvector $\psi$ of
$L_0$ with eigenvalue $m\geq 0$ in
$\cap_{1\leq i\leq n} \overline{B(I)\Omega}$,  $\psi$ is also in
$\overline{B(I)\Omega}$. By Reeh-Schlieder theorem it is sufficient to
show that $\psi\in  \overline{\vee_{I\in \I}B(I)\Omega}$. The proof
is essentially contained on Page 22 of \cite{Xu1} as follows:
Choose
two smooth functions $f_1(z)$ and $ f_2(z)$ on the
unit circle,  with support $f_1\subset I_1
\in \I$, support $f_2\subset I_2
\in \I$ and $f_1+f_2= 1$. Then
$\psi = V(\psi, z^{-1})\Omega =V(\psi, z^{-1}f_1)\Omega+
V(\psi, z^{-1}f_2)\Omega
$
where $ V(\psi,\cdot)$ are the smeared vertex operators as
defined on Page 11 of \cite{Xu1}. By the same proof as in Prop. 2.11 of
\cite{Xu1}, $V(\psi, z^{-1}f_1)$ is a closed operator affiliated with
with $\B_i(I_1), i=1,2,...,n$, and so $V(\psi, z^{-1}f_1)$
is a closed operator affiliated with  $\B(I_1)$, it follows that
$V(\psi, z^{-1}f_1)\Omega\in \overline{B(I_1)\Omega}$. Similarly
$V(\psi, z^{-1}f_2)\Omega\in \overline{B(I_2)\Omega}$, and
we conclude that  $\psi\in  \overline{\vee_{I\in \I}B(I)\Omega}$.
\endproof
\begin{remark}
Due to Lemma \ref{irred2}, we conjecture that assumption (3) of
Cor. \ref{subnet1} is always satisfied.
\end{remark}
We note that by \cite{Xu3} which is based on
\cite{W}, $\A_G$ is completely rational
if $G=SU(N_1)\times SU(N_2)\times...\times SU(N_k)$, and it has been
conjectured
that
$\A_G$ is completely rational for all $G$.\par
\begin{corollary}\label{subnet2}
Let $H_1,..., H_l$ be simply connected Lie
subgroups of $G$, and let $\Gamma_{l+1},...\Gamma_{n}$ be finite
subgroups of $G$.
Let $\B_i=\A_{H_i,G/H_i}, i=1,...,l$ and
$\B_j= \A^{\Gamma_j}, j=l+1,...,n$ be conformal subnets of $ {\A}_{G}$
as described before Lemma \ref{irred2}, and
 let $\B$ be the conformal
subnet of $\A$ such that $\B(I)= \B_1(I)\cap  \B_2(I)...\cap \B_n(I),
\forall I\in \I$. Assume that  each $H_i \subset G, i=1,2,...,l$ is cofinite
and  $\A_G$ is completely rational.
Then $\B(I)$ is completely rational if and only if
$e_1,...,e_n$ generate a finite dimensional algebra where
$e_i, i=1,...,n$ are Jones projections from $\H$
onto $\overline{\B_i(I)\Omega},i=1,...,n .$
\end{corollary}
\proof
Note that since we assume that  each $H_i \subset G, i=1,2,...,l$ is cofinite,
$\B_i(I)\subset \A_G(I)$ has finite index for $i=1,...,n$. By
Th. 24 of \cite{L2} each $\B_i, i=1,...,n$ is completely rational. Hence
assumption (2) of Cor. \ref{subnet1} is satisfied. Note that assumptions
(1) and (3) of Cor. \ref{subnet1}
are satisfied thanks to Lemma \ref{irred2}. Hence the
corollary is proved by Cor.\ref{subnet1}.
\endproof
We note that the nature of the algebra generated by
$e_1,...,e_n$ in Cor. \ref{subnet2} can in principle be determined by
the representation theory information about pairs $LH_i\subset LG$
and $\Gamma_j\subset LG$. In the case $n=2$, it is well-known that
$e_1,e_2$ generate a finite dimensional algebra if and only if
the ``angle operator'' $e_1e_2e_1$ has finite spectrum. It is
an interesting question to see if one can obtain new examples of
completely rational conformal nets by using Cor. \ref{subnet2}.
{\footnotesize

