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\title {Exponents for $B$-stable ideals}

\author{Eric Sommers}
\author{Julianna Tymoczko}
\thanks{E.S. was supported in part by NSF grants DMS-0201826 and DMS-9729992}
\thanks{2000 MSC: 20G05 (14M15, 05E15)}

\address{
Department of Mathematics \& Statistics\\
University of Massachusetts--Amherst\\
Amherst, MA 01003
}

\email{esommers@math.umass.edu}

\address{
Department of Mathematics \\
University of Michigan\\
Ann Arbor, MI  48109-1109
}

\email{tymoczko@umich.edu}

\date{1/06/04; 6/02/04}

\begin{abstract}

Let $G$ be a simple algebraic group over the complex numbers
containing a Borel subgroup $B$.
Given a $B$-stable ideal $I$ in the nilradical of the Lie algebra of $B$,
we define natural numbers $m_1, m_2, \dots, m_k$ which we call
ideal exponents.  We then propose two conjectures where these
exponents arise, proving these conjectures in types $A_n, B_n, C_n$ and some other types.

When $I = 0$, we recover the usual exponents of $G$ by Kostant \cite{kostant:princ}
and one of our conjectures reduces to a
well-known factorization of the Poincar{\' e} polynomial of the Weyl group
(see also \cite{macdonald:poincare}).
The other conjecture reduces to a well-known result of Arnold-Brieskorn
on the factorization of the
characteristic polynomial of the corresponding
Coxeter hyperplane arrangement (see \cite{orlik-terao}).

\end{abstract}
\maketitle

\section{Introduction}

Let $G$ be a simple algebraic group over the complex numbers
containing a Borel subgroup $B$.
The ideals in the nilradical of the Lie algebra of $B$, which are stable
under the action of $B$, have attracted much recent attention.

In this paper we define a sequence of natural numbers
$m_1, m_2, \dots, m_k$ for each $B$-stable ideal $I$, and call them ideal exponents.
The definition is a generalization of the usual exponents of $G$ in the case
where $I = 0$, via Kostant's proof relating the heights of positive roots to the
exponents \cite{kostant:princ}.

We then conjecture (and prove in type $A_n, B_n, C_n$ and in some other cases)
two results about these ideal exponents.
The first concerns a Poincar{\' e} polynomial defined for each ideal which generalizes
the Poincar{\'e} polynomial for the Weyl group.
The conjecture is that this new
polynomial factors according to the ideal exponents
just as the usual polynomial factors according to the usual exponents.
This result is relevant for the study of regular nilpotent Hessenberg varieties
(there is one for each ideal) since the combinatorially-defined Poincar{\' e}
polynomials in this paper should be the actual
Poincar{\' e} polynomials for these varieties.  This is known in many cases, as studied in \cite{T}.

The second occurrence of these new exponents is in the context of
a hyperplane arrangement defined for each ideal.  The hyperplane arrangement
in question consists of those hyperplanes which correspond to the positive roots
whose root space does not belong to the ideal.
Generalizing the known result that the usual exponents are the roots of the
characteristic polynomial for the full Coxeter arrangement,
we conjecture (and prove in
the classical types)
that the characteristic polynomial of this new hyperplane arrangement
has (non-trivial) roots $m_1, m_2, \dots, m_k$.
We also speculate that these arrangements are free
(which we also prove in the classical types).

The paper concludes with speculation linking these two occurrences of the
ideal exponents.

\section{Notation}

Fix a  maximal torus $T$ in $B$ and let
$(X, \ro, Y, \co)$ be the root datum determined
by $G$ and $T$, and let $W$ be the Weyl group.
Let $\Pi \subset \posroots$ denote the simple roots and positive roots
determined by $B$.  As usual, $\langle \ , \ \rangle$ denotes the pairing
of $X$ and $Y$.  Let $\colat$ denote the lattice
in $Y$ generated by $\co$ (the coroot lattice).
We denote the
standard partial order on $\ro$ by $\preceq$; so $\al \preceq \beta$ for
$\al, \beta \in \ro$ if and only if $\beta - \al$ is a sum of positive roots.
As is customary, we write $\al \prec \beta$ if $\al \preceq \beta$ and $\al \neq \beta$.
For $\beta \in \ro$, write $\beta = \sum_{\al \in \Pi} c_{\al} \al$ and
let $\hgt(\beta) = \sum_{\al \in \Pi} c_{\al}$ denote the height of $\beta$.

We define an ideal (also called an upper order ideal)
$\ideal$ of $\posroots$ to be a collection
of roots such that if
$\alpha \in \ideal, \beta \in \posroots$, and $\alpha+\beta \in \posroots$,
then $\alpha + \beta \in \ideal$.
In other words, if $\al \in \ideal$ and $\gamma \in \posroots$ with $\al \preceq \gamma$,
then $\gamma \in \ideal$.

Let $\g, \bo, \tor$ be the Lie algebras of $G$, $B$, $T$, respectively.
It is easy to see that $B$-stable ideals in the nilradical $\nilrad$ of $\bo$
are naturally in bijection with the ideals of $\posroots$.
Namely, if $I$ is a $B$-stable ideal of $\nilrad$, it is stable under the action of $T$,
hence $I$ is a sum of root spaces.  Denote by $\ideal$ the set of roots
whose root space is contained in $I$.  Then $\ideal$ is an ideal of $\posroots$
and this map is a bijection.

\section{Ideal exponents}

In this section, motivated by Kostant's proof relating the heights of the positive
roots and the usual exponents of $G$, we define exponents for each ideal.
Our definition is an easy modification:  we consider only those positive roots which
do not lie in the ideal.

For an ideal $\ideal \subset \posroots$, let
$\ideal^c = \posroots - \ideal$ be the positive roots not in $\ideal$.

Define
$$\lambda_i = \# \{ \al \in \ideal^c \, | \, \hgt( \al) = i \}.$$
We first observe

\begin{prop} \label{partition}
The $\lambda_i$ give a partition of the number of roots in $\ideal^c$.
That is, $$\lambda_1 \geq \lambda_2 \geq \dots $$
In addition, $\lambda_1 > \lambda_2$.
\end{prop}

\begin{proof}
This is easy to check in the classical groups and was checked on a
computer in the exceptional
groups.
\end{proof}

Let $k = \lambda_1$, which is just the number of simple roots in $\ideal^c$.
We define $m^{\ideal}_k \geq  \dots \geq m^{\ideal}_1$ to be the dual partition
of $\lambda_i$.  In other words,
$m^{\ideal}_i = \# \{ \lambda_j \, | \, \lambda_j \geq k - i + 1 \}$.

\begin{defn}
The {\bf ideal exponents} of $\ideal$,
also called $\ideal$-exponents,
are the natural numbers $$m^{\ideal}_k \geq m^{\ideal}_{k-1} \geq \dots \geq m^{\ideal}_1.$$
\end{defn}

It follows from the fact that $\lambda_1 > \lambda_2$ that $m^{\ideal}_1 = 1$.
We also observe, as mentioned previously,
that when $\ideal = \emptyset$ these are the usual exponents
(in this case $k$ equals the rank of $G$) \cite{kostant:princ}.

We suspect that there are many situations where these
new exponents will arise.  We propose two situations in what follows, namely
Theorem \ref{thm1} and Theorem \ref{thm2}.

\section{Poincar{\' e} polynomials for ideals}

Let $R \subset \posroots$ be any subset of the positive roots.
Given $S \subset R$ we say that
$S$ is {\bf R-closed} if
$\al, \beta \in S$ and $\al + \beta \in R$,
then $\al + \beta \in S$.

Given an ideal $\ideal \subset \posroots$,
we are interested in those
subsets $S$  of $\ideal^c$ with the property
that both $S$ and its complement $S^c:= \ideal^c - S$
are $\ideal^c$-closed.

These subsets are analogous to Weyl group elements.  Indeed,
if $\ideal = \emptyset$ and if $w \in W$, then
$$N(w) := \{ \al \in \posroots \, | \, w(\al) \prec 0 \}$$
is $\ideal^c$-closed (in this case, $\ideal^c = \posroots$).
Conversely every subset of $\posroots$ which is $\posroots$-closed
is equal to $N(w)$ for a unique $w \in W$.
This is well-known and goes back to \cite{kostant:borel}.

Given this background, we
define a subset $S$ of $\ideal^c$ to be of {\bf Weyl-type} for $\ideal$
if both $S$ and $S^c$ are $\ideal^c$-closed.
Let ${\mathcal W}^{\ideal}$ denote the subsets of $\ideal^c$ of Weyl-type.
One of the main results of this paper can now be formulated.

\begin{thm} \label{thm1}
Let $\ideal$ be an ideal in $\posroots$.  Then
in types $A_n, B_n, C_n, G_2, F_4, E_6$
\begin{equation} \label{poincare}
\sum_{S \in {\mathcal W}^{\ideal}} t^{|S|}
= \prod_{i=1}^{k} (1 +t + t^2 + \dots + t^{m^{\ideal}_i}),
\end{equation}
where the $m^{\ideal}_{i}$ are the exponents of $\ideal$.
\end{thm}

We conjecture that the theorem also holds in the remaining cases.  We defer the proof,
which is case-by-case, until Section \ref{main_proof}.

In the case where $\ideal^c = \posroots$, the theorem is well-known
\cite{kostant:princ}, \cite{macdonald:poincare}.
On the one hand,
the $\ideal$-exponents become the usual exponents as mentioned previously.
On the other hand, if $l(w)$ denotes the length of $w \in W$ with respect to
the set of simple reflections coming from $\Pi$, then $l(w) = |N(w)|$ and so
the left-hand side of Equation (\ref{poincare}) is equal to
$$ \sum_{w \in W} t^{l(w)}.$$
Then Equation (\ref{poincare}) becomes the well-known factorization of the Poincar{\' e} polynomial of
the Weyl group, which is also the Poincar{\' e} polynomial of the flag variety $G/B$
when $t$ is replaced by $t^2$.

\section{Some old results for general root systems}

Many of the results in this paper rely on
the following lemma and its corollary.
The lemma is well-known, but we could not find a reference and saw no
harm in recording the proof.   We thank Jim Humphreys for simplifying our earlier proof.

\pagebreak
\begin{lem} \label{momma_lemma}
Let $x, y \in W$. Then the following four conditions are equivalent:~
\begin{enumerate}
\item[(i)] $N(x) \subseteq N(yx)$
\item[(ii)] $x^{-1} N(y) \subseteq \posroots$
\item[(iii)] $N(yx) = N(x) \cup x^{-1} N(y)$
\item[(iv)] $l(yx) = l(y) + l(x)$
\end{enumerate}
\end{lem}

\begin{proof}
Certainly (iii) implies (ii) by definition.

