TITLE: Given a primitive root $r$ mod $p$, why is all $r^m, m\in GCD(\phi(p),m)=1$ incongruent primitive root to r?
QUESTION [0 upvotes]: Given a primitive root $r$ mod $p$, why is all $r^m, m\in GCD(\phi(p),m)=1$ incongruent primitive root to r?
So I don't really understand how incongruent primitive roots are found using this strategy is Elementary Number Theory and It's Applications.
The example I'll use is find incongruent primitive roots of 13.
So they do $\phi(\phi(13))=4$, so that means 4 incongruent primitive roots, which makes sense, then they show 2 is a primitive root, which makes sense because a primitive root can be found if $ord_{13}2=\phi(13)=12$, which it is in this case $2^{12}\equiv1(mod13)$, but then the answer skips ahead and says any $2^u$ where u is relatively prime to $\phi(13)=12$ is incongruent primitive root to 2. Why is this? Does this hold in general for all primitive roots?

REPLY [2 votes]: If a primitive root exists,  then this strategy will work.
The relatively prime condition ensures you have a primitive root, since $\mid g^m\mid=\mid g\mid/\operatorname {gcd}(m,\mid g\mid)$.  This is a basic fact from the theory of cyclic groups.
Secondly,  any two powers of a generator less than the order of the group are not congruent,  quite trivially.   That is, if $m,n\lt\mid G\mid$, we have $g^m\ne g^n$. For, if $g^m=g^n$, then $g^{m-n}=e$, contradicting that $g$ is primitive.


TITLE: Using a Lyapunov function to determine stability of equilibria
QUESTION [3 upvotes]: Given
$$\left\{\begin{aligned} x' &= -x^3 + 7xy^2\\
y' &= -3x^2y+y^3\end{aligned}\right.$$
find $a, b > 0$ such that $L(x,y) = a x^2 + b y^2$ obeys $\frac{d}{dt}L \neq 0$ whenever $(x,y) \neq (0,0)$. State whether the origin is a stable or unstable equilibrium.

So we want to find $a,b$ where $L(x,y)$ is a Lyapunov function, and then determine the stability of the equilibria.
First differentiate: $\frac{dL}{dt}=2axx'+2byy'$
Now plug in system: $2ax(-x^3+7xy^2)+2by(-3x^2y+y^3)$
By checking the equilibria, we see that both $x'$ and $y'$ equal zero only when $(x,y)=(0,0)$.
Therefore we need to find when $2ax(-x^3+7xy^2)= -2by(-3x^2y+y^3)$
This is where I am confused, as I am not sure how to find $a$ or $b$ purely in terms of $x,y$ and what restrictions I may need to impose. Any help is appreciated. Thank you.

REPLY [2 votes]: Complete the square in
$$
\dot L=2(-ax^4+(7a-3b)x^2y^2+by^4)=\frac1{2a}\left(-(2ax^2-(7a-3b)y^2)^2+(4ab+(7a-3b)^2)y^4\right).
$$
This means that there will always be points in any neighborhood of $(0,0)$ with positive and negative signs of this type of Lyapunov function.
Checking with Wolfram alpha
streamplot[{-x^3+7xy^2,-3x^2y+y^3}, {x,-12,12}, {y,-9,9}]

confirms the saddle point behavior in the plot


TITLE: Solve$(10+6\sqrt3)^{\frac{1}{3}}-(-10+6\sqrt3)^{\frac{1}{3}}$
QUESTION [1 upvotes]: We need to solve the following equation $y=(10+6\sqrt3)^{\frac{1}{3}}-(-10+6\sqrt3)^{\frac{1}{3}}$ and it is equal to 2 while I am getting the value in excel I am not able to solve it manually eventhough the values are conjugate
I tried $y=a-b$
$a=(10+6\sqrt3)^{\frac{1}{3}}$ & $b=(-10+6\sqrt3)^{\frac{1}{3}}$
$y^3=(a-b)^3$
$y^3=a^3-b^3-3ab(a-b)$ after this step I am struck

REPLY [2 votes]: Substituting the values of $a$ and $b$, and the relation $y=a-b$, into $y^3=a^3-b^3-3ab(a-b)$ yields
$$y^3=10+6\sqrt3+10-6\sqrt3-3((6\sqrt3+10)(6\sqrt3-10))^{1/3}y$$
$$y^3=20-3(3\cdot36-100)^{1/3}y$$
$$y^3=20-6y$$
$$y^3+6y-20=0$$
$$(y-2)(y^2+2y+10)=0$$
Since the quadratic factor has no real roots, $y=2$.


TITLE: Volume of the solid whose base is a triangular region with squares as a cross-section
QUESTION [0 upvotes]: Find the volume of the solid whose base is a triangular region with vertices (0, 0), (2, 0) and (0, 2) if the cross-sections perpendicular to the Y -axis are squares.
My problem is that I can't figure out how the shape will be like... I need some images.
What I think is
Volume = $\int\limits_a^b Area\,dy$
=$\int\limits_a^b base^2\,dy$
= $\int\limits_0^2 y^2\,dy$
= $\frac{8}{3}$
?????
[2nd Edit]
Should be like this?
Volume = $\int\limits_a^b Area\,dy$
=$\int\limits_a^b base^2\,dy$
= $\int\limits_0^2 x^2\,dy$ ; Since y = -x +2 or x = 2-y
=  $\int\limits_0^2 (2-y)^2\,dy$
= $\int\limits_0^2 y^2-4y+4\,dy$
= $[\frac{y^3}{3} - 2y^2 +4y]$ from 0 to 2
= $\frac{8}{3}$

REPLY [1 votes]: If the cross-sections perpendicular to the $Y$-axis are squares, then we know that the cross-sectional area at a height $y$, will be given by the square distance from the Y-axis to your line $y=2-x$. This distance is just $x$. If we integrate over the $y$-direction we can now find the volume of your object.
So, volume = $\int_{0}^{2} x^2 dy$. Here we have the mismatch of a variable in $x$ and our integration being over $y$, we so we the equation of the line to substitute $x = (2-y)$, and we find that the volume is $\int_{0}^{2} (2-y)^2 dy$.
I think you can take it from here.


TITLE: How to write an equation that started with for each where
QUESTION [1 upvotes]: First of all, it was difficult to find the proper title for my question. Basically, I would like to write an equation for the following:
For each branch $b_i$ where i in 1 to n, the BEC value for that branch is
$$\sum_{l=1}^{w} r_l \ \forall \ r_l \in \left \{ 0, 1 \right \}$$
where w is the number of runs and $r_l$ is whether the branch $b_i$ has been executed in that run (i.e., = 1) or not (i.e., = 0).
To write that as an equation, I did the following:
$$\forall \ b_i \ where \ i \in \left \{ 1, \dots, n \right \} BEC_i = \sum_{l=1}^{w} r_l \ \forall \ r_l \in \left \{ 0, 1 \right \}$$
I'm not quite sure if this equation looks fine or not.
Can someone please help me in writing the proper equation?

REPLY [2 votes]: If you're writing for computer science, you can use $[1..n]$ for set notation and "such that" (s.t.) instead of where:
$\forall b_i$ s.t. $i \in [1..n] : $ ...
I would also rearrange the equation so that the quantifier is placed ahead of the sum:
$\forall b_i$ s.t. $i \in [1..n] : \forall r_l \in \{0, 1\} : BEC_i = \sum_{l=1}^w r_l
$
Some computer scientists use colon to specify the domain, and dot to separate the statements:
$\forall b_i: i \in [1..n] \cdot \forall r_l: r_l \in \{0, 1\} \cdot BEC_i = \sum_{l=1}^w r_l
$
Alternatively, the quantifiers can be merged.
$\forall b_i, r_l \cdot BEC_i = \sum_{l=1}^w r_l$, such that $i \in [1..n], r_l \in \{0,1\}$


TITLE: How to find horizontal shift after solving catenary
QUESTION [0 upvotes]: I'm trying to compute the chain/rope/string curve between two points and a given length. I followed the instructions as answered here but I have a last step to accomplish. The equations I'm solving assume the catenary is right on the y-axis.  How do I find where in the curve my initial two points were? I want to plug in my x-coordinates and get the corresponding y-coordinates.

REPLY [0 votes]: Horizontal shifts are accomplished by making a change of variable $u = x - \sigma_r$ where your former variable was $u$, the amount you want to shift right by is $\sigma_r$, and $x$ is the variable as you want it displayed in the future.
For example., the line $y = 2u - 1$ can be shifted 3 units to the right by doing $y = 2(x-3) - 1 = 2x - 7$.
To double check: $(1,1)$ was a point in the original, and $(4,1)$ is a point in the new.
The same method works to horizontal shift any curve, be it a line, a quadratic, a catenary, or whatever.
To shift vertically, it works similarly, just replace your former codomain variable with $y-\sigma_u$ where $\sigma_u$ is how much you shifted up.

tldr: its better not to think about moving the lines themselves as it is to move the axis via a change of variable.


TITLE: How many solutions exists for this equation?
QUESTION [2 upvotes]: $$x_1 + x_2 + x_3 + x_4 = 28$$
I tried to solve it with generating functions.
Is it correct to get to the form of
$${(1 + x + {x^2} + {x^3} + ....)^4}$$
and this equals to:
$${(1 - x)^{ - 4}}$$
and then solve it with binomial expansion ?
Is this correct ?
every question similar to "how many solutions are axsist to this equation?"
can be solved this method?
I know there're a lot of methods and manipulations for this kind of problem.
Can you confirm it's the right path for this kind of problems?
It's like distribute 28 things into 4 objects.
This is why I used generating functions.