config: arxiv


text:
\begin{definition}[Definition:Zariski Topology on Maximal Spectrum of Ring]
Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $\operatorname{MaxSpec}(A)$ be its [[Definition:Maximal Spectrum of Ring|maximal spectrum]].
=== Definition 1 ===
The '''Zariski topology''' on $\operatorname{MaxSpec}(A)$ is the [[Definition:Topology Defined by Closed Sets|topology with as closed sets]] the [[Definition:Maximal Zero Locus of Subset of Ring|maximal zero loci]].
=== Definition 2 ===
The '''Zariski topology''' on $\operatorname{MaxSpec}(A)$ is the [[Definition:Subspace Topology|subspace topology]] induced by the [[Definition:Zariski Topology on Spectrum of Ring|Zariski topology]] on the [[Definition:Spectrum of Ring|spectrum]] $\operatorname{Spec} A$.
\end{definition}

config: wiki


text:
\section{Discussion}\label{Sec:Conclusion}
\begin{comment}
In this paper, we explored the idea of Gaussian sampling for problems with PSD matrix decision variables. We saw that the memory requirement for the intermediate steps and the final solution for the problem was linear. The idea of sampling was extended to randomize over extreme points so that the objective function value in expectation remains the same.

The class of problems to which this can be effectively applied is restricted to having a smooth objective function and a well-defined feasible region (i.e., it is easy to characterize the extreme points).
\end{comment}

\paragraph{Comparison with \citet{yurtsever2019scalable}.}
In Section~\ref{Sec:Methodology}, we saw that it is possible to generate and store samples of a near-feasible, near-optimal solution to~$\eqref{prob:SDPlin}$ using Algorithm~\ref{Algo1} with working memory that is independent of the approximation parameter~$\epsilon$ and limited to $\mathcal{O}(d+n)$. This memory requirement differs from the $\mathcal{O}(d+rn/\epsilon)$ working memory required by the algorithm given by~\citet{yurtsever2019scalable}. The difference arises due the fact that we only aim to provide a sampled representation of the approximate solution rather than generate a near best rank $r$ approximation of the solution, or recover the exact solution matrix. In the special case of \textsc{MaxCut}, our sample-based representation is sufficient to implement the Goemans-Williamson rounding scheme. This allows us to implement an $(1-\epsilon)\alpha_{GW}$ approximation algorithm for~\textsc{MaxCut} using memory linear in~$n$ and independent of~$\epsilon$. \citet{yurtsever2019scalable} have performed numerical experiments which show that their method is capable of handling~\eqref{prob:relaxedMC} with $n\approx 8\cdot 10^6$ on a computer with 16 GB RAM. Our preliminary numerical results (see Section~\ref{sec:ComputationalResults}) are perhaps less promising in terms of practical convergence rate.

\begin{comment}
Furthermore, we can similarly store a $n\times k$ sketch and perform rank-1 update to it using the samples from the output of Algorithm~\ref{Algo1} and use their reconstruction method to generate a rank-$r$ approximate solution. The results in \cite[Theorem~1.1]{yurtsever2019scalable} hold in expectation in this case.
\end{comment}

\paragraph{Alternative algorithms with Gaussian sampling.} The Frank-Wolfe algorithm is well suited for Gaussian and extreme-point sampling when the extreme points have low rank. It is interesting to consider which other algorithms can be modified to track a sampled representation of the decision variable rather than the decision variable itself. For example, \textsc{MaxCut} algorithm given by~\citet*{klein1996efficient} generates a rank-1 update at each iteration and its output is used to produce a factorization of an approximate solution to~\eqref{prob:relaxedMC}. The structure of the updates and the computations required in their algorithm are structurally similar enough to our approach that the matrix iterates in~\cite{klein1996efficient} can be systematically replaced with Gaussian samples.