Then
\begin{equation} \label{equivalence}
N(x) \subseteq N(yx) \text{  if and only if  }
x^{-1} N(y) \subseteq \posroots
\end{equation}
since $\al \in x^{-1}N(y)$ with $\al \prec 0$ if and only if
$-\al \in N(x)$ and $-\al \notin N(yx)$.
This shows the equivalence of (i) and (ii).

It is easy to check from the definitions that
\begin{equation}\label{containment}
N(yx) \subseteq  N(x) \cup  x^{-1}N(y)

\end{equation}
where the union is a disjoint union.
Also
\begin{equation} \label{lengths}
l(yx) \leq l(x) + l(y)
\end{equation}
which implies that
$$|N(yx)| \leq |N(x)| + |N(y)| = |N(x)| + |x^{-1}N(y)|.$$

Hence equality in Equation (\ref{lengths}) is equivalent to equality in Equation (\ref{containment}),
which shows that (iii) and (iv) are equivalent.
Finally, it is clear that
$$x^{-1}N(y) \cap \posroots \subseteq N(yx),$$
so if (i) holds so does (iii) by Equation (\ref{equivalence}) and Equation (\ref{containment}).
\end{proof}

The next corollary follows from the lemma by taking a reduced expression for
$y = w x^{-1} = s_{\beta} \cdots s_{\al}$ where $\al, \beta \in \Pi$.

\begin{cor} \label{momma_cor_2}
Given $x, w \in W$.  If $N(x) \subsetneq N(w)$,
then there exists $\al, \beta \in \Pi$
so that
\begin{eqnarray*}
N(x)  \subsetneq N(s_{\al}x) \subseteq N(w) & \text{ and }\\
N(x)  \subseteq  N(s_{\beta}w) \subsetneq N(w) &
\end{eqnarray*}
\end{cor}

We conclude the section with a well-known lemma
related to the minimal length coset representatives of a parabolic subgroup of $W$.
We include a proof since we refer to the proof in what follows.

Let $\ro' \subset \ro$ be a parabolic subsystem.
In other words, $\ro'$ has a basis of simple roots which is contained in
the simple roots $\Pi$ of $\ro$.
Let $\ro^1 = \ro^+ - \ro'^+$ and let $W' \subset W$ be the Weyl group of $\ro'$.

\begin{lem} \label{coset_reps}
The set $C:= \{ x \in W \ | \ N(x) \subseteq \ro^1 \}$
is a set of distinct coset representatives for $W'$ in $W$.
\end{lem}

\begin{proof}
Take $w \in W$.
The intersection $N(w) \cap \ro'$ is of Weyl-type for $\ro'$.  Thus
$$N(w) \cap \ro' = N(x)$$
for some $x \in W'$, where $N(x)$ is
the same whether computed in $W'$ or $W$.
Now $N(x) \subseteq N(w)$, so by Lemma \ref{momma_lemma},
we have that $y = wx^{-1}$ satisfies
$$x^{-1}N(y) = N(w) - N(x).$$
Hence $x^{-1}N(y) \subseteq \ro^1$ by the definition of $x$.
Consequently $N(y) \subseteq \ro^1$ as $x \in W'$ and $W'$
preserves $\ro^1$.
Certainly, $wW' = yxW' = yW'$, which shows that the elements of $C$ are a set
of coset representatives.

They must be distinct representatives.
Indeed, suppose $y = z x$ for $y,z \in C$ and some $x \in W'$.
Then $x^{-1}N(z) \subseteq \ro^1$ since $W'$ preserves $\ro^1$.
Then Lemma~\ref{momma_lemma}~ implies that
$N(x) \subseteq N(y) \subseteq \ro^1$, forcing $x =1$ since also
$N(x) \subseteq \ro'$ and so $y=z$.

\end{proof}

\section{Some new results for general root systems}

The first new result of this paper is a generalization
of the fact that every subset of $\posroots$ of Weyl-type is of the form $N(w)$ for some $w \in W$
\cite{kostant:borel}.
On the one hand, it is easy to see that that
$N(w) \cap \ideal^c$  for $w \in W$ is always of Weyl-type
in $\ideal^c$.  But the converse is also true.
Namely, every subset $S$ of $\ideal^c$ of Weyl-type
is of the form $N(w) \cap \ideal^c$ for some $w \in W$.
That is,

\begin{prop}
Let $S \in {\mathcal W}^{\ideal}$.  There exists $w \in W$ such that
$S = N(w) \cap \ideal^c$.
\end{prop}

\begin{proof}
This is equivalent to showing that there exists $T \in {\mathcal W}^{\emptyset}$
for which $S = T \cap \ideal^c$ since we already know the result is true when $\ideal = \emptyset$.
In fact, it is enough to prove that given an ideal $\ideal_1$ where
$\ideal_1^c = \ideal^c \cup \{ \delta \}$ for $\delta \in \posroots$ we can find
$T \in {\mathcal W}^{\ideal_1}$ where $S = T \cap \ideal^c$.  Then the result
would follow by induction as there is always a sequence
$$\ideal^c = \ideal_0^c \subset  \ideal_1^c \subset \dots \subset  \ideal_r^c = \posroots,$$
such that
$\ideal_i$ is an ideal and
$|\ideal_{i+1}^c| = |\ideal_{i}^c| + 1$.

There are four possible situations given the above setup:
\begin{enumerate}
\item $S, \ S \cup \{ \delta \} \in {\mathcal W}^{\ideal_1}$
\item $S \in {\mathcal W}^{\ideal_1}$, \ $S \cup \{ \delta \}  \notin {\mathcal W}^{\ideal_1}$
\item $S \notin {\mathcal W}^{\ideal_1}$, \ $S \cup \{ \delta \}  \in {\mathcal W}^{\ideal_1}$
\item $S, \  S \cup \{ \delta \} \notin {\mathcal W}^{\ideal_1}$
\end{enumerate}
Assuming one of the first three possibilities arises, one of
$S$ or $S \cup \{ \delta \}$ (or in the first case both) would suffice for $T$.
The proposition is therefore equivalent to the last possibility never occurring.

The last possibility would only occur if there exists $\al, \beta \in S$
and $\al', \beta' \in \ideal^c - S$ for which
$\al + \beta = \delta = \al' + \beta'$.
If so,
then
$$ -\al' + \al + \beta = \beta'.$$
By Lemma 3.2 in \cite{sommers:b_stable}, either
$-\al' + \al$ or $-\al' + \beta$ lies in $\ro \cup \{ 0 \}$.
Since $\al, \beta \in S$ and $\al' \notin S$, neither
$-\al' + \al$ nor $-\al' + \beta$ can be zero.
Without loss of generality we take
$-\al' + \al \in \ro$, and by possibly interchanging the roles of $\al$ and $\al'$ in what
follows, we may assume $-\al' + \al \in \posroots$.

On the one hand, $(-\al' + \al) + \al' = \al$.
Now since $\ideal$ is an ideal and $\al \in \ideal^c$
and $-\al' + \al \prec \al$, then $-\al' + \al \in \ideal^c$.
It follows that $-\al' + \al \in S$, since $\al' \notin S$ and $\al \in S$
and otherwise $S^c = \ideal^c - S$ would not be $\ideal^c$-closed.

On the other hand, $-\al' + \al = \beta' - \beta \in \ideal^c$.
Clearly, $(\beta' - \beta) + \beta = \beta'$.  Then
$\beta' - \beta \notin S$, since $\beta \in S$ and $\beta' \notin S$
and otherwise $S$ would not be $\ideal^c$-closed.
This contradicts the fact that $-\al' + \al = \beta' - \beta \in S$ from
the previous paragraph.

It follows that $T$ can be chosen to be
one of $S$ or $S \cup \{ \delta \}$ (or possibly both), proving the proposition.
\end{proof}

Although there is not always a unique $w \in W$ satisfying the hypotheses of the
proposition, there is a unique $w$ with the property that
$N(w)$ is contained in $N(w')$ for any other $w' \in W$
satisfying the hypotheses of the
proposition.
More generally,

\begin{prop} \label{minimal}
Let $\ideal' \subseteq \ideal$ be ideals.
Given $S \in {\mathcal W}^{\ideal}$, there exists
$T \in {\mathcal W}^{\ideal'}$ with the property
that $S = T  \cap \ideal^c$ and if
${\widehat T} \in {\mathcal W}^{\ideal'}$ satisfies
$S \subseteq {\widehat T}$, then $T \subseteq {\widehat T}$.  Such a $T$ is clearly unique.
\end{prop}

\begin{proof}
The proof is by induction on the difference in cardinalities
$l = |\ideal| - |\ideal'|$.  The case $l=0$ is trivial, with $T=S$.
If $l>0$, pick an ideal $\ideal_1$
such that
$$\ideal' \subsetneq \ideal_1 \subseteq \ideal$$
where $\ideal'^c = \ideal_1^c \cup \{ \delta \}$
for some $\delta \in \posroots$.
By induction there exists
$T_1 \in {\mathcal W}^{\ideal_1}$ satisfying the hypotheses
of the proposition with respect to $\ideal_1 \subseteq \ideal$
and $S \in {\mathcal W}^{\ideal}$.

Set $T= T_1$ if $T_1 \in {\mathcal W}^{\ideal'}$, or else
set $T= T_1 \cup \{ \delta \}$ if $T_1 \notin {\mathcal W}^{\ideal'}$.
In either case the proof of the previous proposition ensures that
$T \in {\mathcal W}^{\ideal'}$.

Now suppose that ${\widehat T} \in {\mathcal W}^{\ideal'}$ satisfies
$S \subseteq {\widehat T}$.
Since $\ideal_1^c  \subset \ideal'^c$, it is clear that
${\widehat T} \cap \ideal_1^c \in {\mathcal W}^{\ideal_1}$.
Then the minimal property for $T_1$ gives that
$$T_1  \subseteq {\widehat T} \cap \ideal_1^c.$$

Hence $T_1 \subseteq {\widehat T}$.
We deduce that
$T \subseteq {\widehat T}$.
Indeed, $T =  T_1 \cup \{ \delta \}$ only when
there exist $\al, \beta \in T_1$ with $\al + \beta = \delta$.
Since $T_1 \subseteq {\widehat T}$ and ${\widehat T}$ is $\ideal'^c$-closed,
we must have $\delta \in {\widehat T}$.
\end{proof}

There is a nice characterization of the
$w \in W$ for which $T = N(w)$ satisfies
the minimal condition of Proposition \ref{minimal} when $\ideal' = \emptyset$.

\begin{prop} \label{condition}
Given $S \in {\mathcal W}^{\ideal}$
there is a unique $w \in W$ satisfying
both $S = N(w) \cap \ideal^c$ and
\begin{equation} \label{nonempty}
w^{-1}(-\Pi) \cap \posroots \subseteq \ideal^c.
\end{equation}
Furthermore, $N(w) \subseteq N(x)$ for any $x \in W$ with
$S \subseteq N(x)$ as in Proposition \ref{minimal}.
\end{prop}

\begin{proof}
Let $w$ be such that $T = N(w)$
satisfies the minimal property from Proposition \ref{minimal}
for $\ideal'= \emptyset$ and $S$.