REPLY [1 votes]: Yes, the binomial expansion
of $(1-x)^{-4}$ will work.
No, many other type of questions
like this can not be solved
by this method.
Here is a useful expansion:
$(1-x)^{-n}
=\sum_{k=0}^{\infty} \binom{-n}{k}(-x)^k
=\sum_{k=0}^{\infty} (-1)^k\binom{-n}{k}x^k
$.
And,
$\begin{array}\\
\binom{-n}{k}
&=\frac{(-n)(-n-1)(-n-2)...(-n-k+1)}{k!}\\
&=\frac{(-1)^k(n)(n+1)(n+2)...(n+k-1)}{k!}\\
&=(-1)^k\frac{(n-1)!(n)(n+1)(n+2)...(n+k-1)}{k!(n-1)!}\\
&=(-1)^k\binom{n+k-1}{k}\\
\end{array}
$
Therefore,
noting that the
two $(-1)^k$s cancel each other out,
$(1-x)^{-n}
=\sum_{k=0}^{\infty} \binom{n+k-1}{k}x^k
$.


text:
\section{Dijkgraaf-Witten Hamiltonian Model\label{sec:DW}}
\emph{In this section, we first review the definition of the Dijkgraaf-Witten theory and the construction of its Hamiltonian realisation. We then generalise the construction to include gapped boundaries.}
\subsection{Partition function}\label{sec:DWpartitionfunction}

The input for the ($d$+1)-dimensional \emph{Dijkgraaf-Witten} theory is given by a pair $(G,[\omega])$ where $G$ is a finite group and $[\omega]\in H^{d+1}(G,\rU(1))$ is a ($d$+1)-cohomology class.\footnote{Here $\rU(1)$ denotes the circle group as a $G$-module with action $\triangleright:G\times \rU(1)\rightarrow \rU(1)$ given by $g\triangleright u=u$ for all $g\in G$ and $u\in \rU(1)$.} Given a closed manifold, this theory can be conveniently expressed as a sigma model with target space the classifying space $BG$ of the group $G$. In order to extend the definition of the partition function to open manifolds, it is necessary to endow the manifold with a triangulation, in which case the partition function is obtained by summing over $G$-labellings of the 1-simplices  that satisfy compatibility constraints. Ultimately, we are interested in lattice Hamiltonian realisations of such theory, for which we need the expression of the partition function that the Dijkgraaf-Witten theory assigns to a special class of open manifolds referred to as \emph{pinched interval cobordisms}. We shall directly define the partition function for this special class of manifolds. Details regarding more basic aspects of this theory can be found in \cite{dijkgraaf1990topological, Bullivant:2019fmk}.

Let $\Xi$ be a compact, oriented $d$-manifold with a  possibly non-empty boundary. We define the \emph{pinched interval cobordism} $\Xi\times_{\rm p}\I$ over $\Xi$ as the quotient manifold
\begin{align}
	\Xi\times_{\rm p}\I\equiv\Xi\times \I \; /\sim \; ,
\end{align}
where $\I\equiv [0,1]$ denotes the unit interval, and the equivalence relation $\sim$ is such that $ (x,i)\sim (x,i')$, for all $(x,i),(x,i')\in \partial\Xi\times \I $. By definition, we have $\partial (\Xi\times_{\rm p}\I)=\overline{\Xi}\cup_{\partial\Xi} \Xi$ and $\overline{\Xi}\cap \Xi=\partial\Xi$, where $\overline{\Xi}$ is the manifold $\Xi$ with reversed orientation. In contrast, the boundary of the \emph{interval cobordism} $\Xi \times \I$ over $\Xi$ reads $\partial (\Xi\times \I)=\overline{\Xi}\cup \Xi\cup (\partial\Xi\times \I)$. To illustrate this distinction, we can consider the following simple examples:
\begin{equation*}
	[0,1]\times_{\rm p}[0,1]=
	\pinchedinterval{}
	\q , \q
	[0,1]\times [0,1]=
	\interval{} \; .
\end{equation*}
Naturally, if $\partial \Xi=\varnothing$, then we have the identification $\Xi\times_{\rm p}\I=\Xi\times \I$.

In order to define the Dijkgraaf-Witten partition function, we shall further require our pinched interval (spacetime) manifold be equipped with a choice of \emph{triangulation}, i.e. a $\Delta$-complex whose geometric realisation is homeomorphic to the manifold. We shall further assume that every triangulation has a chosen total ordering of its $0$-simplices (vertices), referred to as a \emph{branching structure}. A choice of branching structure for a triangulation naturally encodes the structure of a directed graph on the corresponding one-skeleton. By convention, we choose the 1-simplices (edges) to be directed from the lowest ordered vertex to the highest ordered vertex. Given a compact, oriented $d$-manifold $\Xi$, we notate a triangulation of the pinched interval cobordism $\Xi\times_{\rm p}\mathbb{I}$ by ${_{\triangle'}\Xi_{\triangle}}$, such that $\partial({_{\triangle'}\Xi_{\triangle}})=\overline{\Xi_{\triangle}}\cup_{\partial \Xi_{\triangle'}} \Xi_{\triangle'}$, where $\Xi_{\triangle}$ and $\Xi_{\triangle'}$ denote two possibly different triangulations of $\Xi$. Let us remark that by definition, we have $\partial (\Xi_{\triangle})=\partial( \Xi_{\triangle'})$.

Let $\Xi\times_{\rm p}\mathbb{I}$ be a ($d$+1)-dimensional pinched interval cobordism endowed with a triangulation ${_{\triangle'}\Xi_{\triangle}}$. We define a $G$-colouring of ${_{\triangle'}\Xi_{\triangle}}$ as an assignment of group elements $g_{v_i v_j} \in G$ to every oriented 1-simplex $(v_{i}v_{j})\subset {_{\triangle'}\Xi_{\triangle}}$, with $v_{i}<v_{j}$, such that for every 2-simplex $(v_{i}v_{j}v_{k})\subset {_{\triangle'}\Xi_{\triangle}}$, with $v_{i}<v_{j}<v_{k}$, the condition $g_{v_i v_j}g_{v_j v_k}=g_{v_i v_k}$ is satisfied. The set of $G$-colourings on ${_{\triangle'}\Xi_{\triangle}}$ is notated by ${\rm Col}({_{\triangle'}\Xi_{\triangle}},G)$. Given a $G$-colouring $g\in {\rm Col}({_{\triangle'}\Xi_{\triangle}},G)$ and an $n$-simplex $\triangle^{(n)}=(v_{0}v_{1}\ldots v_{n})\subset {_{\triangle'}\Xi_{\triangle}}$, we denote by $g[v_{0}v_{1}\ldots v_{n}]\equiv(g_{v_0v_1},\ldots,g_{v_{n-1}v_n})\in G^{n}$, the $n$ group elements specifying the restriction of $g$ to a $G$-colouring of $(v_{0}v_{1}\ldots v_{n})$. Using this notation, we further define the evaluation of a ($d$+1)-cocycle $\omega \in Z^{d+1}(G,\rU(1))$ on a $G$-colouring $g\in {\rm Col}({_{\triangle'}\Xi_{\triangle}},G)$ restricted to a ($d$+1)-simplex $(v_{0}\ldots v_{d+1})\subset {_{\triangle'}\Xi_{\triangle}}$ as
\begin{align*}
	\omega(g[v_{0}\ldots v_{d+1}])
	\equiv
	\omega(g_{v_0v_1},\ldots,g_{v_dv_{d+1}})\;.
\end{align*}
Equipped with the above, let us now define the partition function that the ($d$+1)-dimensional Dijkgraaf-Witten theory assigns to a given pinched interval cobordism. Letting $\Xi$ be a compact, oriented $d$-manifold and ${_{\triangle'}\Xi_{\triangle}}$ a triangulation of $\Xi\times_{\rm p}\mathbb{I}$, the partition function defines a linear operator
\begin{align*}
	\Z[{_{\triangle'}\Xi_{\triangle}}]
	:
	\Hilb{\Xi_{\triangle}}
	\rightarrow
	\Hilb{\Xi_{\triangle'}} \; ,
\end{align*}
where the Hilbert spaces $\Hilb{\Xi_\triangle}$ and $\Hilb{\Xi_{\triangle'}}$ are defined according to
\begin{align}
	\label{eq:microhilbpartitionfunction}
	\Hilb{\Xi_{*}}\equiv \bigotimes_{\triangle^{(1)}\subset \Xi_{*}}
	\mathbb{C}[G] \; .
\end{align}
In the equation above, the tensor product is over all 1-simplices $\triangle^{(1)}$ in the corresponding triangulation, and $\mathbb{C}[G]$ denotes the Hilbert space spanned by $\{\ket{g} \}_{\forall \, g\in G}$ with inner product $\langle g|h\rangle=\delta_{g,h}$, $\forall \, g,h\in G$. Explicitly, the linear operator $\Z[{_{\triangle'}\Xi_{\triangle}}]$ reads
\begin{align*}
	\Z[{{_{\triangle'}\Xi_{\triangle}}}]
	\equiv
	\frac{1}{|G|^{\#({_{\triangle'}\Xi_{\triangle}})}}
	\!\!
	\sum_{g\in {\rm Col}({{_{\triangle'}\Xi_{\triangle}}},G)}
	\prod_{\triangle^{(d+1)}\subset {{_{\triangle'}\Xi_{\triangle}}}}
	\!\!\!\!
	\omega(g[\triangle^{(d+1)}])^{\epsilon(\triangle^{(d+1)})}
	\!\!\!
	\bigotimes_{\triangle^{(1)}\subset \Xi_{\triangle'}}
	\!\!\!\!
	\ket{g[\triangle^{(1)}]}
	\!\!
	\bigotimes_{\triangle^{(1)}\subset \Xi_{\triangle}}
	\!\!\!\!
	\bra{g[\triangle^{(1)}]} \; ,
\end{align*}
where $\#({_{\triangle'}\Xi_{\triangle}}) := |{_{\triangle'}\Xi_{\triangle}}^{(0)}|-\frac{1}{2}|\partial{_{\triangle'}\Xi_{\triangle}}^{(0)}|-\frac{1}{2}|\partial\Xi_{\triangle}^{(0)}|$ and $\epsilon(\triangle^{(d+1)})\in\pm 1$ denotes the orientation of the ($d$+1)-simplex $\triangle^{(d+1)}\subset {_{\triangle'}\Xi_{\triangle}}$.