Another method where this could be done is the Matrix Multiplicative Weights (MMW) method, where the update to the variable takes the form $x_{t+1} \leftarrow x_t\exp(c(x))$ with $c(x)$ being a feedback function from the previous iterates. In the case of SDPs, this method requires computing the matrix exponential to generate the updates. \citet{carmon2019rank} provide an algorithm~\cite[Algorithm~1]{carmon2019rank} for solving~\eqref{prob:SDPlin} using MMW method which relieves the computational burden of generating the matrix exponential. They do so by restricting the update to be a rank-1 sketch which is the result of multiplying the matrix exponential with a random vector drawn from a standard Gaussian distribution. This rank-1 sketch is computed using Lanczos algorithm without actually generating the matrix exponential. Again we expect that it should be possible to use the idea of randomized extreme-point sampling to get linear working memory while implementing this algorithm.

\paragraph{Using Gaussian sampling for other rounding schemes.} There are a number of other approximation algorithms that involve solving~\eqref{prob:relaxedMC} (with appropriate cost matrix $C$) and then rounding the Gaussian samples with covariance given by the SDP solution. Examples include \textsc{Max-2SAT} \cite{GW} and the maximization of indefinite binary quadratic forms studied by \citet{charikar2004maximizing} and \citet{megretski2001relaxations}. Our approach allows these approximation algorithms to also be solved in $\mathcal{O}(n)$ working memory. It would be interesting to investigate which other rounding schemes can be implemented in a memory-efficient way using modifications of our approach.

\begin{comment}
Another application to which we can extend Algorithm~\ref{Algo1} is the matrix completion problem which involves estimating a low-rank factorization of the matrix from partial observed noisy entries $b_{ij}$ of $B\in \mathbb{R}^{m\times n}$. The following formulation is the convex relaxation of matrix completion problem,
\begin{equation}
\begin{split}
\min_{X\in \mathbb{R}^{m\times n}} &f(X) = \sum_{ij\in E} (x_{ij} - b_{ij} )^2\\
\textup{s.t.}\ &\| X\|_{*} \leq \alpha.
\end{split}
\end{equation}

The constraint is the relaxation of the bound on the rank of matrix $X$ and the parameter $\alpha$ is selected so that rank of $X$ is bounded by a predetermined value $r$. When the distribution of observed entries in nonuniform,~\citet*{srebro2010collaborative} showed that the standard nuclear norm relaxation behaves poorly. Such nonuniform distribution is observed in case of applications such as Netflix problem, where the observation of entries depends on the user ratings which varies based on the user as well as the movie. Another convex relaxation to rank constraint is the max-norm based approach which sometimes outperforms the nuclear norm based relaxation. Max-norm $\|X\|_{\textup{max}}$ of the matrix is defined as
\begin{equation}
X_{\textup{max}} = \min_{X= UV^T} \| U\|_{2,\infty}\|V\|_{2,\infty},
\end{equation}
where the minimization is carried out over all the factorizations $U\in \mathbb{R}^{m\times k}, V\in \mathbb{R}^{n\times k}$ with $k = 1,\dotsc, \textup{min}(m,n)$. The max-norm based matrix completion problem then involves minimizing $f(X)$ over the set $\mathcal{K}(\alpha, \alpha\sqrt{r}) = \{X: \| X\|_{\infty} \leq \alpha, \|X\|_{\textup{max}} \leq \alpha \sqrt{r}\}$ and its SDP reformulation can be written as~\cite{cai2016matrix}
\begin{equation}
\begin{split}
\min_{X\in \mathbb{R}^{m\times n}} &f(X)\\
\textup{s.t.}\ &\| X\|_{\infty} \leq \alpha\\
&\textup{diag}(W_1) \leq \alpha \sqrt{r}\\
&\textup{diag}(W_2) \leq \alpha \sqrt{r}\\
&\left( \begin{matrix} W_1&X\\ X^T&W_2 \end{matrix} \right) \succeq 0.
\end{split}
\end{equation}