Suppose there exists a simple root $\al \in \Pi$ for which
$$w^{-1}(-\al) \in \posroots - \ideal^c.$$
Consider $x = s_{\al} w$.  Then
$$w^{-1}(-\al) = x^{-1}(\al) \in \posroots$$
and so Lemma \ref{momma_lemma} implies that
$$N(w) = N(x) \cup \{ w^{-1}(-\al) \}.$$

But $w^{-1}(-\al) \notin \ideal^c$ and
therefore
$$N(x) \cap \ideal^c = N(w) \cap \ideal^c,$$
contradicting the minimal property of $N(w)$.  Hence
we must have
$$w^{-1}(-\Pi) \cap \posroots \subseteq \ideal^c.$$

For the uniqueness, take $y \in W$ with
$S = N(y) \cap \ideal^c$ and $y \neq w$.
Then $N(w) \subsetneq N(y)$ by Proposition \ref{minimal}.
By Corollary \ref{momma_cor_2}

there exists $\al \in \Pi$ such that
$$N(w) \subseteq N(s_{\al} y) \subsetneq N(y)$$
and this implies that $N(y)= N(s_{\al} y) \cup \{ y^{-1}(-\al) \}$.
It follows that $y^{-1}(-\al) \notin \ideal^c$ since
$$N(y) \cap \ideal^c = N(w) \cap \ideal^c$$ and hence
$y^{-1}(-\Pi) \cap \posroots \not\subset \ideal^c$.
\end{proof}

\section{Further results for types $A_n, B_n, C_n$}

In this section we explore some properties which are particular to
types $A_n, B_n, C_n$ and which are used in the proof of Theorem \ref{thm1}.

We label the simple roots in types $B_n$ and $C_n$ so that $\al_n$ is the only simple root of its length.

For each root system of type $X_n$, we embed
$X_{n-1}$ in $X_{n}$ via the simple roots $\al_2, \dots, \al_n$.
Denote by $\ro_{n-1}$ the roots of $X_{n-1}$ and
let
$$\ro^1 = \posroots - \posroots_{n-1}.$$
Let $W' \subset W$ be the Weyl group of $X_{n-1}$.
One of the key facts about these root systems
is that $\ro^1$ is linearly ordered under $\prec$ and (thus) there
is only root of each height from $1$ to the largest height ($n$
in type $A_n$ and $2n-1$ in types $B_n, C_n$).

More precisely,

\begin{lem} \label{abc_lemma}
Given $\al, \beta \in \ro^1$ with $\al \prec \beta$,
we always have
\begin{equation} \label{key_one}
\beta - b  \al = c  \gamma
\end{equation}
for some $\gamma \in \ro_{n-1}$ and
some $b, c \in \{ 1, 2 \}$.

\end{lem}

\begin{proof}
The roots in $\ro^1$ (in a standard basis)
are
\begin{eqnarray*}
&\{ e_1 - e_j  \  | \ j = 2, 3, \dots, n \} & \text{ for } A_{n-1}  \\
&\{ e_1 \pm e_j  \  | \ j = 2, 3, \dots, n\} \cup \{e_1\}  & \text{ for } B_n \\
&\{ e_1 \pm e_j  \  | \ j = 2, 3, \dots, n\} \cup \{2e_1\}  & \text{ for } C_n  \\
\end{eqnarray*}
These roots are ordered as follows:
$$e_1 - e_2 \prec \dots \prec e_1 - e_n \prec e_1
\prec e_1+e_n \prec \dots \prec e_1 +e_2 \prec 2e_1,$$
whenever the given root is present in the appropriate root system.

It is easy to see in type $A_{n-1}$ that $\beta - \al \in \ro^+_{n-2}$ (as desired,
given the shift in subscript).

In type $B_n$, $\beta - \al \in  \ro^+_{n-1}$, unless $\beta = e_1 + e_j$
and $\al = e_1 - e_j$, in which case $\beta - \al = 2 \gamma$
for some $\gamma \in \ro^+_{n-1}$.

In type $C_n$, we have $\beta - \al \in \ro^+_{n-1}$, except when $\beta = 2e_1$,
the highest root.  In that case, we can say that $\beta - 2 \al \in \ro_{n-1}$.
\end{proof}

\begin{lem} \label{key_lemma}
In types $A_n, B_n, C_n$,
for each $k \in \{0, 1, \dots, |\ro^1| \}$,
there is a unique element $x \in W$ satisfying
$N(x) \subseteq \ro^1$ and $|N(x)| = k$.
\end{lem}

\begin{proof}
Certainly there exists a unique $w^1$
with $N(w^1) = \ro^1$ since $\ro^1 \in {\mathcal W}^{\emptyset}$.
Explicitly, $w^1$ is the product of the long element of $W$ and the long element of $W'$,
using, for example, Lemma \ref{momma_lemma}.

It follows that there exists at least one $x \in W$ satisfying
$N(x) \subseteq \ro^1$ and $|N(x)| = k$
from Corollary \ref{momma_cor_2} by taking any reduced expression for $w^1$.

We still need to show uniqueness.
Let $x \in W$ satisfy $N(x) \subsetneq \ro^1$ and
assume uniqueness is true when $k > |N(x)|$.
Since $N(x) \subsetneq N(w^1)$,
Corollary \ref{momma_cor_2} implies the existence of $\al \in \Pi$
such that
$$N(x) \subsetneq N(s_{\al}x) \subseteq N(w^1),$$
where $N(s_{\al}x) = N(x) \cup \{ x^{-1}(\al) \}$.
Since $N(s_{\al}x) \subseteq N(w^1) = \ro^1$, we have
$$x^{-1}(\al) \in \ro^1.$$

We claim that $\al$ is unique.  Indeed, assume that
$\beta \in \Pi$ and $x^{-1}(\beta) \in \ro^1$ with $\beta \neq \al$.
Without loss of generality,
$x^{-1}(\al) \prec x^{-1}(\beta)$.  By Lemma \ref{abc_lemma},
$x^{-1}(\beta) - b x^{-1}(\al) = c \gamma$ for some $b,c$ and $\gamma \in \ro_{n-1}$.
Applying $x$ to both sides yields
$\beta - b  \al = c  x(\gamma)$.  This is impossible
since the right side is a linear combination of simple roots with
either all positive or all negative coefficients, whereas the left side
is a combination of two simple roots whose coefficients have opposite signs.
Since $s_{\al}x$ is unique by induction, $x$ is unique and the result follows.

\end{proof}

\section{Proof of Theorem \ref{thm1} for types $A_n, B_n, C_n$} \label{main_proof}

Assume the factorization is true for $\ro_{n-1}$.
Clearly $\ideal^c \cap  \ro_{n-1}$ is equal to $\ideal'^c$ for
some ideal $\ideal'$ for $\ro_{n-1}$.
Let $m_1, \dots, m_{k-1}$ be the $\ideal'$-exponents.

As noted in the previous section, the roots of
$\ro^1 = \posroots - \posroots_{n-1}$
are linearly ordered and so contain one root of each height.
It follows that
$\ideal^c \cap \ro^1$ contains one root of height
$1, 2, \dots, m_k$ for some natural number $m_k$,
and that
the $\ideal$-exponents are $m_k$ together with the $\ideal'$-exponents
$m_1, m_2, \dots, m_{k-1}$ (in this indexing, $m_k$ need not be
the largest exponent).

Now it is certainly true that if $S \in {\mathcal W}^{\ideal}$, then
$S \cap \ro_{n-1} \in {\mathcal W}^{\ideal'}$.

We have a strong converse which holds in types $A_n ,B_n, C_n$:

\begin{lem}
Let $S' \in {\mathcal W}^{\ideal'}$.
For each $j \in \{ 0, 1, \dots, |\ro^1| \}$, there exists
a unique $S \in {\mathcal W}^{\ideal}$
with $S \cap \ro_{n-1} = S'$ and $|S \cap \ro^1| = j$.
\end{lem}

\begin{proof}
Let $w' \in W'$  be the unique element with
the property that $N(w') \cap \ideal'^c = S'$
and $N(w') \subseteq N(x)$ for any $x \in W$ with $N(x) \cap \ideal'^c = S'$
as in Proposition \ref{minimal}.

Let $\{ x_0=1, x_1, \dots, x_i, \dots \}$ be the elements from Lemma \ref{key_lemma}
with $|N(x_i)|=i$.  Note that
\begin{equation*}
N(x_i) \subsetneq N(x_{i+1})
\end{equation*}
from the existence part of the
proof of that lemma.

Consider the elements $x_i w'$.
As in the proof of Lemma \ref{coset_reps},
\begin{equation} \label{yeah}
N(x_i w')  = N(w') \cup w'^{-1} N(x_i)
\end{equation}
where $w'^{-1} N(x_i) \subseteq \ro^1$.

It follows that $N(x_{i} w') \subsetneq N(x_{i+1} w')$
and the two sets differs by a single element of $\ro^1$.

Next, consider the intersection
$$N(x_i w') \cap \ideal^c \cap \ro^1.$$
This intersection is empty for $i=0$ and has $m_k$ elements
when $i=n$ in type $A_n$ and when $i=2n-1$ in types $B_n$ and $C_n$.
From the previous paragraph,
we know that
$N(x_{i+1} w') \cap \ideal^c \cap \ro^1$ and
$N(x_i w') \cap \ideal^c \cap \ro^1$ can differ by at most one
element.
Consequently, for some $i$ we have that $S:=N(x_i w') \cap \ideal^c$
satisfies $S \cap \ro_{n-1} = S'$ and  $|S \cap \ro^1|=j$
has the desired cardinality.
This gives existence.  It would also give uniqueness if we
knew that every $S$ is of the form $N(x_i w') \cap \ideal^c$
for some $i$.

To that end, suppose that
$S \in {\mathcal W}^{\ideal}$ and $S \cap \ro_{n-1} = S'$.
Let $w \in W$ be the unique element with
the property that $N(w) \cap \ideal^c = S$
and $N(w) \subseteq N(x)$ for any $x \in W$ with $S \subseteq N(x)$
as in Proposition \ref{minimal}.