Before concluding this section, let us describe some of the salient features of the partition function above. Firstly, given a pinched interval cobordism $\Xi\times_{\rm p}\mathbb{I}$ and two choices of triangulation ${_{\triangle'}\Xi_{\triangle}}$ and ${_{\triangle'}\tilde{\Xi}_{\triangle}}$ such that $\partial({_{\triangle'}\Xi_{\triangle}}) = \partial({_{\triangle'}\tilde{\Xi}_{\triangle}})$, we find the operators $\Z[{_{\triangle'}\Xi_{\triangle}}]=\Z[{{_{\triangle'}\tilde{\Xi}_{\triangle}}}]$ to be equal. This property follows directly from the ($d$+1)-cocycle condition satisfied by $\omega$, i.e. $d^{(d+1)}\omega=1$. This implies that the operator $\Z$ is \emph{boundary relative triangulation independent}, i.e. it remains invariant under retriangulation of the interior ${\rm int}({_{\triangle'}\Xi_{\triangle}}) := {_{\triangle'}\Xi_{\triangle}} \backslash \partial({_{\triangle'}\Xi_{\triangle}})$ of ${_{\triangle'}\Xi_{\triangle}}$ but does depend on a choice of boundary triangulation. Using this boundary relative triangulaton independence, we find the crucial relation
\begin{align*}
	\Z[{{_{\triangle''}\Xi_{\triangle'}}}] \,
	\Z[{{_{\triangle'}\Xi_{\triangle}}}]
	=
	\Z[{{_{\triangle''}\Xi_{\triangle}}}] \;.
\end{align*}
Secondly, given a $d$-manifold $\Sigma$ equipped with a triangulation $\Sigma_\triangle$ and $\Xi_{\triangle}$ a subcomplex of ${\rm int}(\Sigma_{\triangle})$, there is a natural action of $\Z[{{_{\triangle'}\Xi_{\triangle}}}]$ on $\Hilb{\Sigma_{\triangle}}$ such that
\begin{align*}
	\Z[{{_{\triangle'}\Xi_{\triangle}}}]
	:
	\mathcal{H}^G_\omega[\Sigma_{\triangle}]
	\rightarrow
	\mathcal{H}^G_\omega[\Sigma_{\triangle'}]
\end{align*}
where $\Sigma_{\triangle'}$ is a triangulation of $\Sigma$ induced from $\Sigma_{\triangle}$ by replacing the subcomplex $\Xi_{\triangle}\subset {\rm int}(\Sigma_{\triangle})$ with $\Xi_{\triangle'}$, while keeping the remaining triangulation the same. On the subspace
\begin{align}
	\label{eq:physSpace}
	\gsopen{\Sigma_{\triangle}}:= {\rm Im} \, \Zo{{_{\triangle'}\Xi_{\triangle}}}
	\subset
	\Hilb{\Xi_{\triangle}} \; ,
\end{align}
the operator $\Z[{{_{\triangle'}\Xi_{\triangle}}}]$ further defines a unitary isomorphism
\begin{align}
	\label{eq:isoTriangulations}
	\Z[{{_{\triangle'}\Xi_{\triangle}}}]
	:
	\gsopen{\Sigma_{\triangle}}
	\xrightarrow{\sim}
	\gsopen{\Sigma_{\triangle'}} \; .
\end{align}
This follows directly from the boundary relative triangulation independence of $\Z$ as well as the Hermicity condition
\begin{align}\label{eq:Hermicity}
	\Zo{{{_{\triangle'}\Xi_{\triangle}}}}^{\dagger}
	=
	\Zo{{{_{\triangle}\Xi_{\triangle'}}}}
	\;.
\end{align}

\subsection{Hamiltonian realisation of Dijkgraaf-Witten theory}\label{sec:DWHam}

Let us now construct an exactly solvable model that is the lattice Hamiltonian realisation of Dijkgraaf-Witten theory in $d$ spatial dimensions \cite{Hu:2012wx,Wan:2014woa,Bullivant:2019fmk}. The input of the model is a pair $(G,\omega)$ where $G$ is a finite group and $\omega$ a normalised representative of a cohomology class in $H^{d+1}(G,\rU(1))$. Given an oriented (possibly open) $d$-manifold $\Sigma$ representing the spatial manifold of the theory, and a choice of triangulation $\Sigma_{\triangle}$, the microscopic Hilbert space of the model is given by
\begin{align*}
	\Hilb{\Sigma_{\triangle}}\equiv \bigotimes_{\triangle^{(1)}\subset \Sigma_{\triangle}}\mathbb{C}[G] \; ,
\end{align*}
as in \eqref{eq:microhilbpartitionfunction}.
A natural choice of basis for $\Hilb{\Sigma_{\triangle}}$ is given by an assignment of $g_{v_{i}v_{j}}\in G$ for each oriented edge $(v_{i}v_{j})\subset\Sigma_{\triangle}$ defined by the vertices $v_{i}<v_{j}$. Henceforth, we shall refer to such states as \emph{graph-states}.

The bulk Hamiltonian is obtained as a sum of mutually commuting projectors that come in two families. Firstly, to every $2$-simplex $(v_0v_1v_2) \subset {\rm int}(\Sigma_{\triangle})$ of the interior of $\Sigma_\triangle$, we assign an operator $\mathbb{B}_{(v_0v_1v_2)}$ that is defined via the following action on a graph-state $\ket{g}\in\Hilb{\Sigma_{\triangle}}$:
\begin{equation*}
	\mathbb B_{(v_0v_1v_2)} : | g \rangle \mapsto \delta_{g_{v_0v_1} g_{v_1v_2}\, , \,g_{v_0v_2}} | g \rangle \; .
\end{equation*}
This definition can be extended linearly to an operator on any state $\ket{\uppsi}\in\Hilb{\Sigma_{\triangle}}$. Secondly, to every 0-simplex $(v_0) \subset {\rm int}(\Sigma_\triangle)$, we assign an operator $\mathbb A_{(v_0)}$ which acts on a local neighbourhood of $(v_0)$ defined as the subcomplex $\Xi_{v_{0}}:={\rm cl} \circ {\rm st}(v_0)\subset \Sigma_{\triangle}$. Here ${\rm st}(-)$ and ${\rm cl}(-)$ are the \emph{star} and the \emph{closure} operations, respectively, so that $\Xi_{v_0}$ corresponds to the smallest subcomplex of $\Sigma_\triangle$ that include all the simplices of which $(v_0)$ is a subsimplex.  The definition of $\mathbb A_{(v_0)}$ requires the triangulated pinched interval cobordism $\cob{\Xi_{v_{0}}\!}{\Xi}{\, \Xi_{v_{0}}}$ defined as
\begin{align*}
	\cob{\Xi_{v_{0}}\!}{\Xi}{\, \Xi_{v_{0}}}:=
	(\snum{v_0'}) \sqcup_{\rm j} {\rm cl} \circ {\rm st}(v_0)\; ,
\end{align*}
where $ {-}\sqcup_{\rm j} {-}$ denotes the \emph{join} operation.  Given two simplices $ \triangle^{(n)} \equiv (v_0 v_1 \ldots v_n) $ and $\triangle^{(n')} \equiv (v_{n+1}v_{n+2} \ldots v_{n+n'+1})$, the join operation creates the new simplex $\triangle^{(n)} \sqcup_{\rm j} \triangle^{(n')} \equiv (v_0v_1 \ldots v_{n+n'+1})$. In the definition above, $(v_0')$ refers to an  auxiliary vertex such that $v_{0}<v_{0}'<v_1$, and which follows the ordering of $(v_{0})$ with respect to the other vertices in $\Sigma_{\triangle}$. For the sake of concreteness, we illustrate these various definitions with the following two-dimensional example:
\begin{equation*}
	\Sigma_{ \triangle} = \clst{1} \q\;\; \text{and} \q\;\;
	\snum{(0')} \sqcup_{\rm j} {\rm cl} \circ {\rm st} (0) =
	\snum{(0')} \sqcup_{\rm j} \; \clst{2} \; =  	\clst{3} \; .
\end{equation*}
Finally, given a state $\ket{\uppsi}\in\Hilb{\Sigma_{\triangle}}$, the action of the operator $\mathbb{A}_{(v_{0})}$ is defined via
\begin{align}
	\label{eq:opA}
	\mathbb{A}_{(v_{0})}: \ket{\uppsi}
	\mapsto
	\Zo{(\snum{v_0'}) \sqcup_{\rm j} {\rm cl} \circ {\rm st}(v_0)} \, \ket{\uppsi} \; .
\end{align}
For instance, in (3+1)d the action of the operator $\mathbb{A}_{(4)}$ on a vertex $\snum{(4)}$ shared by four 3-simplices explicitly reads
\begin{align*}
	\mathbb{A}_{(4)}  \Bigg| \! \HamONE{0.9}{1} \!\! \Bigg\rangle
	&= \mathcal{Z}_\pi^G\Bigg[ \! \HamTWO{0.9}{1} \! \Bigg]  \, \Bigg| \! \HamONE{0.9}{1} \!\! \Bigg\rangle \\[-1.6em] \nn
	& = \frac{1}{|G|}\sum_{k \in G}
	\frac{\pi(g_{01}g_{12},g_{23},g_{34},g_{44'}) \, \pi(g_{01},g_{12},g_{23}g_{34},g_{44'})}{\pi(g_{12},g_{23},g_{34},g_{44'}) \, \pi(g_{01},g_{12}g_{23},g_{34},g_{44'})}
	\Bigg| \! \HamONE{0.9}{2} \!\! \Bigg\rangle  ,
\end{align*}
where $\pi \in Z^4(G,\rU(1))$.
The lattice Hamiltonian is finally obtained as
\begin{align}
	\label{eq:Ham}
	\Hambulk{\Sigma_{\triangle}}
	=-\sum_{\triangle^{(2)}\subset {\rm int}(\Sigma_{\triangle})} \!\! \mathbb{B}_{\triangle^{(2)}}
	\, -\sum_{\triangle^{(0)}\subset{\rm int}(\Sigma_{\triangle})} \!\!\mathbb{A}_{\triangle^{(0)}}\;,
\end{align}
where the sums run over all the 2-simplices and 0-simplices in the interior of $\Sigma_{\triangle}$, respectively. It follows from the definitions and the boundary relative triangulation independence that the operators $\{\mathbb{A}_{\triangle^{(0)}},\mathbb{B}_{\triangle^{(2)}}\}_{\forall \,  \triangle^{(0)}\! ,\triangle^{(2)}\subset {\rm int}(\Sigma_{\triangle})}$
satisfy the algebra
\begin{gather*}
	\mathbb{A}_{(v_{i})}\mathbb{A}_{(v_{i})}=\mathbb{A}_{(v_{i})}
	\; , \q
	\mathbb{A}_{(v_{i})}\mathbb{A}_{(v_{j})}=\mathbb{A}_{(v_{j})}\mathbb{A}_{(v_{i})} \; ,
	\\
	\mathbb{B}_{(v_{j}v_{k}v_{l})}\mathbb{B}_{(v_{j}v_{k}v_{j})}=\mathbb{B}_{(v_{j}v_{k}v_{l})}
	\; , \q
	\mathbb{B}_{(v_{j}v_{k}v_{l})}\mathbb{B}_{(v_{j}'v_{k}'v_{l}')} = \mathbb{B}_{(v_{j'}v_{k'}v_{l'})}\mathbb{B}_{(v_{j}v_{k}v_{l})} \; ,
	\\
	\mathbb{A}_{(v_{i})}\mathbb{B}_{(v_{j}v_{k}v_{l})}
	=
	\mathbb{B}_{(v_{j}v_{k}v_{l})}\mathbb{A}_{(v_{i})} \; ,
\end{gather*}
for all $(v_{i}), (v_{i'}),(v_{j}v_{k}v_{l}),(v_{j}'v_{k}'v_{l}')\subset \Sigma_{\triangle}$. All the operators are mutually commuting projectors and the Hamiltonian is exactly solvable. It follows that the \emph{ground state projector} $\mathbb{P}^{\rm bulk}_{\Sigma_{\triangle}}$ simply reads
\begin{align}
	\label{eq:projGS}
	\mathbb{P}^{\rm bulk}_{\Sigma_{\triangle}}:= \prod_{\triangle^{(0)}\subset {\rm int}(\Sigma_{\triangle})}\mathbb A_{\triangle^{(0)}} \prod_{\triangle^{(2)}\subset {\rm int}(\Sigma_{\triangle})}\mathbb B_{\triangle^{(2)}} \;.
\end{align}
Notice that the ordering in the product is superfluous by the commutativity of the operators. Furthermore it follows from inspection that
\begin{align}
\mathbb{P}^{\rm bulk}_{\Sigma_{\triangle}}=\mc{Z}^{G}_{\omega}[{_{\triangle}}{\Sigma}_{\triangle}]\,,
\end{align}
such that the ground state subspace of $\Hambulk{\Sigma_{\triangle}}$ is given by
\begin{align}
	\label{eq:idSub}
	{\rm Im} \, \mathbb{P}^{\rm bulk}_{\Sigma_{\triangle}}
	= {\rm Im} \, \mc{Z}^{G}_{\omega}[{_{\triangle}}{\Sigma}_{\triangle}] \,
	\equiv \gsopen{\Sigma_{\triangle}} \; ,
\end{align}
with the last equality following from \eqref{eq:physSpace}. This is the space spanned by linear superpositions $| \uppsi \ra$ of graph-states fulfilling the stabiliser constraints $\mathbb A_{\triangle^{(0)}} | \uppsi \ra = | \uppsi \ra$ and $\mathbb B_{\triangle^{(2)}}| \uppsi \ra = | \uppsi \ra$ at every $\triangle^{(0)}, \triangle^{(2)} \subset {\rm int}(\Sigma_{\triangle})$.