The constraints bounding the diagonal entries of $W_1$ and $W_2$ are linear inequality constraints and can be penalized as given in Section~\ref{sec:LintoBounded}. This problem differs from the problem studied in Section~\ref{sec:LintoBounded} because of the addition of the nonlinear constraint $\| X\|_{\infty} \leq \alpha$. It needs to be seen how to handle this constraint. If we take an approximation of the feasible set by bounding only the entries $x_{ij}$ such that $b_{ij}\neq 0$, then we can solve the above problem in low memory. However, the approximate solution to the problem might not necessarily be feasible in this case and we do not know if feasibility guarantees exist in this case.
\end{comment}

\subsection{Preliminary Computational Results}\label{sec:ComputationalResults}
We conclude with some preliminary computational experiments for \textsc{MaxCut}. The algorithms we propose are simple to implement, and offer a scope for modification and improvement. Our aim here is to illustrate this simplicity and to identify possible areas for future algorithmic developments. We generated random graphs with sizes $n = 800$ to $n = 2000$ and with average degree of $10$. Note that all the graphs had unweighted edges. The computations were performed using MATLAB R2018b on a machine with 8GB RAM and 4 cores. The input parameter values for \textsc{MaxCut} were set as $d = n$, $\alpha = n$, $\omega = 1$, $\epsilon = 0.1$ $\beta = 4\textup{Tr}(C)$ and $M = 4\frac{\log(2d)}{\epsilon}$.

The key observation during the implementation was that the working memory at any time during the algorithm was restricted to at most $20n$, which is linear in the size of the graph as claimed. We tracked the change in infeasibility, which also determines an upper bound on sub-optimality of the solution, with each iteration. The plot of $\log(\| \mathcal{A}(X) - b\|_{\infty})$ vs $\log(t)$ (iteration number) for $n =800$ and $n=2000$ is shown in Figure~\ref{fig:ConstrViol}.

\begin{figure}[htb]
\centering
  \begin{subfigure}[b]{.49\linewidth}
    \centering
    \includegraphics[width=.99\textwidth]{ConstrViolLog800.png}
    \caption{$n=800$}\label{fig:1a}
  \end{subfigure}
  \begin{subfigure}[b]{.49\linewidth}
    \centering
    \includegraphics[width=.99\textwidth]{ConstrViolLog2000.png}
    \caption{$n=2000$}\label{fig:1b}
  \end{subfigure}
 \caption{Plot of $\log(\parallel \mathcal{A}(X_t) - b\parallel_{\infty})$ vs $\log(t)$.}\label{fig:ConstrViol}
\end{figure}

By comparing the two plots, we also noticed that the rate of the change in error observed was similar for problems with different sizes. For both problems, it was observed that during the initial phase, the rate of change in infeasibility is small. However, after a fixed percentage of total iterations, there is a steady decrease in the error and finally, it converges to a value about 10 times smaller than $\epsilon = 0.1$ in both cases.

Algorithm~\ref{Algo1} was implemented exactly as given in Section~\ref{Sec:Methodology} without any enhancements with the exception of using \texttt{eigs} command in MATLAB instead of the power method to compute eigenvectors in the \texttt{LMO} subroutine at each iteration. Note that the bounds in Theorem~\ref{thm:MaxCutResult} are satisfied by the iterates. As the convergence is quite slow for first $10^3$ iterations for $n=800$, there is room for improvement in the design of the algorithm. We conjecture that this slow initial convergence is due to the approach taken to penalize the constraints and identify this as a natural direction for future algorithmic work. Currently, each iteration also requires computing the leading eigenvector. This could be improved with warm start if we know the approximate subspace in which the eigenvector lies which might become clearer as the algorithm reaches the near-feasible, near-optimal solution to the input problem. These improvements can potentially lead to a more practical low memory method, based on the ideas presented in this paper.