Write $w = x_i y$ for $y \in W'$ by Lemma \ref{coset_reps}.
By Equation (\ref{yeah}) and the line following it,
$N(y) = N(w) \cap \ro_{n-1}$.
The latter contains $S'$ by the definition of $w$.
Thus there exists $i'$ so that
$$S \subseteq N(x_{i'} w'),$$
simply by taking the largest possible value of $i'$.
By Proposition \ref{minimal}, we get $N(x_i y) \subseteq N(x_{i'} w')$
and thus $N(y) \subseteq N(w')$ after intersecting with $\ro_{n-1}$.

Now Proposition \ref{minimal} applied to $w'$ gives
the equality $N(y)= N(w')$ and so $w' = y$.
It follows that $w = x_i w'$, showing uniqueness of $S$.
\end{proof}

\noindent {\bf Proof of Theorem \ref{thm1}: }
By the previous lemma, if we consider the sum
$\sum_{S} t^{|S|}$ over all $S \in {\mathcal W}^{\ideal}$
with $S \cap \ro_{n-1} = S'$ for some $S' \in {\mathcal W}^{\ideal'}$, then the sum
equals
$$t^{|S'|}(1 + t +t^2 + \dots + t^{m_k}).$$

Thus
\begin{eqnarray*}
\sum_{S \in {\mathcal W}^{\ideal}} t^{|S|} =
\sum_{S' \in {\mathcal W}^{\ideal'}} t^{|S'|}(1 + t +t^2 + \dots + t^{m_k}) = \\
(1 + t +t^2 + \dots + t^{m_k}) \prod_{i=1}^{k-1} (1 +t + t^2 + \dots + t^{m_i}),
\end{eqnarray*}
where the last step is by induction.  This completes the proof
of the theorem in types $A_n, B_n, C_n$.  This proof also works in type $G_2$.  In types $F_4$ and $E_6$,
the theorem was checked on a computer, running through all possible ideals.  There are $105$ ideals in $F_4$ and
$833$ of them in $E_6$.

\section{A uniform proof for the penultimate ideal} \label{uniform}

The goal for this section is to prove Theorem \ref{thm1} uniformly
when $\ideal = \{ \theta \}$, where $\theta$ is the highest root of $\posroots$.

The next result is a special case of Theorem 2.8 in \cite{macdonald:poincare}.
For $\gamma$ in $X$ (the weight lattice),
let $e^{\gamma}$ denote the corresponding element of the
group algebra $\Z[X]$ of $X$.
\begin{prop}  \cite{macdonald:poincare}
Let $R \subset \posroots$ be any subset of the positive roots.
The following identity holds:
\begin{equation} \label{macky}
\sum_{w \in W} \prod_{\al \in R} \frac{1 - t e^{w \al}}{1 - e^{w \al}}
=
\sum_{w \in W} t^{|N(w) \cap R|}
\end{equation}
\end{prop}

\begin{proof}
In Theorem 2.8 of
\cite{macdonald:poincare},
set $u_{\al} = t$ if $\al \in R$ and set $u_{\al} = 1$ if $\al \not\in R$.
\end{proof}

When $R = \ideal^c$ for some ideal $\ideal$, the right-side of
(\ref{macky}) is the Poincar{\'e} polynomial (after replacing $t$ by $t^{2}$)
of a regular semisimple Hessenberg variety
(see the next section) by work of
\cite{MPS}.  In that case, the identity can be proven by a fixed-point formula
as in \cite{macdonald:poincare} since these Hessenberg
varieties are smooth and projective (generalizing
the role of the flag variety in Macdonald's fixed-point formula proof).

We now use this
identity to prove Theorem \ref{thm1} uniformly for any root system
when $\ideal = \{ \theta \}$.

\begin{thm} \label{uniform1}
In all types when $\ideal = \{ \theta \}$,
$$\sum_{S \in {\mathcal W}^{\ideal}} t^{|S|}
= \prod_{i=1}^{n} (1 +t + t^2 + \dots + t^{m^{\ideal}_i})$$
where the $m^{\ideal}_i$ are the exponents of $\ideal$.
\end{thm}
\begin{proof}

In Equation (\ref{macky}), put $R = \ideal^c = \posroots - \{ \theta \}$
and specialize $e^{\al}$ to
$t^{\hgt(\al)}$ as in \cite{macdonald:poincare}.  Then we have
$$\sum_{w \in W} \prod_{\al \in \ideal^c} \frac{1 - t^{ \hgt(w \al) + 1 }}{1 - t^{ \hgt(w \al)}}
=
\sum_{w \in W} t^{|N(w) \cap \ideal^c|}  .$$

We will break apart the sum on the left side into two parts, according to whether
$w \in W$ satisfies Equation (\ref{nonempty}).
Let $W_{min}$ denote those elements of $W$ satisfying Equation (\ref{nonempty})
for $\ideal$.

If $w \in W_{min}$ and $w \neq 1$, then
$w^{-1}(\al) \in \posroots$ for some $\al \in -\Pi$ since $w \neq 1$.
Then $\beta := w^{-1}(\al) \in \ideal^c$ by Equation (\ref{nonempty}).
Since $\hgt(w \beta) = -1$ , the term for $w$ vanishes in the sum.
This leaves only the identity term as the contribution from the elements in $W_{min}$.

Therefore the sum on the left side reduces to
\begin{equation} \label{another_eq}
\prod_{\al \in \ideal^c} \frac{1 - t^{\hgt(\al)+1}}{1 - t^{\hgt(\al)}}
+ \sum_{w \not\in W_{min}} \prod_{\al \in \ideal^c} \frac{1 - t^{\hgt(w \al)+1}}{1 - t^{\hgt(w \al)}}
\end{equation}

The isolated product is exactly the right side of Theorem \ref{uniform1}.
To finish the proof we must, by Proposition \ref{condition}, show that the sum
in Equation (\ref{another_eq}) is equal to
$$\sum_{w \not\in W_{min}} t^{|N(w) \cap \ideal^c|} \ .$$

We divide $\ideal^c$ into two parts:
let $$\ideal^c_i = \{ \gamma \in \ideal^c \ | \ \langle \gamma, \theta^{\vee} \rangle = i \}$$
for $i=0,1$ (the only two possibilities since $\theta$ is the highest root).
The roots of $\ideal^c_0$ are the positive roots of a parabolic subsystem of $\ro$, with
corresponding Weyl group $W_{\theta}$.
These are exactly the elements of $W$ which fix $\theta$.

Given $\gamma \in \ideal^c_1$, we have
$s_{\theta}(-\gamma) = \theta - \gamma$, which is a positive root.
Hence also $s_{\theta}(-\gamma) \in \ideal^c_1$
and $\gamma + s_{\theta}(-\gamma) = \theta$.
This shows that elements of $\ideal^c_1$ come in pairs which sum up to $\theta$.

Take $w \not\in W_{min}$.
Then $w \theta \in -\Pi$ and thus $\hgt(w \theta)=-1$.
Suppose that $\al + \beta = \theta$ for $\al, \beta \in \ideal^c_1$.
Then
\begin{equation} \label{pairs}
\hgt(w \al) + \hgt(w \beta) = -1.
\end{equation}

Consequently exactly one of $\al$ and $\beta$ belongs
to $N(w) \cap \ideal^c$
and thus
$$|N(w) \cap \ideal^c_1| = \frac{1}{2} |\ideal^c_1|$$
for all $w \not\in W_{min}$.

On the other hand,
the identity
$$\frac{1 - t^{a+1}}{1-t^a} \cdot \frac{1 - t^{-a}}{1-t^{-a-1}}
= t$$
and Equation (\ref{pairs}),
imply that
$$\prod_{\al \in \ideal^c_1} \frac{1 - t^{\hgt(w \al)+1}}{1 - t^{\hgt(w \al)}}
= t^{\frac{1}{2} |\ideal^c_1|}$$
for $w \not\in W_{\min}$.

Therefore the proof will be completed
if we can show that
$$\sum_{w \not\in W_{min}} \prod_{\al \in \ideal_0^c} \frac{1 - t^{\hgt(w \al)+1}}{1 - t^{\hgt(w \al)}} = \sum_{w \not\in W_{min}} t^{|N(w) \cap \ideal^c_0|} \ .$$

We can do this by using Equation (\ref{macky}) for the Weyl group $W_{\theta}$.
First we observe that the action of $W_{\theta}$ on $W$ preserves $W_{min}$
and so $W_{min}$ and its complement are a union of left cosets of $W_{\theta}$.

Pick $x \not\in  W_{min}$.
Then

\begin{equation} \label{coset}
\sum_{w \in x W_{\theta}}  \prod_{\al \in \ideal_0^c} \frac{1 - t^{\hgt(w \al)+1}}{1 - t^{\hgt(w \al)}}
=
\sum_{y \in W_{\theta}} \prod_{\al \in \ideal^c_0} \frac{1 - t^{\hgt(x y (\al))+1}}{1 - t^{\hgt(x y (\al))}}

\end{equation}

In Equation (\ref{macky}) applied now for the case of the Weyl group $W_{\theta}$,
set $e^{\al} = t^{\hgt(x \al)}$ where the height
is still calculated with respect to $W$ (since $x \al$ need
not belong to the root system of $W_{\theta}$).
The positive roots for $W_{\theta}$ are $\ideal_0^c$, so
Equation (\ref{coset}) is equal to
$$\sum_{y \in W_{\theta}} t^{|N(y) \cap \ideal_0^c|}$$
which is the same thing as
$$\sum_{w \in x W_{\theta}} t^{|N(w) \cap \ideal_0^c|}$$
for any $x \in W$.
Indeed, we may choose $x$ such that $N(x) \subseteq \posroots - \ideal_0^c$
by Lemma \ref{coset_reps}.  Then Lemma \ref{momma_lemma}
implies that for $w_1, w_2 \in W_{\theta}$ that
$N(xw_1) \cap \ideal_0^c = N(xw_2) \cap \ideal_0^c$ if and only
if $w_1 = w_2$, which is what we needed.
The proof is completed since $W_{min}$ is a union of left cosets of $W_{\theta}$.

\end{proof}

\section{Poincar{\' e} polynomials of regular nilpotent Hessenberg varieties}

The combinatorial Poincar{\' e} polynomials from Theorem \ref{thm1}
should arise as the actual topological Poincar{\' e} polynomials
of certain projective subvarieties of the flag variety.
This section defines these subvarieties, called
regular nilpotent Hessenberg varieties, and lists some of their main
properties.

Write $B^-$ for the Borel subgroup opposite to $B$ and $\mathfrak{b}^-$
for its Lie algebra.

Given an ideal $\ideal$, we define
\[H_{\ideal} = \mathfrak{b}^- \oplus \bigoplus_{\alpha \in \ideal^c} \mathfrak{g}_{\alpha},\]
where $\mathfrak{g}_{\alpha} \subset \g$ is the $\al$-weight space.
The subspace $H_{\ideal}$ is stable for the action of $B^-$ and
it is easy to see that each subspace with this property is of the above
form.  Such a subspace is called a Hessenberg space.