Let us conclude this construction by making two observations. The first one is that we showed in \eqref{eq:isoTriangulations} how given two triangulations $\Sigma_\triangle$ and $\Sigma_{\triangle'}$ of $\Sigma$ such that $\partial \Sigma_\triangle = \partial \Sigma_\triangle'$, the subspaces $\gsopen{\Sigma_{\triangle}}$ and $\gsopen{\Sigma_{\triangle'}}$ were unitarily isomorphic. This signifies that it is always possible to perform local changes of the triangulation in the interior of $\Sigma$ while remaining in the same gapped phase. This will turns out to be very useful when performing explicit computations. In particular, we shall often apply unitary isomorphisms obtained from pinched interval cobordisms describing so-called \emph{Pachner moves}. The second observation is that the Hamiltonian operators do not mix ground states with differing boundary $G$-colourings, so that there exists a natural decomposition of the Hilbert space as
\begin{align}\label{eq:boundarydecomp}
	\gsopen{\Sigma_{\triangle}}=\bigoplus_{a\in {\rm Col}(\partial\Sigma_{\triangle},G)}
	\!\!
	\gsopen{\Sigma_{\triangle}}_{a}
\end{align}
where $\gsopen{\Sigma_{\triangle}}_{a}\subseteq \gsopen{\Sigma_{\triangle}}$ denotes the subspace of states identified by the boundary colouring $a\in {\rm Col}(\partial\Sigma_{\triangle},G)$. More details regarding the construction up to that point can be found in \cite{Bullivant:2019fmk}.

\subsection{Gapped boundary partition function}
Given an open $d$-dimensional surface $\Sigma$ endowed with a triangulation $\Sigma_\triangle$, we reviewed above how to define an exactly solvable model as the Hamiltonian realisation of Dijkgraaf-Witten theory whose input data is a finite group $G$ and normalised ($d$+1)-cocycle in $H^{d+1}(G,\rU(1))$. The lattice Hamiltonian $\mathbb H^G_\omega[\Sigma_\triangle]^{\rm bulk}$ was obtained as a sum of mutually commuting projectors that act on the interior of $\Sigma_\triangle$. We would like to extend this Hamiltonian to $\partial \Sigma_\triangle$ while preserving the gap of the system, giving rise to the notion of gapped boundaries. In order to do so, we shall first  define a generalisation  of the partition function introduced in sec.~\ref{sec:DWpartitionfunction} for spacetime ($d$+1)-manifolds presenting two types of boundaries.

Let us begin by introducing the notion of \emph{relative pinched interval cobordisms}. Let $\Xi$ be a compact, oriented, $d$-manifold with non-empty boundary and $\Omega \subseteq \partial \Xi$ a choice of ($d$$-$1)-dimensional submanifold of the boundary. The relative pinched interval cobordism $\Xi \times^{\Omega}_{\rm p}\mathbb{I}$ over $\Xi$ with respect to $\Omega$ is defined as the quotient manifold
\begin{align}
	\Xi\times^{\Omega}_{\rm p}\mathbb{I} \equiv \Xi \times \mathbb{I}/\sim_{\Omega},
\end{align}
where $\sim_{\Omega}$ is defined such that $(x,i)\sim_{\Omega}(x,i')$, for all $(x,i),(x,i')\in (\partial \Xi\backslash {\rm int}(\Omega))\times \mathbb{I}$. By definition, we have $\partial(\Xi \times^{\Omega}_{\rm p}\mathbb{I})=\overline{\Xi\cup_{\Omega}(\Omega \times_{\rm p}\mathbb{I})}\cup_{\partial\Xi}\Xi$ and $\overline{\Xi}\cap\Xi=\partial\Xi\backslash{\rm int}(\Omega)$. To illustrate this definition we consider the following simple examples:
\begin{equation*}
	[0,1]\times_{\rm p}[0,1]=
	\pinchedinterval{}
	\q , \q
	[0,1]\times^{\Omega}_{\rm p} [0,1]=
	\begin{tikzpicture}[scale=0.6,baseline={([yshift=-.5ex]current bounding box.center)}]
	\def\a{1.5}
	\def\b{1.5}
	\coordinate (1) at (0,0);
	\coordinate (2) at (\a,0);
	\coordinate (3) at (0,\b);
	\coordinate (4) at (\a,\b);
	\draw[] (1) to node[anchor=north]{} (2);
	\draw[] (1) to (3);
	\draw[] (3) to (2);

	\end{tikzpicture} \; ,
\end{equation*}
with $\Omega \equiv 0\subset \{0,1\}=\partial\mathbb{I}$. Henceforth, we shall utilise the convention that $\Xi\times^{\Omega}_{\rm p}\mathbb{I}$ defines a cobordism
\begin{align}
	\Xi\times^{\Omega}_{\rm p}\mathbb{I}:\Xi\rightarrow \Xi\,,
\end{align}
and refer to ${\Omega \times_{\rm p}\mathbb I}\subset \partial(\Xi\times^{\Omega}_{\rm p}\mathbb{I})$ as a \emph{time-like boundary}. A triangulation of $\Xi\times^{\Omega}_{\rm p}\mathbb I$ can be constructed as follows: Let $\Xi_{\triangle}$, $\Xi_{\triangle'}$ be a pair of triangulations of $\Xi$ such that $\Omega_{\triangle}\subset\partial \Xi_{\triangle}$ and $\Omega_{\triangle'}\subset\partial \Xi_{\triangle'}$ define two possibly different triangulations of $\Omega$ satisfying
\begin{align}
	\partial\Xi_{\triangle}\backslash{\rm int}(\Omega_{\triangle})
	=
	\partial\Xi_{\triangle'}\backslash{\rm int}(\Omega_{\triangle'}) \; .
\end{align}
Considering a triangulation ${_{\triangle'}\Omega_{\triangle}}$ of the time-like boundary $\Omega\times_{\rm p} \mathbb I$, we define ${_{\triangle'}{\Xi^{\Omega}}_{\triangle}}$ as the triangulation of the relative pinched interval cobordism $\Xi\times^{\Omega}_{\rm p} \mathbb I$ whose boundary reads
$\overline{
	\Xi_{\triangle}\cup_{\Omega_{\triangle}} {_{\triangle'}\Omega_{\triangle}}
}
\cup_{\partial \Xi_{\triangle'}}\Xi_{\triangle'}$.