TITLE: prime numbers and natural numbers
QUESTION [1 upvotes]: Prove that $n^4+4^n$ is never prime. Here $n$ is any natural number greater than $1$.
I have tried by induction hypothesis but to no avail.
Can it be done by considering cases when $n$ is odd and when it is even?

REPLY [0 votes]: on binomial expansion of (2k+1)^2,all terms are multiples of 4 but one term is 1.so 4 cannot be taken common.


TITLE: The probability that Student $1$ performs better than Student $2$ on an examination
QUESTION [2 upvotes]: Suppose there's a exam with 5 questions.
If the probability that Student $1$ correctly answers question $i$ is $P_{1.i}$, then
$P_{1.1} = 0.3$ , $P_{1.2} = 0.4$ , $P_{1.3} = 0.9$, $P_{1.4} = 0.7$ , $P_{1.5} = 0.1$
For Student $2$,
$P_{2.1} = 0.4$ , $P_{2.2} = 0.5$ , $P_{2.3} = 0.2$, $P_{2.4} = 0.8$ , $P_{2.5} = 0.1$
What is the probability that Student $1$ performs better than Student $2$ ?
How to solve something like that? I want an expression to do this.

REPLY [0 votes]: Comment: I wish you success with @HennoBrandsma's approach (+1). It seems there will be some bookkeeping in considering
all the possibilties. In case it is of any use (e.g., for checking intermediate results), here are simulated
distributions for Student 1, Student 2, and Difference scores.
It is
reasonable to expect the simulated probabilities to be accurate to two
or three places.
Although Student 1 will do slightly better on average, it seems that there is slightly less than a 50:50 chance for
Student 1 to do better than Student 2. However, there are about 3 chances in 4 Student 1 will do as well or better.
set.seed(429)
m = 10^6;  p1=c(3,4,9,7,1)/10;  p2=c(4,5,2,8,1)/10
s1 = replicate(10^6, sum(rbinom(5, 1, p1)))
round(table(s1)/m,3)
s1
    0     1     2     3     4     5
0.011 0.142 0.398 0.340 0.101 0.008
s2 = replicate(10^6, sum(rbinom(5, 1, p2)))
round(table(s2)/m,3)
s2
    0     1     2     3     4     5
0.043 0.260 0.407 0.236 0.051 0.003
d = s1 - s2
round(table(d)/m,3)
d
   -5    -4    -3    -2    -1     0     1     2     3     4     5
0.000 0.001 0.011 0.060 0.172 0.285 0.272 0.148 0.044 0.006 0.000

mean(s1 > s2)
[1] 0.471404
mean(s1 >= s2)
[1] 0.756087

par(mfrow=c(1,3))
 hist(s1, prob=T, br=(0:6)-.5, col="skyblue2", main="Student 1 Scores")
 hist(s2, prob=T, br=(0:6)-.5, col="skyblue2", main="Student 2 Scores")
 hist(d, prob=T, br=(-5:6)-.5, col="skyblue2", main="Difference in Scores")
  abline(v = .5, col="red", lwd=3, lty="dashed")
par(mfrow=c(1,1))


TITLE: Number of integer solutions to the equation $x_1+x_2+x_3+x_4=100$
QUESTION [4 upvotes]: $x_1+x_2+x_3+x_4 = 100$ with
$1 \le x_1\le10$
$2\le x_2\le15$
$x_3\ge5$
$0\le x_4\le10$
Apparently this is the same as
$y_1 + y_2 + y_3 + y_4 = 92$ with
$y_1 \le 9$
$y_2 \le13$
$y_4 \le10$
I understand the $3$ conditions, but what has happened to $x_3$?
I can see that it has been subtracted but what has happened to the conditions?

REPLY [5 votes]: We are defining $y_1=x_1-1, y_2=x_2-2,y_3=x_3-5,y_4=x_4$ and demanding that all the $y$'s be $\ge 0$.  Once we defined $y_3=x_3-5$, there is no constraint on $y_3$ except that it be nonnegative, so it is not listed in the constraints.