Fix an

element $X \in \mathfrak{g}$ and a Hessenberg space $H = H_{\ideal}$.
The Hessenberg variety $\H_{X,H}$
is the subvariety of the flag variety $\H = G/B^{-}$ defined by
\[\H_{X,H} = \{gB^{-} \ | \ \Ad(g^{-1})(X) \in H\}.\]
This is a closed subvariety of $\H$ and hence is projective.  In general
a Hessenberg variety is not smooth.
Hessenberg varieties were first defined in \cite{MPS}.

When $\ideal = \posroots$ and thus $H = \bo^{-}$,
the Hessenberg variety reduces to a Springer variety, a well-studied and important
object in representation theory.
At the other end of the spectrum, when $\ideal = 0$
then the Hessenberg variety is the whole flag variety, independent of $X$.
In between, when $\ideal^c$ is the set of simple roots
and $X$ is regular nilpotent,
the Hessenberg variety is called the Peterson variety and
has been used to give geometric constructions for the
quantum cohomology of the flag variety (see \cite{Ko2}, \cite{R}).
Other Hessenberg varieties have been used in \cite{GKM} to give a partial
proof of the fundamental lemma of the Langlands program.

The following proposition about $\H_{X,H}$ follows from work in \cite{T}.

Let $B^{-} w B^{-} \subset \H$ be the Schubert cell containing
the point $wB^{-}$ where $w \in W$.
Here we do not distinguish between $w \in W$ and a representative of $w$
in $G$.

\begin{prop} \label{tymoczko thesis}
Let $G$ be of classical type and let $X$ be a sum of negative simple
root vectors.
Let $C_w := B^{-} w B^{-} \cap \H_{X,H_{\ideal}}$.  Then $C_w$ is
non-empty if and only if $w$ satisfies Equation (\ref{nonempty}) for $\ideal$.
If $C_w$ is non-empty, then it is an affine space of dimension
$|N(w) \cap \ideal^c|$.

This yields an affine paving of $\H_{X,H_{\ideal}}$.
\end{prop}

This proposition allows us to demonstrate that the Poincar{\' e} polynomials
for regular nilpotent Hessenberg varieties are the
polynomials which arise in Theorem \ref{thm1}.

\begin{thm}
Let $\mbox{P}_{\ideal}(t)$ denote the Poincar{\' e} polynomial of the
Hessenberg variety $\H_{X,H_{\ideal}}$ for $X$ a regular nilpotent element.
In types $A_n$, $B_n$, and $C_n$ this can be factored
\[\mbox{P}_{\ideal}(t^{\frac{1}{2}})
= \prod_{i=1}^k (1 + t + \ldots + t^{m_i^{\I}}).\]
\end{thm}

\begin{proof}
Proposition \ref{condition} shows that for each
$S \in \mathcal{W}^{\I}$ there is exactly one
$w \in W$ satisfying Equation (\ref{nonempty})
and $S = N(w) \cap \I^c$.
Proposition \ref{tymoczko thesis} shows that the
dimension of the affine cell $C_w$ is $|N(w) \cap \I^c|=|S|$
and that $C_w$ is empty if $w$ does not satisfy Equation (\ref{nonempty}).
The proof follows from the fact
that these cells give an affine paving of the variety together with Theorem \ref{thm1}.
\end{proof}

We should mention that if one knew that Proposition \ref{tymoczko thesis}
were true in all types
then a result of Peterson announced in \cite{BC} would be equivalent to
Theorem \ref{thm1}.
Unfortunately Peterson's proof is not given.  One could imagine
something along the lines of \cite{carrell:kostant-macdonald}
if the regular nilpotent Hessenberg varieties were smooth.  But this is not
the case, already in type $A_2$.

\section{Hyperplane arrangement defined by an ideal}

The second venue where the $\ideal$-exponents arise is in the context of hyperplane arrangements.
Let $V := \colat \otimes \mathbf{R}$ be the ambient vector space containing the coroot lattice
$\colat$.
For each $\al \in \posroots$, let $H_{\al} \subset V$ be the hyperplane
$$H_{\al} = \{ v \in V \ | \ \langle \al, v \rangle = 0 \}.$$
We are interested in the hyperplane arrangement in $V$
given by the hyperplanes $H_{\al}$ where $\al \in \ideal^c$.
We will denote this arrangement by $\hyp_{\ideal}$ and call it an arrangement of ideal-type
in $\ro_n$.

In general,
given a hyperplane arrangement,
one is interested in whether the arrangement is free and if so,
what are the roots of its characteristic polynomial, which are also called
exponents \cite{orlik-terao}.

We briefly recall the basic definitions and theorems about hyperplane arrangements
from Chapters 2 and 4 of \cite{orlik-terao}.
Let $\A$ be an arrangement of hyperplanes in the $\mathbb R$-vector space $V$.
Let $S(V^*)$ denote the symmetric algebra on the dual space $V^*$ of $V$.
Given $H \in \A$, let $\al_H \in V^*$ be a linear functional vanishing on $H$.
Set
$$Q(\A) = \prod_{H \in \A} \al_H.$$
Let $D(\A)$ denote the $\mathbb R$-linear
derivations of $S(V^*)$ which preserve the ideal generated
by $Q(\A)$.
Then the hyperplane arrangement $\A$ is said to be free if $D(\A)$ is a free $S(V^*)$-module.

Let $L=L(\A)$ denote the set of nonempty intersections of the elements
of $\A$.  This is a poset with a partial order given by reverse inclusion, with minimal element
$V$.  Define a function $\mu$ on $L$ as follows.  Set $\mu(V)=1$ and define
$\mu(X)$ recursively for $X \in L$ by the formula
$$\mu(X) = - \sum_{V \leq Z < X} \mu(Z),$$
where the sum is over all $Z \in L$ with $V \leq Z < X$.
Then the characteristic polynomial $\chi(\A, t)$ of $\A$ is defined
as
$$\chi(\A, t) = \sum_{X \in L} \mu(X)t^{\text{dim}(X)}.$$

The factorization result of Terao (see Theorem 4.137 in \cite{orlik-terao}) states
that if
$\A$ is free, then all of the roots of $\chi(\A, t)$ are nonnegative integers, called
the exponents of $\A$.  These exponents coincide with the polynomial degrees of a set of
homogeneous generators of $D(\A)$ as an $S(V^*)$-module.

There is another key property of hyperplane arrangements.
Given $H_0 \in \A$,
let $\A'$ denote the arrangement in $V$ obtained
by omitting the hyperplane $H_0$ from $\A$.
This is the deleted arrangement given by $H_0$.
Let $\A''$ denote the arrangement in $H_0$ given by the nonempty intersections $H \cap H_0$
for $H \in \A$ with $H \neq H_0$.
This is the restricted arrangement given by $H_0$.
The three arrangements $(\A, \A', \A'')$
is called a triple of arrangements.

We will use the following direction
of the Addition-Deletion Theorem of Terao in what follows
(see Theorem 4.51 in \cite{orlik-terao}).
If $\A'$ is free with exponents $b_1, \ldots, b_{k-1}, b_k-1$
and if $\A''$ is free with exponents $b_1, \ldots, b_{k-1}$,
then $\A$ is free with exponents $b_1, \ldots, b_k$.
If we wished only to know about the implication involving the exponents,
this result goes back to Brylawski and Zaslavsky (see Theorem 2.56 in \cite{orlik-terao}).

\begin{thm} \label{thm2}
Except possibly in types $F_4, E_6, E_7, E_8$,
the hyperplane arrangement $\hyp_{\ideal}$ is free and its
non-zero exponents are $m^{\ideal}_1, \dots, m^{\ideal}_k$.
There are also $n-k$ exponents equal to $0$.
\end{thm}

\begin{proof}
We will show that $\hyp_{\ideal}$ is free with the desired exponents by
using the Addition-Deletion Theorem.
First assume that $X_n$ is of type $A_n, B_n, C_n$ and assume the
result for any ideal $\ideal_1$ properly containing $\ideal$.
Furthermore assume
the result for root systems of smaller rank of these types.  The theorem is clearly
true for the base case where $\ideal = \posroots$, since the arrangement is empty.

Let $\ideal_1$ be
the unique ideal for which

$\ideal_1 = \ideal \cup \{ \delta \}$
where $\delta$ is the maximal root
in $\ro^1 \cap \ideal^c$.
Of course, if the latter intersection is empty, then
we are already done since the arrangement is the direct product
of the one-dimensional empty arrangement and an arrangement of ideal-type
in $\ro_{n-1}$.

Now by induction $\hyp_{\ideal_1}$ is free
and its non-zero exponents are
$$m^{\ideal}_1, \dots, m^{\ideal}_{k}-1$$ (we do not order the exponents)
where we have assigned $m^{\ideal}_k = \hgt(\delta)$.

Next consider the restricted arrangement defined
by $H_{\delta}$.  This is the arrangement
in $H_{\delta}$ defined by the hyperplanes
$H_{\al} \cap H_{\delta}$ for $\al \in \ideal^c$ and $\al \neq \delta$.
We denote this restricted arrangement by $\mathcal A^{\delta}$.

For $\beta \in \ro^1$ and $\beta \prec \delta$
Equation (\ref{key_one}) says that
$\delta - b \beta = c \gamma$ where $\gamma \in \ro_{n-1}$.
It follows that either $\gamma$ or $-\gamma$ is in $\ideal^c$ since $\delta \in \ideal^c$.
Set $\ideal' = \ideal \cap \ro_{n-1}$.
Then the hyperplane arrangement
$\hyp_{\ideal'}$ (defined for $\ro_{n-1}$)
is isomorphic to $\mathcal A^{\delta}$.  Indeed,
for each $\beta \in \ro^1$ with $\beta \neq \delta$ we have
$$H_{\beta} \cap H_{\delta} = H_{\gamma} \cap H_{\delta}$$
for some $\gamma \in \ideal'^c$.
Thus the hyperplanes $H_{\gamma} \cap H_{\delta}$
for $\gamma \in \ideal'^c$ yield the distinct
hyperplanes in $\mathcal A^{\delta}$.

Thus $\mathcal A^{\delta}$ is free and its non-zero exponents
are $$m^{\ideal}_1, m^{\ideal}_2, \dots, m^{\ideal}_{k-1},$$
since these are the exponents of $\ideal'$.

Consequently by Addition-Deletion applied to the triple
of arrangements $(\hyp_{\ideal}, \hyp_{\ideal_1}, \mathcal A^{\delta})$,
the arrangement $\hyp_{\ideal}$
is free and its non-zero exponents
are equal to
$$m^{\ideal}_1, m^{\ideal}_2, \dots, m^{\ideal}_{k},$$
as desired.