Given a triangulation ${_{\triangle'}{\Xi^{\Omega}}_{\triangle}}$ of $\Xi\times^{\Omega}_{\rm p} \mathbb I$, let us now define a generalisation of the ($d$+1)-dimensional Dijkgraaf-Witten theory with input data $(G,\omega)$ such that the corresponding partition function evaluated on ${_{\triangle'}{\Xi^{\Omega}}_{\triangle}}$ remains invariant under triangulation changes of both the interior of
${_{\triangle'}{\Xi^{\Omega}}_{\triangle}}$ and the interior of the time-like boundary ${_{\triangle'}{\Omega}_{\triangle}}$.
Let $\Omega = \sqcup_{i} \Omega_i$ be a decomposition of $\Omega$ into connected components $\Omega_{i}$, each with triangulations $\Omega_{\triangle,i}\subset\partial \Xi_{\triangle}$ and $\Omega_{\triangle',i}\subset\partial \Xi_{\triangle'}$. The generalised theory associates to each connected component $\Omega_i$ a pair $(A_{i},\phi_{i})$, where $A_{i}\subset G$ is a subgroup and $\phi_{i}\in C^{d}(A_{i},\rU(1))$ a normalised group $d$-cochain such that $d^{(d)}\phi_{i}=\omega^{-1}{\sss |}_{A_{i}}$. We refer to the data $(A_{i},\phi_{i})$ as a choice of \emph{gapped boundary condition}.\footnote{In sec.~\ref{sec:Highercat}, we shall revisit gapped boundary conditions from a category theoretical point of view.} We define a $(G,\{A_i\})$-colouring $g$ of ${_{\triangle'}{\Xi^{\Omega}}_{\triangle}}$ as a $G$-colouring such that $g[{_{\triangle'}{\Omega}_{\triangle,i}}]\in {\rm Col}({_{\triangle'}{\Omega}_{\triangle,i}},A_{i})$. The set of $(G,\{A_i\})$-colourings on ${_{\triangle'}{\Xi^{\Omega}}_{\triangle}}$ is denoted by ${\rm Col}({_{\triangle'}{\Xi^{\Omega}}_{\triangle}},G,\{A_i\})$. Equipped with such choices, we define the generalised partition function as follows:
\begin{align}
	\label{eq:relativepartitionfunction}
	\mc{Z}^{G,\{A_{i}\}}_{\omega,\{\phi_{i}\}}[{_{\triangle'}{\Xi^{\Omega}}_{\triangle}}]
	&  =
	\frac{1}{|G|^{\#({_{\triangle'}{\Xi^{\Omega}}_{\triangle}}) } \prod_i |A_i|^{\#({_{\triangle'}{\Omega}_{\triangle,i}})} }
	\\
	\nn
	&\hspace{-2.2em} \sum_{g\in{\rm Col}({_{\triangle'}{\Xi^{\Omega}}_{\triangle}},G,\{A_{i}\})}
	\;\prod_{i}\bigg(
	\prod_{\triangle^{(d)}\subset {_{\triangle'}{\Omega}_{\triangle,i}} }
	\!\!\! \phi_{i}(g[\triangle^{(d)}])^{\epsilon(\triangle^{(d)})}
	\bigg)
	\prod_{\triangle^{(d+1)}\subset {_{\triangle'}{\Xi^{\Omega}}_{\triangle}}}
	\!\!\!\!
	\omega(g[\triangle^{(d+1)}])^{\epsilon(\triangle^{(d+1)})}
	\\
	\nn
	&\hspace{5.3em}\bigotimes_{\triangle^{(1)}\subset \Xi_{\triangle'}}
	\!\!\!\!
	\ket{g[\triangle^{(1)}]}
	\!\!
	\bigotimes_{\triangle^{(1)}\subset \Xi_{\triangle}}
	\!\!\!\!
	\bra{g[\triangle^{(1)}]} \; ,
\end{align}
where $\#({_{\triangle'}{\Xi^{\Omega}}_{\triangle}}):=|{\rm int}({_{\triangle'}{\Xi}_{\triangle}})^{(0)}|+\frac{1}{2}|{\rm int}(\Xi_{\triangle'})^{(0)}|+\frac{1}{2}|{\rm int}(\Xi_{\triangle})^{(0)}|$ and $\#({_{\triangle'}{\Omega}_{\triangle,i}}):=|{\rm int}({_{\triangle'}{\Omega}_{\triangle,i}})^{(0)}|+\frac{1}{2}|{\rm int}(\Omega_{\triangle',i})^{(0)}|+\frac{1}{2}|{\rm int}(\Omega_{\triangle,i})^{(0)}|$.

As stated previously, the partition function remains invariant under retriangulation of the interior of ${_{\triangle'}\Omega_{\triangle}}$ as well as the interior of ${_{\triangle'}{\Xi^{\Omega}}_{\triangle}}$. In this manner, the partition function  $\mc{Z}^{G,\{A_{i}\}}_{\omega,\{\phi_{i}\}}[{_{\triangle}{\Xi^{\Omega}}_{\triangle}}]$ defines a projection operator and we associate to the triangulation $\Xi_\triangle$ the following Hilbert space:
\begin{align}
	\mc{V}^{G,\{A_{i}\}}_{\omega,\{\phi_{i}\}}[\Xi_{\triangle}]
	:=
	{\rm Im} \; \mc{Z}^{G,\{A_{i}\}}_{\omega,\{\phi_{i}\}}[{_{\triangle}{\Xi^{\Omega}}_{\triangle}}]\; .
\end{align}
Furthermore, akin to equations \eqref{eq:isoTriangulations} and \eqref{eq:Hermicity}, the triangulation invariance properties of the partition function together with the Hermicitiy condition
	\begin{align}
	\mc{Z}^{G,\{A_{i}\}}_{\omega,\{\phi_{i}\}}[{_{\triangle'}{\Xi^{\Omega}}_{\triangle}}]^{\dagger}
	=
	\mc{Z}^{G,\{A_{i}\}}_{\omega,\{\phi_{i}\}}[{_{\triangle}{\Xi^{\Omega}}_{\triangle'}}]
\end{align}
demonstrate that the operator
\begin{align}
	\mc{Z}^{G,\{A_{i}\}}_{\omega,\{\phi_{i}\}}[{_{\triangle'}{\Xi^{\Omega}}_{\triangle}}]
	:
	\mc{V}^{G,\{A_{i}\}}_{\omega,\{\phi_{i}\}}[\Xi_{\triangle}]
	\xrightarrow{\sim}
	\mc{V}^{G,\{A_{i}\}}_{\omega,\{\phi_{i}\}}[\Xi_{\triangle'}]
\end{align}
defines a unitary isomorphism of Hilbert spaces.

\subsection{Hamiltonian model in the presence of gapped boundaries}\label{sec:bdryHam}

In sec. \ref{sec:DWHam}, we described the Hamiltonian realisation $\mathbb H^G_\omega[\Sigma_\triangle]^{\rm bulk}$ of the Dijkgraaf-Witten theory in $d$ spatial dimensions in the presence of open boundary conditions. Utilising the partition function \eqref{eq:relativepartitionfunction} introduced in the previous section, we shall now define an extension of the Hamiltonian model to include gapped boundary conditions \cite{Bullivant:2017qrv,wang2018gapped}.

Let us consider an oriented $d$-manifold $\Sigma$ with non-empty boundary and a choice of triangulation $\Sigma_\triangle$. The input of the model is a pair $(G,\omega)$ and a choice of gapped boundary conditions $\{(A_i ,\phi_i)\}$ for each connected component $\partial \Sigma_{\triangle,i} \subset \Sigma_\triangle$, where $A_i \subset G$ is a subgroup and $\phi_{i}\in C^{d}(A_{i},\rU(1))$ is a normalised group $d$-cochain satisfying the condition $d^{(d)}\phi_{i}=\omega^{-1}{\sss |}_{A_{i}}$. In the interior of $\Sigma_\triangle$, the (bulk) Hamiltonian was defined in eq.~\ref{eq:Ham}. Given such a choice of gapped boundary conditions, let us now define an operator that acts on a local neighbourhood of a boundary vertex $(v_{0})\subset \partial\Sigma_{\triangle,i}$. Mimicking the definition of the bulk vertex operator, we consider the subcomplex $\Xi_{v_{0}}:={\rm cl}\circ {\rm st}(v_{0})$, which corresponds to the smallest subcomplex that includes all the simplices of which $(v_0)$ is a subsimplex. We next define the triangulated relative pinched interval cobordism over $\Xi_{v_0}$ with respect to $\Omega := {\rm cl}\circ {\rm st}(v_{0})\cap \partial \Sigma_{\triangle,i}$
\begin{align}
	\label{eq:pinchedBdryOp}
	\cob{\Xi_{v_{0}}\!}{\Xi}{\, \Xi_{v_{0}}}:=(v'_{0})\sqcup_{\rm j}  {\rm cl}\circ {\rm st}(v_{0}) \;,
\end{align}
whose boundary is given by
\begin{align}
	\partial (\cob{\Xi_{v_{0}}\!}{\Xi}{\, \Xi_{v_{0}}})
	=
	\overline{
	\Xi_{v_{0}}\cup_{\Omega}(\cob{\Omega_{v_{0}}\!}{\Omega}{\, \Omega_{v_{0}}})
	}
	\cup_{\partial\Xi_{v_{0}}}\Xi_{v_{0}}
\end{align}
where  $\Omega_{v_0} := (v'_{0})\sqcup_{\rm j}\Omega$. Given this relative pinched interval cobordism, we define the action of the operator $\mathbb{A}^{A_{i},\phi_{i}}_{(v_{0})}$ on a state $| \uppsi \ra \in \mathcal H^{G,A_i}_{\omega,\phi_i}[\Sigma_\triangle]$ via
\begin{align}
	\mathbb{A}^{A_{i},\phi_{i}}_{(v_{0})}:\ket{\uppsi}
	\mapsto
	\mc{Z}^{G,A_{i}}_{\omega,\phi_{i}}[\cob{\Xi_{v_{0}}\!}{\Xi}{\, \Xi_{v_{0}}}]
	\ket{\uppsi} \; .
\end{align}
The gapped boundary Hamiltonian is finally defined as
\begin{align}
	\mathbb{H}^{G,\{A_{i}\}}_{\omega,\{\phi_{i}\}}[\Sigma_{\triangle}]
	=
	\Hambulk{\Sigma_{\triangle}}
	+
	\sum_{\partial\Sigma_{\triangle,i}\subset\partial \Sigma_{\triangle}}
	\mathbb{H}^{G,A_{i}}_{\omega,\phi_{i}}[\partial \Sigma_{\triangle,i}]^{\rm bdry} \; ,
\end{align}
where
\begin{align}
	\mathbb{H}^{G,A_{i}}_{\omega,\phi_{i}}[\partial \Sigma_{\triangle,i}]^{\rm bdry}
	:=
	-\sum_{\triangle^{(0)}\subset \partial\Sigma_{\triangle,i}}\mathbb{A}^{A_{i},\phi_{i}}_{\triangle^{(0)}}\; .
\end{align}
From the triangulation invariance properties of the partition function $\mc{Z}^{G,\{A_{i}\}}_{\omega,\{\phi_{i}\}}$ follows that the Hamiltonian is a sum of mutually commuting projection operators, and as such it is still exactly solvable. Furthermore, analogously to  the bulk Hamiltonian, we can identify the ground-state subspace $\mc{V}^{{G,\{A_{i}\}}}_{\omega,\{\phi_{i}\}}[\Sigma_{\triangle}]$ with
\begin{align}
	{\rm Im}\; \mc{Z}^{{G,\{A_{i}\}}}_{\omega,\{\phi_{i}\}}[{_{\triangle}{\Sigma^{\partial \Sigma}}_{\triangle}}]
	\equiv
	\mc{V}^{{G,\{A_{i}\}}}_{\omega,\{\phi_{i}\}}[\Sigma_{\triangle}] \; ,
\end{align}
and verify that the unitary isomorphism
\begin{align}
	\mc{Z}^{{G,\{A_{i}\}}}_{\omega,\{\phi_{i}\}}[{_{\triangle'}{\Sigma^{\partial \Sigma}}_{\triangle}}]:
	\mc{V}^{{G,\{A_{i}\}}}_{\omega,\{\phi_{i}\}}[\Sigma_{\triangle}]
	\xrightarrow{\sim}
	\mc{V}^{{G,\{A_{i}\}}}_{\omega,\{\phi_{i}\}}[\Sigma_{\triangle'}]
\end{align}
commutes with the Hamiltonian. This last statement implies that we can always replace a given triangulated subcomplex $\Omega_\triangle \subset \partial \Sigma_\triangle$ by $\Omega_{\triangle'}$ while remaining in the ground state sector.

Note finally that in the subsequent discussion, we shall also refer to \emph{gapped interfaces} between several gapped boundaries. However, we will not require an explicit form of the Hamiltonian for such interfaces, and as such we omit here the explicit definition. Despite such an omission, the corresponding Hamiltonian can be explicitly defined in close analogy with the construction of the gapped boundary Hamiltonian presented in this section.

\bigskip \noindent
In order to illustrate the definition and some properties of the gapped boundary Hamiltonian, let us now specialize to two dimensions (see also \cite{Bullivant:2017qrv}). We consider a two-dimensional surface $\Sigma$ endowed with a triangulation $\Sigma_\triangle$ and a single connected boundary component $\partial\Sigma_{\triangle}$. The input data for the bulk Hamiltonian is a finite group $G$ and a normalised group $3$-cocycle $\alpha$. Furthermore, we define on $\partial \Sigma_\triangle$ a gapped boundary whose input data is a pair $(A, \phi)$, where $A \subset G$ is a subgroup and $\phi$ a group 2-cochain satisfying $d^{(2)}\phi=\alpha^{-1}{\sss |}_{A}$ which is explicitly expressed via
\begin{equation}
\alpha^{-1}(a,a',a'') \stackrel{!}{=} d^{(2)}\phi(a,a',a'')
=
\frac{\phi(a',a'') \, \phi(a, a'a'')}{\phi(aa',a'') \, \phi(a,a')} \; ,
\end{equation}
for every $a,a',a'' \in A \subset G$. We consider the following situation:
\begin{equation*}
\exampleTwoD{1.2}{1}
\end{equation*}
where the dashed area $\snippetBulk{0.6}{1}$ represents the bulk of the manifold, whereas the coloured line stands for the gapped boundary. The black lines represent the 1-simplices on the interior $\Sigma_\triangle$ that are included in ${\rm cl}\circ {\rm st}\snum{(1)}$. We first want to write down the action of the boundary operator at the vertex $\snum{(1)}$ on graph-states of the form
\begin{equation}
\label{eq:exampleHilb}
{\rm Span}_{\mathbb C}\bigg\{\bigg| \mathfrak{g} \bigg[ \exampleTwoD{1.2}{2}  \bigg] \bigg\ra \bigg\}_{\substack{\forall \, \mathfrak{g} \in {\rm Col}({\rm cl} \circ {\rm st}(1), G,A) }}
\!\!\! \equiv
{\rm Span}_{\mathbb C}\bigg\{\bigg|  \exampleTwoD{1.2}{3} \! \bigg\ra \bigg\}_{\substack{\hspace{-0.9em} \forall \, g \in G \\ \forall \, a,a' \in A }} \!\!\!\!  .
\end{equation}
The boundary vertex operator $\mathbb{A}^{A,\phi}_{(1)}$ boils down to evaluating the partition function \eqref{eq:relativepartitionfunction} on the relative pinched interval cobordism $\snum{(023)}\times_{\rm p}^{(02)} \mathbb I$ defined by
\begin{equation}
\raisebox{-1.9em}{\exampleTwoD{1.2}{4}} ,
\end{equation}
such that $\snum{0} < \snum{1} < \snum{\tilde 1} < \snum{2} < \snum{3}$ and the orange edges represent the time-like boundary.
Explicitly, the action of this boundary vertex operator reads
\begin{align}
	 \mathbb{A}^{A,\phi}_{(1)} \bigg | \exampleTwoD{1.2}{3}  \! \bigg\ra  = \frac{1}{|A|}\sum_{\tilde a \in A} \frac{\alpha(a, \tilde a, \tilde a^{-1}g) \, \phi(\tilde a, \tilde a^{-1}a')}{\alpha(\tilde a ,\tilde a^{-1}a',a'^{-1}g) \, \phi(a,\tilde a)}
	\bigg|  \exampleTwoD{1.2}{5} \! \bigg\ra \; .
\end{align}
Let us now compute a triangulation changing boundary operator on a graph state \eqref{eq:exampleHilb}. More specifically, let us construct the isomorphism that replaces the boundary subcomplex $\snum{(01)} \cup \snum{(12)}$ by a single 1-simplex $\snum{(02)}$. The corresponding operator is conveniently obtained by evaluating the partition function \eqref{eq:relativepartitionfunction} on the relative pinched interval cobordism
\begin{equation}
\raisebox{3.7em}{\exampleTwoD{1.2}{7}} ,
\end{equation}
with time-like boundary $\snum{(012)}$, implementing the isomorphism
\begin{equation}
\bigg| \exampleTwoD{1.2}{3} \! \bigg\ra \simeq
\frac{1}{|A|^\frac{1}{2}}  \frac{\alpha(a,a',a'^{-1}g)}{\phi(a,a')}
\bigg| \exampleTwoD{1.2}{6} \! \bigg\ra \; .
\end{equation}
We can now confirm that this triangulation changing operator does commute with the Hamiltonian operator. This follows from the cocycle relations $d^{(2)}\phi(a, \tilde a, \tilde a^{-1}a') = \alpha^{-1}(a,\tilde a, \tilde a^{-1}a')$ and $d^{(3)}\alpha(a,\tilde a, \tilde a^{-1}a',a'^{-1}g)$ = 1.

config: arxiv


TITLE: Is $f(x)=\log(1+x^2)$ uniformly continuous on $(0,\infty)$?
QUESTION [4 upvotes]: Is $f(x) = \log(1+x^2)$ uniformly continuous on $(0,\infty)$?

My work:
Looking at the graph and knowing that $\log$ considered a "slow-growing" function, my guess is that $f(x)$ is uniformly continuous.
Let $\varepsilon > 0$ and  $\left| {x - y} \right| < \delta $ where $\delta$ will be defined later.
$$\left| {f(x) - f(y)} \right| = \left| {\log (1 + {x^2}) - \log (1 + {y^2})} \right| = \left| {\log \left( {{{1 + {x^2}} \over {1 + {y^2}}}} \right)} \right| \le {{1 + {x^2}} \over {1 + {y^2}}}$$
The problem is that the last expression is too large.
I tried assuming $x < y$ (without loss of generality).
Also, I tried multiplying by $(1-y^2)$ but nothing good came out from that.
What am I missing?
Update:
I'm not allowed to use the fact that the derivative is bounded which implies uniform continuity. I'm suppose to prove it "directly".

REPLY [4 votes]: Hint: $f\in\mathcal{C}^1$ (i.e., is continuously differentiable), and its derivative is bounded. Hence, it is Lipschitz.
Another approach (following the edit of the original question -- "Lipschitz-ness is not allowed"):
 suppose wlog that $\epsilon \leq \frac{1}{2}$, and let $\delta\stackrel{\rm{def}}{=}\frac{\epsilon}{2}$. Fix any $x \geq y$ with $\lvert x-y \rvert \leq \delta$. Then
$$
1 \leq \frac{1+x^2}{1+y^2} \leq \frac{1+(y+\delta)^2}{1+y^2} = 1+\delta \frac{2y+\delta}{1+y^2} \leq 1+\delta \frac{1+2y}{1+y^2}
$$
so that
$$
\ln 1 \leq \ln \frac{1+x^2}{1+y^2} \leq \ln\!\left( 1+\delta \frac{1+2y}{1+y^2} \right)
$$
Now, using the "fact" (easy to prove) that for every $y > 0$ one has $\frac{1+2y}{1+y^2} \leq \frac{1+\sqrt{5}}{2} \leq 2$, you get
$$
0 \leq \ln \frac{1+x^2}{1+y^2} \leq \ln(1+2\delta)\leq 2\delta = \epsilon.
$$


TITLE: Orbital Hyperloop for Approach to $c$?
QUESTION [2 upvotes]: We have a bunch of "orbital catapult" ideas to get things into orbit - most involve some sort of loop with mag levitation in vacuum. There is a practical limit on the top speed here due to whatever is being shot having to go through the atmosphere.
What if we built this loop in orbit for imparting initial velocity on a spacecraft / payload? (Other than removing the concern of burning up through the atmosphere)
What limits the speed of an object in a hyperloop? What keeps us from accelerating the object close to c?
Edit: Unrelated to bounty, but the comments got me thinking - put an object in orbit, beam energy to it to get it faster and faster while it's still within a certain proximity to earth, then let it escape. I am guessing this would waste a lot of energy to stay in orbit at higher speeds though.

REPLY [1 votes]: Launch loops should be attached to a planet, which supplies practically infinite reaction mass and tensile strength, though limited angular momentum (mass times velocity times radius), necessary for high circular orbits around a planet.  A large asteroid can be a reaction mass anchor, but won't directly supply tensile strength.  Rotating tethers is the best bet for smaller bodies.
Energy is cheap; the Sun emits 380 trillion terawatts.  Some of that can be beamed from space solar power satellites (SSPS) to rectennas powering launch loops;  indeed, the beam polarity can be rotated to cross-polarized rectennas, directly powering the launch loop drive motors.  Beam energy to vehicle kinetic energy conversion efficiency will be rectenna-limited;  if that is 80%, then 10 terawatts of beam energy can launch 150 tonnes per second to a constellation of construction ports, more than 4 billion tonnes per year.  That power can be generated with less than 10 million tonnes of SSPS.
Given the current global launch market of less than 3000 tonnes per year, that won't happen for a while, but it's nice to know there is room for growth.
Slingshot around the Moon is one way to add angular momentum to a loop-launched vehicle, but the Moon is in the proper position for this only twice a month, while a loop can launch thousands of vehicles per day.
An amusing kludge is to launch some of the vehicles to a very high apogee, fire a small rocket, and add lots of angular momentum up there (small velocity times huge radius).  If the perigee of the resulting orbit intersects the apogee of a direct launch from Earth, you can arrange to exchange the momentum there, with rotating tethers or an Arnold/Kingsbury spaceport.  See http://launchloop.com/ConstructionPort for work in progress.
Want to go somewhere at the speed of light?  Digitize yourself and send the information to a distant receiver.  The atoms you are made of are as common as stardust, and tend to disperse in less than a century.  The receiver can be sent as a small replicator, along with software to build the receiver, at relatively slow speeds.  This is an enormous technical challenge, but tiny compared to moving anything sizeable at at relativistic speeds (CERN LHC moves nanograms of protons for gigabucks - scale from that).   Humans are, after all, the end product of tiny replicators.