Type $G_2$ is trivial.  We consider the case of type $D_n$.
Here,
$$\ro^1 = \{ e_1 \pm e_j \ | \ 2 \leq j \leq n \},$$
using the standard notation for roots in $D_n$.
Let
\begin{eqnarray*}
\gamma_1 = & e_1 + e_n \\
\gamma_2 = & e_1 - e_n .
\end{eqnarray*}

The above proof carries over perfectly well as long as $\gamma_1$ and $\gamma_2$
do not both belong to $\ideal^c \cap \ro^1$
since Equation (\ref{key_one}) would hold for $\al, \beta \in \ideal^c \cap \ro^1$.

Suppose that $\gamma_1, \gamma_2 \in \ideal^c$.
First, assume that both $\gamma_1$ and $\gamma_2$ are maximal elements of $\ideal^c \cap \ro^1$.
Then $\ideal^c$ consists of the $n$ non-empty sets:
\begin{equation} \begin{split} \label{roots}
& \{e_{1+i} - e_j \ | \ 2+i \leq j \leq n \} \cup
\{e_{1+i} + e_ j \ | \ a_i \leq  j \leq n-1\} \text{ with } 1 \leq i \leq n-2  \\
& \{e_1 - e_j | \ 2 \leq j \leq n \} \\
& \{e_j +  e_n | \ 1 \leq j \leq n-1 \}
\end{split} \end{equation}
where $a_i$ is some natural number satisfying $2+i \leq a_i \leq n-1$ .
The former sets contain one root of each height $1, \dots, m^{\ideal}_i$,
where $m^{\ideal}_i$ depends on $a_i$.  The latter two sets contain
one root of each height $1, \dots, n-1$ and so the ideal exponents
of $\ideal$ can be written as
$$m^{\ideal}_1, m^{\ideal}_2, \dots, m^{\ideal}_{n-2}, n-1, n-1.$$

Let $\ideal_1 = \ideal \cup \{ \gamma_1 \}$.
By induction $\hyp_{\ideal_1}$ is free
and its non-zero exponents are
$$m^{\ideal}_1, \dots, m^{\ideal}_{n-2}, n-1, n-2.$$

Consider the restricted arrangement  $\mathcal A^{\gamma_1}$ defined
by $H_{\gamma_1}$.  We want to show it is free
with non-zero exponents $m^{\ideal}_1, \dots, m^{\ideal}_{n-2}, n-1$.

In order to do this, consider the deleted and restricted arrangements
of $\mathcal A^{\gamma_1}$ defined by $H_{\gamma_2}$.
The deleted arrangement $\mathcal (A^{\gamma_1})'$ is isomorphic
to the arrangement defined by $\ideal' = \ideal \cap \ro_{n-1}$ in $\ro_{n-1}$
by the same proof as in the other classical cases.
Thus $\mathcal (A^{\gamma_1})'$  is free and its
non-zero exponents
$$m^{\ideal}_1, \dots, m^{\ideal}_{n-2}, n-2$$
by inspection of the heights of roots in (\ref{roots}).

On the other hand, the restricted arrangement $\mathcal (A^{\gamma_1})''$
defined by $H_{\gamma_2}$ is more complicated.
This arrangement lives in $H_{\gamma_1} \cap H_{\gamma_2}$
which coincides with the intersection of the null spaces of $e_1$ and $e_n$.
The hyperplanes defining $\mathcal (A^{\gamma_1})''$
are given by the null spaces of
$$\{e_{1+i} - e_j \ | \ 2+i \leq j < n \} \cup \{e_{1+i} \} \cup
\{e_{1+i} + e_ j \ | \ a_i \leq  j \leq n-1\},$$
for $i = 1, \dots, n-2$.
This arrangement is precisely an ideal arrangement in $B_{n-2}$
which is free with non-zero exponents equal to
$$m^{\ideal}_1, \dots, m^{\ideal}_{n-2}.$$
By the Addition-Deletion Theorem,
${\mathcal A}^{\gamma_1}$ is free
with non-zero exponents equal to
$$m^{\ideal}_1, \dots, m^{\ideal}_{n-2}, n-1,$$
and using the theorem a second time,
it follows that $\hyp_{\ideal}$
is free and its non-zero exponents
are $$m^{\ideal}_1, \dots, m^{\ideal}_{n-2}, n-1, n-1,$$
as desired.

Finally, consider the case where $$\delta = e_1 + e_{2n - 1 - k}$$
is the maximal element of $\ideal^c \cap \ro^1$
for  $k > n-1$.
In this case $\hyp_{\ideal_1}$ is free
with non-zero exponents
$$m^{\ideal}_1, \dots, m^{\ideal}_{n-2},  n-1, k-1,$$
by induction.

It suffices to complete the proof by showing that
the restricted arrangement
of $\hyp_{\ideal}$ defined by $H_{\delta}$
is isomorphic to ${\mathcal A}^{\gamma_1}$ above.
Consider the element $w$ of the Weyl group of $D_n$ given
by exchanging $e_n$ and $e_{2n - 1 - k}$ and fixing all other $e_i$.
It is not hard to check that the hyperplanes defining ${\mathcal A}^{\gamma_1}$
in $H_{\gamma_1}$ are mapped to the hyperplanes defining this restricted arrangement
in $H_{\delta}$, yielding the isomorphism and completing the proof in type $D_n$.

\end{proof}

We conclude this section by noting that a uniform proof in all types
for the case $\ideal = \{ \theta \}$ is easy.
On the one hand, the full Coxeter arrangment is free
with exponents the usual exponents
$m_1 \leq \dots \leq m_n$.
On the other hand,
the restricted arrangement for $H_{\theta}$ is free with exponents
$m_1, \dots, m_{n-1}$ by \cite{terao}.
Thus $\hyp_{\ideal}$ is free with the desired exponents.

\section{Speculation}

We speculate that the two conjectures are equivalent by a general principle.
Namely, suppose a hyperplane arrangement $\hyp$ is free
with exponents
$m_1, \dots, m_n$.
Suppose further that the arrangement is central, meaning each hyperplane contains
the origin (what we will speculate is false otherwise).

Let $\mathcal C_{\hyp}$ denote the set of components
of the complement $V - \cup_{H \in \hyp} H$.
Fix one component $A \in \mathcal C_{\hyp}$.  Then for each component
$B \in \mathcal C_{\hyp}$ we can define $l(B)$ to be the least number
of hyperplanes needed to be crossed to move from $B$ to $A$.

Define a polynomial
$$P_{\hyp} (t) =  \sum_{B \in \mathcal C_{\hyp}} t^{l(B)}.$$
This is equivalent to the left-hand side of Equation (\ref{poincare}) in the case
when $\hyp$ is of ideal type and the component $A$
is chosen to contain the dominant Weyl chamber.

One can speculate that there always exists a component $A$ such that
$$P_{\hyp} (t) =  \prod_{i=1}^{n} (1 +t + t^2 + \dots + t^{m_i}).$$

One can also wonder if there exists a complex projective variety naturally associated
to $\hyp$ such that $P_{\hyp} (t^{\frac{1}{2}})$ is its Poincar{\' e} polynomial.
This would be the analogue of the regular nilpotent Hessenberg varieties.


TITLE: Documentaries about mathematics and mathematicians
QUESTION [3 upvotes]: Possible Duplicate:
List of Interesting Math Videos/ Documentaries

I have watched "Fermat's last theorem" a documentary about  Andrew Wiles proof of the theorem, it was a great show. and i am asking if there is some other good documentaries about mathematics and mathematicians out there (not movies).

REPLY [4 votes]: I liked documentary named "Dangerous Knowledge" about four brilliant mathematicians – Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing.


TITLE: Filling a barrel, using small containers
QUESTION [0 upvotes]: In how many ways, using containers, one with 2 liters and other with 7 liters, can you fill a barrel of 1234 liters?
What's the fastest and what's the slowest way to fill the barrel?
Should I use formula for permutation n!/(n−r)! or the other formula n!/r!-(n−r)! . In similar tasks I don't know which formula should be used.
Can anyone help? Thank you in advance!

REPLY [2 votes]: If the order of using the $2$ and $7$ liter barrels does not matter then you do not need combinations or permutations. You simply need to find the number of solutions to
$2a + 7b = 1234$
where $a$ and $b$ are non-negative integers (you can use $0$ barrels of either size, but you cannot use $-1$ barrels).
So the minimum value of $a$ is $0$. And the maximum value is $\frac{1234}{2}=617$. So we have
$0 \le a \le 617$
In addition, to make $7b=1234 - 2a$ a multiple of $7$, we must have $a \mod 7 = 1$.
So $a$ can be $1, 8, 15, 22, \dots, 610, 617$. The corresponding values of $b$ are $176, 174, 172, 170, \dots, 2, 0$.
From this you should be able to work out how many possible values there are for $a$ and $b$.


text:
\begin{document}

\begin{center}
\vspace{24pt}
{ \large \bf Scale-dependent Hausdorff dimensions in 2d gravity}

\vspace{30pt}

{\sl J. Ambj\o rn}$\,^{a,c}$,
{\sl T. Budd}$\,^{a}$,
and {\sl Y. Watabiki}$\,^{b}$

\vspace{48pt}
{\footnotesize

$^a$~The Niels Bohr Institute, Copenhagen University\\
Blegdamsvej 17, DK-2100 Copenhagen \O , Denmark.\\
{ email: ambjorn@nbi.dk, budd@nbi.dk}\\

\vspace{10pt}

$^b$~Tokyo Institute of Technology,\\
Dept. of Physics, High Energy Theory Group,\\
2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan\\
{email: watabiki@th.phys.titech.ac.jp}

\vspace{10pt}

$^c$~Institute for Mathematics, Astrophysics and Particle Physics (IMAPP)\\
Radbaud University Nijmegen,\\
Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands

}
\vspace{96pt}
\end{center}

\begin{center}
{\bf Abstract}
\end{center}

By appropriate scaling of coupling constants a one-parameter family of ensembles of two-dimensional geometries is obtained, which interpolates between the ensembles of (generalized) causal dynamical triangulations and ordinary dynamical triangulations.
We study the fractal properties of the associated continuum geometries and identify both global and local Hausdorff dimensions.

\vspace{12pt}
\noindent

\vspace{24pt}
\noindent
PACS: 04.60.Ds, 04.60.Kz, 04.06.Nc, 04.62.+v.\\
Keywords: quantum gravity, lower dimensional models, lattice models.

\newpage

\section{Introduction}\label{intro}

In dynamical triangulations (DT) the path integral of two-dimensional Euclidean quantum gravity
is discretized by summing over equilateral triangulations.
If the topology of the two-dimensional manifold is kept fixed,
the Einstein curvature term in the action is trivial
(being a topological invariant) and can be safely ignored.
In that case the DT path integral takes the form
\beq\label{1.1}
Z = \sum_{T \in \cT} \frac{1}{C_T} \;\e^{-\m N_T},
\eeq
where the sum is over all combinatorial triangulations of the desired topology, $C_T$ is the order of its automorphism group, $N_T$ its number of triangles, and $\mu$ is a coupling constant. The matrix
model representation of eq.\  \rf{1.1} is
\beq\label{1.2}
\cZ = \int d \phi \; e^{-N \tr \left(\oh \phi^2 -\frac{\kpdt}{3} \phi^3\right)},
~~~Z= \log \cZ,~~~~\kpdt = e^{-\m},
\eeq
where the integration is over the Hermitian $N\times N$ matrices $\phi$.
The partition function $Z$ allows for an expansion in $1/N^2$ and the Feynman diagrams contributing to the coefficient of $N^{\chi}$ are precisely the cubic graphs dual to triangulations of Euler characteristic $\chi$ appearing in \rf{1.1}.
In this paper we will only deal with the leading term
in the $1/N$ expansion, i.e.\
cubic graphs with spherical topology (or, in cases where
a boundary is present, the topology of the disk).

The lattice action has a critical point at $\kpdt\to(\kpdt)_c$ and the corresponding continuum limit can be identified with quantum Liouville
field theory with central charge $c_{ \rm liouville}=26$.
Universality of the scaling
limit ensures that one obtains the same continuum limit
for any potential
\beq\label{1.3}
V(\phi) = \frac{1}{g}\Big(\oh \phi^2 -\sum_{n} \kp_n \phi^n\Big),
\eeq
instead of the cubic potential used in \rf{1.2}, provided\footnote{Some
 mild constraint has to be imposed if an infinite number of $\kp_n \neq 0$.}
the $\kp_n \geq 0$ and at least one $\kp_n >0$ for $n \geq 3$. Thus
one can replace the set of triangulations $\cT$ in \rf{1.1} (or rather the dual cubic graphs) with
a much larger class of graphs if desired \cite{ajm}.
Independent of the precise class of graphs and as long as one keeps the couplings in the potential fixed as $N\to\infty$, the geometry of a randomly sampled very large graph has a number of universal properties, one of these
being that its fractal dimension is $d_h=4$
rather than the naively expected $d_h=2$ \cite{kawai,watabiki,aw,ajw}.
In the following we will refer to this universal continuum limit, which in the mathematical literature is known as the Brownian map \cite{brownianmap}, as the DT continuum limit.

However, it is possible to define a different scaling limit
for which $d_h=2$ by scaling the coupling $g$ in
\rf{1.3} non-trivially as function of $N$. Scaling
$g \to 0$ as $g = G a^3$ while keeping the continuum volume $\propto N a^2$ fixed, where $a$ may be interpreted as the length of a link in the graph,
leads to a scaling limit known as generalized causal dynamical triangulations (GCDT)
\cite{gcdt}.
It generalizes the original model of CDT in the continuum \cite{al}, which arises as the $G\to 0$ limit of GCDT and is different from Liouville quantum gravity.\footnote{Instead, continuum CDT was shown to correspond to two-dimensional Ho\v{r}ava-Lifshitz gravity \cite{aggsw}.}

In \cite{ab} the GCDT was shown to arise explicitly as a scaling limit of random quadrangulations with a fixed number of local maxima of the distance function to a distinguished vertex.
These quadrangulations were shown to be in bijection with general planar graphs with a fixed number of faces has been given, of which the continuum limit is therefore also described by GCDT.

The purpose of the present article is to investigate scalings that interpolate between the
two ``extremes'' mentioned above, DT and GCDT. For that purpose we will restrict our
attention to a simple potential \rf{1.3} of the form
\beq\label{1.4}
V_0(\phi) =
\frac{1}{g} \Big( -\kp \phi + \oh \phi^2 -\frac{\kp}{3} \phi^3\Big).
\eeq

\section{The scaling limit}

The disk amplitude for a general potential \rf{1.3} has the form:
\beq\label{2.1}
w(z) = \frac{1}{2} \Big(V'(z)- A(z)\sqrt{(z-c)(z-d)}\Big),
\eeq
where $V(z)$ is the potential \rf{1.3} with the matrix $\phi$ replaced
by the complex number $z$, and $V'(z)$ denotes the derivative with
respect to $z$. The polynomial $A(z)$ and the numbers $b,c,d$ (with $c\geq d$) are uniquely determined by the requirement that
$w(z) \to 1/z$ for $|z| \to \infty$. For the potential \rf{1.4} we write
\beq\label{2.2}
w(z) = \frac{1}{2g} \Big(-\kp +z-\kp z^2 + \kp (z-b) \sqrt{(z-c)(z-d)}\Big),
\eeq
where $z = e^{\lam_{B}}$, $\lam_B$ having  the interpretation as
a boundary cosmological constant associated with the disk-boundary
for positive $z$.

For a fixed $g$ the critical point $\kp_c$ is determined by the
condition that $b(\kp_c) = c(\kp_c)$, where $b(\kp),c(\kp)$ and $d(\kp)$
are determined by the required asymptotics of $w(z)$, as mentioned.
The solution can be written as follows (denoting $b(\kp_c)$ by $b_c$ etc.):
\beq\label{2.3}
b_c = c_c,~~~(b_c-d_c)^3 = \frac{32g}{\kp_c},~~~~~
\kp_c b_c = \oh+\oh (4\kp_c^2\,g)^{1/3}
\eeq
\beq\label{2.4}
\left(1-4\kp_c^2\right)^{3/2}=3^{3/2}\,4\kp_c^2\, g.
\eeq
From these equations one observes that if we scale
$g$ to 0  as $g= G\, a^3$ one obtains to lowest order
\beq\label{2.5}
\kp^2_c = \frac{1}{4} -\frac{3}{4} G^{2/3} a^2,~~~
b_c = \oh+ \frac{3}{4} G^{1/3}a,~~~b_c-d_c= 4 G^{1/3} a,
\eeq
as discussed in \cite{gcdt}.

We will now show how to understand this scaling limit in
a simple way which also allows us to define more
general scaling limits.

Consider the partition function \rf{1.2} with the potential
$V_0(\phi)$ defined by eq.\ \rf{1.4}. Expanding the exponential
in powers of $\kp$ and performing the Gaussian integrals
can be viewed as generating a certain set of graphs. We can view
these graphs as $\phi^3$ graphs decorated with tadpoles coming
from the linear $\phi$ term, see Fig. \ref{tadpoles}.
\begin{figure}

\centerline{\scalebox{0.35}{\rotatebox{0}{\includegraphics{tadpoles.pdf}}}}
\caption{Left: A typical graph generated by the action \rf{1.4}. Right:
Summing over the trees one obtains a cubic ``skeleton'' graph.}
\label{tadpoles}
\end{figure}
The shift in integration variables by
\beq\label{2.6}
\phi = \vph+ \al(\kp),~~~~~~\al(\kp) = \frac{1-\sqrt{1-4\kp^2}}{2\kp}
\eeq
eliminates the tadpole term:
\beq\label{2.7}
V_1(\vph) =V_0(\vph +\al(\kp)) = \frac{1}{g}
\left( \frac{\sqrt{1-4\kp^2}}{2}\,\vph^2-
\frac{\kp}{3} \,\vph^3 \right) +{\rm const.}
\eeq
The constant will play no role when we calculate expectation
values of observables. In terms of
graphs it means that we are introducing a ``dressed''
propagator  by first summing over all tadpole terms.
This is illustrated in Fig.\ \ref{tadpoles} and
in more detail in Fig.\ \ref{propagator}. We call the graph left
after summing over the tadpole terms for the skeleton
graph.

The constant $\al(\kp)$
is precisely the summation over all connected planar, rooted tree diagrams,
where each line has weight $g$ and each vertex weight $\kp/g$, as dictated
by the action $V_0(\phi)$. This summation is shown in the upper part
of Fig.\ \ref{propagator}. Next, the lines remaining
in the skeleton graph have  the weight $g\bt(\kp)$ as
illustrated in the lower part of Fig.\ \ref{propagator}, where
\beq\label{2.7a}
\bt(\kp) = 1 +  [2\kp \al(\kp)] +  [2\kp \al(\kp)]^2 + \cdots
= \frac{1}{\sqrt{1-4\kp^2}}.
\eeq
This explains the form of $V_1(\vph)$ from the point of view
of graph re-summation. Both for a graph $G$ generated from
$V_0(\phi)$ or for its corresponding skeleton graph  generated
from $V_1(\vph)$, the power of $g$ associated with the graph is
\beq\label{2.7b}
g^{F(G)-2},~~~F(G)=\mbox{number  of faces of $G$}.
\eeq
Thus it is clear that when we take $g \to 0$ we will suppress
the number of faces of the graph.

Finally we can perform a rescaling
\beq\label{2.8}
\vph = \sqrt{\frac{g}{\sqrt{1-4\kp^2}}} \; \Phi,
\eeq
such that
\beq\label{2.9}
V_1(\vph) = V_2(\Phi) = \oh\, \Phi^2 -\frac{\kpdt}{3}\,\Phi^3,~~~~~~
\kpdt= \frac{\sqrt{g}\, \kp}{(1-4\kp^2)^{3/4}}.
\eeq
In a graph $G$ this change of variables corresponds to absorbing
the weight $g/\sqrt{1-4\kp^2}$
given to each link by $V_1(\vph)$ into  the two vertices associated with
the link. Thus the weight of each link is 1, but the coupling
constant associated with a vertex is changed from $\kp/g$ to $\kpdt$.
Again, this rescaling will not affect expectation values
of observables. The potential $V_2(\Phi)$ is the standard
potential used to represent dynamical triangulations
using matrix models, and the critical coupling is known
to be $(\kpdt)_c = 3^{1/4}/6$. Using eq.\ \rf{2.4} we can write:
\beq\label{2.10}
\frac{\kpdt}{(\kpdt)_c} = \frac{\kp}{\kp_c}\,
\left( \frac{1-4\kp_c^2}{1-4\kp^2}  \right)^{3/4}.
\eeq
\begin{figure}

\centerline{\scalebox{0.5}{\rotatebox{0}{\includegraphics{propagator.pdf}}}}
\caption{Top figure: the graphic equation defining the summation
over all rooted trees. Bottom figure: The graphic equation for
the dressed propagator which appears in Fig.\ \ref{tadpoles} after summing