TITLE: Equivalence between Hölder norms
QUESTION [0 upvotes]: Let us consider the following two norms:
$$
\left\lVert f\right\rVert_\alpha = \left\lVert f\right\rVert_\infty +  \displaystyle{\sup_{\substack{x,y \in U \\ x \neq y}} \frac{\left| f(x) - f(y))\right|}{\left| x-y \right|^\alpha}},
$$
$$
\left\lVert f\right\rVert_{\alpha • } = |f(0)|+ \displaystyle{\sup_{\substack{x,y \in U \\ x \neq y}} \frac{\left| f(x) - f(y))\right|}{\left| x-y \right|^\alpha}}.
$$
I need to show that those two norms are equivalent on $\mathcal{C}^\alpha(U,\mathbb{R})$, where $U=[0,T]$ that is the space of $\alpha$-Hölder functions from $U$ to $\mathbb{R}$, then I have to find two constants, for instance let us call them $a,b \in \mathbb{R}$, such that:
$$
a \left\lVert f\right\rVert_{\alpha • } \leq \left\lVert f\right\rVert_{\alpha } \leq b \left\lVert f\right\rVert_{\alpha • }.
$$
I have issues in handling the $\infty$-norm and readapting the constant to the Hölder seminorm. Any suggestions?

REPLY [2 votes]: \begin{eqnarray}
|f(x)| &\le& |f(0)| + |f(x)-f(0)| \\
&\le& |f(0)| + \|f\|_{\alpha • }|x-0|^\alpha \\
&\le&  \|f\|_{\alpha • } + T^\alpha \|f\|_{\alpha • } \\
&=& (1+T^\alpha)\|f\|_{\alpha • }
\end{eqnarray}
And so $\|f\|_\infty \le (1+T^\alpha) \|f\|_{\alpha • }$ and then
$\|f\|_{\alpha} \le (2+T^\alpha) \|f\|_{\alpha • }$.
The other direction is immediate since $|f(0)| \le \|f\|_\infty$ and so $\|f\|_{\alpha • } \le \|f\|_{\alpha}$.

REPLY [2 votes]: It is clear that $\|f\|_{\alpha • } \leq \|f\|_\alpha$. Essentially the only difficult part in showing the other inequality is controlling $\|f\|_\infty$ in terms of $\|f\|_{\alpha • }$ so I will do this.
For $x \in [0,T]$ we have
\begin{align*}
|f(x)| &\leq |f(0)| + |f(x) - f(0)|
\\ & \leq |f(0)| + \|f\|_{\alpha • } |x|^\alpha \\&
\leq \|f\|_{\alpha • } + \|f\|_{\alpha • } T^\alpha \\&
= (1+ T^\alpha) \|f\|_{\alpha • }
\end{align*}
Taking the $\sup$ over $U$ gives an inequality of the desired form.


TITLE: Why is $\frac{d}{dx}\ln(x) = \frac1x$ when the domain of $\frac1x$ includes $(-\infty,0)$?
QUESTION [7 upvotes]: On Khan Academy, they factor the derivative:
$$
\frac{d}{dx}\frac{x^2 + x - 2}{x-1} = \frac{d}{dx}(x+2) = 1 \text { where $x \ne 1$}
$$
Khan says:

Recall the derivative is equal to $1$ for all $x$ values where the function is defined. Since the function is undefined for $x=1$, so is the derivative.

I did some Google searching to see if this was really a rule, and I found this Quora question.  One of the answers provides a seeming counterexample:
$$
\ln(x) \text{ has a domain } (0,∞) \\
\frac{d}{dx}\ln(x)=\frac{1}{x} \\
\frac{1}{x} \text { has the domain} (-∞,0) \text{ and } (0,∞) \\
$$
So is Khan stating an incorrect rule?  Obviously it works in the case of his problem, but I'd like to understand what the accurate rule is here.  Maybe the key question is what type of discontinuity we're dealing with?

REPLY [5 votes]: By definition,$$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a},$$and therefore $f'(a)$ being defined only makes sense if $a$ is in the domain of $f$.
In the example in which $\log'(x)=\frac1x$, what happens is that the analytic expression for $\log'(x)$ makes sense for numbers $x$ which don't belong to the domain of $\log$. But that does not mean that $\log$ is differentiable at those points.

REPLY [4 votes]: Technically you should consider your function not just as a single expression, but also together with its domain and codomain. Since the log function is defined for every $x>0$, it has no sense to consider the analytic expression of its derivative only. You are deriving a function that is defined for strictly positive $x$, so the derivative has to be defined at most on this domain as well.
Even if the geometric meaning of the derivative is not always the best intuition, here it may help. Think about the derivative as the angular coefficient tangent line at a point for your function. It makes sense to be considered only at the points where the function itself is defined. Of course you can draw lines also on the negative $x$-axis, but you are no longer considering it as a derivative of your starting function.


text:
\begin{document}
\maketitle

\def\theenumi{\roman{enumi}}
\def\labelenumi{(\theenumi)}

\def \ch {{\rm ch}}
\def \gr {{\rm gr}\,}
\def \g  {{\mathfrak g}}
\def \b  {{\mathfrak b}}
\def \h  {{\mathfrak h}}
\def \a  {{\mathfrak a}}
\def \rkg {{{\rm rk}(\g)}}
\def \gs {{\mathfrak sl}}
\def \gl {{\mathfrak gl}}
\def \CC {{\mathbb C}}
\def \QQ {{\mathbb Q}}
\def \NN {{\mathbb N}}
\def \cpf {\CC{\rm PF}}
\def \da {{\mathcal A}}
\def \prt {\partial/\partial t}
\def \sgr {{\rm Sign}}
\def \v {{\bf m}}
\def \ve {\varepsilon}

\begin{abstract}
This paper is based on the observation that dimension of weight spaces of multi-variable Weyl modules
depends polynomially on the highest weight (Conjecture~\ref{cmain}). We support this conjecture
by various explicit answers for up to three variable cases and discuss the underlying combinatorics.
\end{abstract}

\section*{Introduction}

Representations of one-variable current algebras are now well studied and have a lot of applications
to algebraic geometry and mathematical physics. Also there are several approaches
to develop a similar theory for multi-variable currents. Let us briefly outline some  of them.

The first one is an affinization of affine Kac-Moody algebras (that is studying
Laurent polynomials in one variable with values in an affine Lie algebra).
It provides a lot of structures, in particular,
the universal central extension and a canonical way to quantize such algebras,
but makes sense only for the two-variable case.

The second approach starts from multi-variable, usually toroidal settings (that is studying
Laurent polynomials in several variables with values in a finite-dimensional Lie algebra). It is more universal,
but very few known notions and methods works in this generality, so other tools should
be used (see e.g. \cite{MRY}).

The third one is adding a variable using the fusion product introduced in \cite{FL1}. Namely,
for a set of representations of a Lie algebra $\a$ it produces a graded action
of the current algebra $\a \otimes \CC[t]$ on their tensor product (see \cite{FL1}, \cite{FF1} how it
works for $\a = sl_2$ and \cite{FF2} for $\a = \widehat{sl_2})$.

Recently we were faced with a rather simply stated question which appeared to be related to fusion product
in the one variable case, related to double affine algebras for two variables, promising for three variables and
pretty challenging by now for a higher number.

Suppose we have even the simplest Lie algebra $sl_2$ with the standard basis $e$, $h$, $f$. Consider
the Lie algebra of $sl_2$-valued polynomials $sl_2 \otimes \CC[x^1, \dots, x^d]$.
Let us follow the most naive notion of highest weight vector, namely let $e \otimes P$
acts on it by zero, and  $h\otimes P$ acts by the scalar $nP(0)$ for each polynomial $P$.
In order to obtain a finite-dimensional module let $n$ be a non-negative integer.
Following \cite{CP} introduce $W_{n}$ to be the module,
generated by a highest weight vector $v$ with the additional integrability
relation $(f\otimes 1)^{n+1}v=0$. Then it appears (see \cite{FL2}) that $W_{n}$ is finite-dimensional
and that there are some pretty formulas for its dimension and dimensions of weight spaces.

The next table is organized as follows. First column indicates the number of variables.
Second column contains the dimensions of Weyl modules together with the number of
the sequence in the On-Line Encyclopedia of Integer Sequences \cite{Sl}. In the third column
there are the dimensions of weight spaces organized as polynomials. Last column
provides a reference.

\medskip

\noindent
\begin{tabular}{|c|c|c|c|}
\hline
$d$ & $\dim W_{n}$ & $\dim W_{n}^{n\omega - k \alpha} = P(n)/P(k)$ & reference \cr
\hline\hline
$0$ & $n+1$ & $1$ for $0\le k \le n$ & well known \cr
\hline
$1$ & $2^n$ &
{\bf Binomial Coefficients} $\bin{n}{k}$ & \cite{CP2} \cr
&
&
$P(n) = \prod\limits_{i=0}^{k-1}(n-i)$ &  \cr
\hline
$2$ & {\bf Catalan Number}  & {\bf Narayana Numbers} & \cite{FL2}\cr
& $ \left.\bin{2n+2}{n}\right/ (n+1)$ &
$\left. \bin{n+1}{k}\bin{n+1}{n-k}\right/ (n+1)$ & using  \cr
& A000108
& $P(n) = \prod\limits_{i=-1}^{k-2}(n-i)\prod\limits_{i=0}^{k-1}(n-i)$ &
\cite{H} \cr
\hline
$3$ & $\left.\bin{3n+3}{n}\right/\bin{n+2}{2}$ & $\frac{\bin{n+k+2}{2k+1}\bin{2n-k+1}{k}}{(k+1)(n+k+2)}^{\vphantom{s}}$
 & Conjec- \cr
& A000139 \rule[-0.7cm]{0cm}{1.6cm} & $P(n) = \prod\limits_{i=2}^{2k+1}(n+i-k) \prod\limits_{i=-1}^{k-2} (2n-i-k)$ &
 ture~\ref{triag}\cr
\hline
$\ge 4$  & Huge prime factors & $P(n)$ is not factorizable /${\mathbb Q}$ & not known \cr
\hline
\end{tabular}

\bigskip

Note that in this table for a fixed $k$ the dimensions of $W_{n}^{n\omega - k \alpha}$
are values of a polynomial in $n$. It seems to be a general phenomenon and the paper is
build around this polynomiality conjecture. In Section 1 we introduce the notation, discuss Weyl modules in full generality and state the polynomiality conjecture (Conjecture~\ref{cmain}).
In Section 2 we produce several explicit answers and
support this conjecture in various particular cases (up to three variables).