over all tree-outgrows.}
\label{propagator}
\end{figure}

This formula captures the  different ways one can take the scaling
limit for the model given by  $V_0(\phi)$.
The  tree sub-graphs shown in the left part of
Fig.\ \ref{tadpoles} and in the top part of Fig.\ \ref{propagator}
have the partition function $\al(\kp)$ given in \rf{2.6}, and become critical
for $\kp^2 \to 1/4$. The average number of vertices in a tree is
\beq\label{2.11}
\la n \ra_\kp = \frac{\kp}{\al} \,\frac{d \al}{d \kp}
\sim \frac{1}{\sqrt{1-4\kp^2}}.
\eeq
Similarly, the average number of vertices $\la n_{prop}\ra_\kp$ associated
with trees  attached to the dressed ``propagator''
shown in Fig.\ \ref{propagator} diverges when  $\kp^2 \to 1/4$.
The partition function for the number of such vertices is $\bt(\kp)$
defined in \rf{2.7a}, and

\beq\label{2.12}
\la n_{prop} \ra_{\kp} = \frac{\kp}{\bt}\, \frac{d\bt}{d \kp}
\sim \frac{1}{1-4\kp^2}.
\eeq
Thus the total average number of trees attached to a dressed propagator
is proportional to $1/\sqrt{1-4\kp^2}$.
Since $\kp \leq  \kp_c$ we conclude from eq.\ \rf{2.4} that
if we keep $g >0$ fixed when taking the scaling limit
$\kp \to \kp_c$ the trees will not be critical since
$\kp_c^2 < 1/4$. Thus the trees can  basically be ignored
in the scaling limit $\kp\to \kp_c$.  Eq.\ \rf{2.10} captures this: for
$g>0$ it tells us that
\beq\label{2.13}
\frac{\kpdt}{(\kpdt)_c} = \frac{\kp}{\kp_c}\, \Big( 1+ O( (\kp_c-\kp)^2)\Big).
\eeq
The critical behavior is thus the standard one of DT and the graphs
responsible for this are the standard $\phi^3$ graphs. In this scaling
limit we write
\beq\label{2.14}
\frac{\kp}{\kp_c} = 1-a^2 \Lam + o(a^2),
\eeq
where $\Lam$ may be interpreted as the cosmological
constant.

Clearly, to obtain  a different scaling limit we have to scale
$g$ to zero when $a \to 0$. The GCDT limit was obtained
by the scaling $g = G \,a^3$, and using \rf{2.5} we obtain from \rf{2.10}
\beq\label{2.15}
\frac{\kpdt}{(\kpdt)_c} = \frac{\kp}{\kp_c}\,
\left(\frac{ 3G^{2/3}}{ 3G^{2/3} +2\Lam}\right)^{3/4}.
\eeq
This shows that $\kpdt$ does not become critical as $\kp\to\kp_c$. Thus there is
only a finite average number of vertices and links and faces
in the skeleton graph. The
critical behavior for $\kp \to \kp_c$ is entirely determined
by the trees.

In order to obtain a new limit, let us consider the scaling
\beq\label{2.16}
g = G_\al a^\al,~~~~0<\al < 3,
\eeq
while maintaining \rf{2.14}, which states that
we view the total number of vertices as proportional with
the continuum area of the graph. With this scaling
of $g$ we obtain from \rf{2.5} and \rf{2.10}
\beq\label{2.17}
\frac{\kpdt}{(\kpdt)_c} =
\left( 1 -\frac{\Lam}{2 G_\al^{2/3}}\, a^{2-2\al/3} +
o( a^{2-2\al/3})\right).
\eeq
By a scaling like \rf{2.16} we thus obtain that both the trees and
the skeleton graphs are critical. The average number of graph vertices
per link in the skeleton graph is
\beq\label{2.18}
\la n_{prop} \ra_\kp \sim \frac{1}{1-4\kp^2} \sim \frac{1}{a^{2\al/3}},
\eeq
while the average number of skeleton vertices\footnote{For a planar
$\phi^3$ graph we have $3V=2L$ and $3F-L=6$, where $V,L$ and $F$ denotes
the number of vertices, links and faces in the graph.}  will be
\beq\label{2.19}
\la n_{skel} \ra_{\kp} \sim \frac{1}{a^{2-2\al/3}},
\eeq
implying that the  total number vertices in a typical graph
scales as $a^{-2}$, in accordance with \rf{2.14}.
Notice also that the average length of a skeleton link scales as $1/\sqrt{1-4\kappa^2}\sim a^{-\alpha/3}$.

In order to study the fractal properties of this ensemble of
graphs one may calculated the so-called two-point function \cite{aw,ajw},
i.e.\ the partition function with two distinguished vertices
separated by a given link distance $r$. It can be calculated using
the methods in \cite{ab} or \cite{aw,ajw}. For small $g$, $\kp$ close to $\kp_c$
and $r\gg g^{-1/3}$ one finds (up to numerical constants)
\beq\label{2.19a}
Z_\mu(r,g) \sim \Big(g^{2/3} \mu\Big)^{3/4} \;
\frac{\cosh \Big((g^{2/3} \mu)^{1/4} \, r\Big)}{
\sinh^3 \Big((g^{2/3} \mu)^{1/4} \, r\Big)},~~~~\mu = \frac{\kp_c-\kp}{\kp_c}.
\eeq
Let us now take the scaling limit prescribed by eqs.\
\rf{2.14}  and \rf{2.16}. Insisting on keeping $G_\al$ and $\Lam$ fixed
and eliminating the scaling parameter $a$ in favor of $\mu$ leads to
\beq\label{2.19b}
Z_\mu(r) \sim (K \mu^{1/d_H})^3\,
\frac{\cosh (K \mu^{1/d_H} r)}{ \sinh^3 (K \mu^{1/d_H} r)},~~~
K  = \Big(\frac{G}{\Lam^{\al/2}}\Big)^{1/6},
\eeq
which holds for $r \gg \mu^{-\alpha/6}$ and where\footnote{A similar scaling was anticipated in \cite{threepoint}, Section 6.}
\beq\label{2.20}
d_H = \frac{4}{1+\al/3}.
\eeq
Since $(1-\mu)$ is a generating variable for the number $N$ of vertices in the graph, we find that the (canonical) two-point function $Z_N(r)$ for fixed $N$ is of the form ,
\beq
Z_N(r) \sim F(r/N^{1/d_H})\quad \text{for }r \gg N^{\alpha/6}
\eeq
where $F(R)$ is some function that goes to zero fast for $R\gg 1$.
In particular, this implies that the average distance $\langle r\rangle$ between arbitrary vertices is of the order $N^{1/d_H}$.
For this reason one may call the exponent $d_H$ the ``global'' Hausdorff dimension \cite{aw,ajw}.

\begin{figure}
\centerline{\scalebox{1.4}{\rotatebox{0}{\includegraphics{dimension.pdf}}}}
\caption{The scale-dependent Hausdorff dimension for $0<\alpha<3$.}
\label{dimension}
\end{figure}

The dimension $d_H$ should be contrasted with the ``local'' Hausdorff dimension $d_h$, which is associated with the growth $V(R)\sim R^{d_h}$ for small $R$ of the expected volume $V(R)$ of a disk as function of its radius $R$ within a fixed continuous geometry.
In order to associate a well-defined Hausdorff dimension $d_h$ to our ensemble of graphs, one first has to specify how to scale the distance $r$ in the continuum limit.
If one defines the continuum distance $R$ as $R = r a^{2/d_H}$ eq. \rf{2.19b} reduces to
\beq
Z_\mu(r) \sim \frac{\cosh(\Lambda^{1/4}G_{\alpha}^{1/6} R)}{\sinh^3(\Lambda^{1/4}G_{\alpha}^{1/6} R)},
\eeq
which, up to the factor $G_{\alpha}^{1/6}$ is precisely the DT two-point function and therefore $d_h=4$.
If, on the other hand, we scale $R = r a^{\alpha/3}$, the skeleton edges will maintain a finite length in the continuum, meaning that the local Hausdorff dimension is determined by that of the trees, i.e. $d_h=2$.

These observations can be summarized by looking at the ``scale-dependent'' Hausdorff dimension
\beq
d_h(N;r) := \frac{\rmd \log\langle N(r)\rangle}{\rmd \log(r)}
\eeq
for a fixed large $N$, where $\langle N(r)\rangle$ is the expected number of vertices within graph distance $r$ from a randomly chosen vertex.
A qualitative plot of $d_h(N;r)$ as function of $\log(r)/\log(N)$ for some $0<\alpha<3$ is shown in Fig.\ \ref{dimension}.
The local Hausdorff dimensions, $d_h=2$ and $d_h=4$, appear as plateaus, while the global Hausdorff dimension corresponds to the scale at which $d_h(r)$ drops to zero.

\section{Discussion}

The partition function
\beq\label{3.1}
\cZ = \int d \phi \;
\e^{- \frac{N}{g} \tr [-\kp \phi + \oh \phi^2 -\frac{\kp}{3} \phi^3]}
\eeq
generates a statistical ensemble of graphs of the kind shown in
Fig.\ \ref{tadpoles}.
The coupling constant $g$ in the action \rf{1.4} can be viewed as
the   temperature $k T$ of this statistical system. Thus a scaling limit
where $g > 0$ corresponds to a finite temperature, and this finite temperature
limit can be identified with the standard scaling limit of 2d Euclidean
quantum gravity: i.e. the typical geometry of the ensemble is fractal
with Hausdorff dimension $d_h=4$.

The scaling limit $g \to 0$ has certain analogues with the
annealing, quenching and tempering of alloys and metals, in the sense that the
precise way we take this zero temperature limit decides
the smoothness of a typical geometry dominating at zero
temperature. The parameter controlling this is the
exponent $\al$ when writing $g= G_\al a^{\al}$, $0 \leq \al \leq 3$,
which tells us how ``fast'' we cool to zero temperature.
The so-called global Hausdorff dimension $d_H$ of a typical graph
in such an $\al$-ensemble is given by
\beq
d_H = \frac{4}{1+\alpha/3},
\eeq
while, depending on the chosen scaling of the geodesic distance, the local Hausdorff dimension
is either $d_h=4$ or $d_h=2$.

Here we have only considered the simplest situation, that
of spherical (or disk) topology of the graphs and the associated
geometries. It remains to be seen if there
exists a complete perturbative expansion in topology and
in number of boundaries for an arbitrary value of $\al$, as
is the case in the two limits $\al =0$ and $\al = 3$.

\subsection*{Acknowledgments}

The authors acknowledge support from the ERC-Advance grant 291092,
``Exploring the Quantum Universe'' (EQU). JA acknowledges support
of FNU, the Free Danish Research Council, from the grant
``quantum gravity and the role of black holes''.
In addition JA was supported
in part by Perimeter Institute of Theoretical Physics.
Research at Perimeter Institute is supported by the Government of Canada
through Industry Canada and by the Province of Ontario through the
Ministry of Economic Development \& Innovation.

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