{\bf Acknowledgements.}
 It is a pleasure to dedicate this paper to the 65-th birthday of Pierre Deligne, who discovered
many wonderful results, in particular, a polynomial extrapolation of representations \cite{D1},
\cite{D}.
I am very grateful to A.Berenstein, P.Etingof, B.Feigin, V.Ostrik, A.Postnikov
for very useful and stimulating discussions. Part of this work
was done during a visit to MSRI and UC Riverside. The author was partially
supported by RF President Grant N.Sh-3035.2008.2, grants RFBR-08-02-00287, RFBR-CNRS-07-01-92214,
RFBR-IND-08-01-91300 and P.Deligne 2004 Balzan prize in mathematics.

\section{Multi-variable Weyl modules}

\subsection{Notation}

Let $\g$ be a reductive Lie algebra. By $\b$ and $\h$ denote its Borel and Cartan subalgebras, by
$R$ and $R_+$ denote the sets of roots and positive roots, by $\alpha_i$, $i=1\dots \rkg$, denote simple roots,
by $\omega_i$ denote simple weights,
by $Q$ and  $Q^+$ denote the root lattice and its positive cone, by $P$ and $P^+$ denote
the weight lattice and the dominant weight cone.

For $\g = \gl_r$ by $\ve_i$, $i=1\dots r$, denote
the standard basis in the weight space. Then partitions
$\xi = (\xi_1 \ge \dots \ge \xi_r)$ correspond to
dominant weights $\sum \xi_i \ve_i$ of  $\gl_r$.
For each $\xi$ by $\xi^t$ denote the {\em transposed} partition
$\xi^t_j = |\{i | \xi_i \ge j\}|$ corresponding to the reflected Young diagram.

For a representation $U$ by $U^\mu$ denote the corresponding weight space, that is the common
eigenspace of $\h$ where $h \in \h$ acts by the scalar $\mu(h)$.

In addition let $A$ be a commutative finitely-generated algebra with a unit $1$ and
a co-unit (augmentation) $\epsilon: A \to \CC$.
Note that $A$ can be treated as the algebra of functions on an affine scheme $M$,
so $\epsilon$ is the evaluation at a certain closed point $p$ of $M$.

By $A_\epsilon$ denote the augmentation ideal, that is, the kernel of $\epsilon$.
Note that the infinitesimal neighborhood spaces $A/A_\epsilon^n$ are always finite-dimensional.

\subsection{Highest weight modules}

Here we study finite-dimensional representations of $\g \otimes A$, that is, the Lie algebra
of $\g$-valued functions on the scheme $M$.

\begin{defn}
Let $U$ be a representation of $\g \otimes A$.
We say that a vector $v_{\lambda} \in U$ is a {\em highest weight vector}
of weight $\lambda \in \h^*$ if
$$(g \otimes P) v_{\lambda} = \lambda(g) \epsilon(P) v_{\lambda}\quad \mbox{for}\ g \in \b,\ P \in A.$$
\end{defn}

\begin{thm}\cite{FL2}\label{wex}
\begin{enumerate}
\item There exists a universal finite-dimensional module $W^{A}_\epsilon(\lambda)$,
such that any finite-dimensional module
generated by $ v_{\lambda}$ is a quotient of $W^A_\epsilon(\lambda)$.

\item We have  $W^A_\epsilon(\lambda)\ne 0$ if and only if $\lambda \in P^+$.

\item For any $\lambda$ there exists $N$ such that $\g \otimes A_\epsilon^N$ acts on
$W^A_\epsilon(\lambda)$ by zero.

\item We have $W^A_\epsilon(\lambda) \cong  \oplus_\mu W^A_\epsilon(\lambda)^\mu$, and
$W^A_\epsilon(\lambda)^\mu \ne 0$ if and only if
$w(\mu) - \lambda \in Q^+$ for any element $w$ of the Weyl group.

\item Any finite-dimensional module, generated by a common eigenvector of $\b \otimes A$,
is a quotient of $W^A_{\epsilon_1}(\lambda^1) \otimes \dots \otimes W^A_{\epsilon_k}(\lambda^k)$
for some $\lambda^i$, $\epsilon_i$.
\end{enumerate}
\end{thm}

\begin{defn}\cite{CP}\cite{FL2}
The module $W^A_\epsilon(\lambda)$ is called {\em Weyl module}. For $A= \CC[x^1, \dots, x^d]$ and
$\epsilon(P) = P(0)$ let us denote it by $W^d(\lambda)$ as well.
\end{defn}

\begin{prop}
\begin{enumerate}
\item For each homomorphism $f: A_1 \to A_2$ we have a natural map
$f^*: W^{A_1}_{\epsilon \circ f}(\lambda) \to W^{A_2}_{\epsilon}(\lambda)$.
\item If $\epsilon$ corresponds to a non-singular point of $M$ then we have
$W^A_\epsilon(\lambda) \cong W^d(\lambda)$, where $d=\dim M$.
\end{enumerate}
\end{prop}

\begin{proof}
For (i) note that due to the map $f$ there is an action of $\g \otimes A_1$ on $W^{A_2}_{\epsilon}(\lambda)$,
and the $\g \otimes A_1$-submodule, generated by the highest weight vector, is a quotient
of $W^{A_1}_{\epsilon \circ f}(\lambda)$.

For (ii) note that for
$\epsilon$ corresponding to a non-singular point
the quotient $A/ A_\epsilon^N$ is isomorphic to the same factor for $\CC[x^1, \dots, x^d]$, so
by Theorem~\ref{wex}~(iii) the Weyl modules are isomorphic.
\end{proof}

In other words, characters of Weyl modules can be considered as a functorial invariant of algebraic singularities.

\begin{cnj}\label{cmain}
Let $\lambda = \sum \lambda_i \omega_i$. Fix $\mu = \sum \mu^i \alpha_i$.
\begin{enumerate}
\item
For $d>0$ we have $\dim W^d(\lambda)^{\lambda-\mu}$
is a polynomial in $\lambda_1, \dots, \lambda_\rkg$ of degree $\mu^i d$ in the variable $\lambda_i$,
$i=1\dots \rkg$.
\item
Even for a singular point there exists $\nu$, such that for $\lambda - \nu \in P^+$ we have
$\dim W^A_\epsilon(\lambda)^{\lambda-\mu}$
is a polynomial in $\lambda_1, \dots, \lambda_\rkg$ of degree $\mu^i d$ in the variable $\lambda_i$,
$i=1\dots \rkg$, where $d=\dim M$.
\end{enumerate}
\end{cnj}

\begin{ex} Let $\g = sl_2$, $\mu = \alpha$. Then $W^A_\epsilon(n\omega)^{n\omega-\alpha} \cong A / A_\epsilon^n$
(see \cite{FL2}).
And for a big enough $n$ the integers
$\dim A / A_\epsilon^n - \dim A/ A_\epsilon^{n+1} = \dim  A_\epsilon^n/ A_\epsilon^{n+1}$
are values of the Hilbert polynomial for the completion
of $A$ at $\epsilon$.
\end{ex}

In this paper we support and refine this conjecture in several particular cases.

\subsection{Schur-Weyl duality}

First let us describe the classical construction.

Let $V$ be the $r$-dimensional vector representation of
$\gl_r$. For the classical settings note that $V^{\otimes n}$ inherits both the
action of the Lie algebra $\gl_r$ and the symmetric group $\Sigma_n$, and that
these actions commute. Decomposition of this bi-module provides a reciprocity
between the representations of these objects and it can be described as the
following functor.

\begin{defn}
Let $\pi$ be a representation of $\Sigma_n$. By $\nabla_n^r(\pi)$ denote
the representation $\left(V^{\otimes n}\otimes \pi\right)^{\Sigma_n}$ of $\gl_r$.
\end{defn}

\begin{rem}
Note that at the level of characters $\nabla_n^r$ is known as the {\em Frobenius
characteristic map}.
\end{rem}

This functor can be generalized for our settings as follows.

\begin{defn}
For an associative algebra $A$ and an integer $n$ by
{\em wreath product} $\sdp{n}{A}$
denote the associative algebra, generated by $A^{\otimes n}$
and the elements of $\Sigma_n$ under the relation
\begin{equation}\label{comm}
(a_1\otimes \dots\otimes a_n) \cdot \sigma = \sigma \cdot (a_{\sigma(1)}\otimes \dots
\otimes a_{\sigma(n)})
\end{equation}
and identification of the unit $e \in \Sigma_n$ with $1\otimes \dots \otimes 1
\in A^{\otimes n}$.
\end{defn}

In other words, we have $\sdp{n}{A}\cong \CC[\Sigma_n] \otimes A^{\otimes n}$
as a vector space with the multiplication given by the relation~\eqref{comm}.
Let us show that $\nabla_n^r(\pi)$ is indeed a functor from representations of
$\sdp{n}{A}$ to the representations
of the Lie algebra $\gl_r \otimes A$.
Note that for this observation $A$ is not necessarily commutative, and  $\gl_r \otimes A$
can be defined as
the space of matrices with entries from $A$ with the usual commutator.

\begin{prop}
Let $\pi$ be a representation of $\sdp{n}{A}$. Then $\gl_r \otimes A$
acts on
$\nabla_n^r(\pi)$
as follows
\begin{equation}\label{swact}
(g \otimes P) (u_1\otimes \dots \otimes u_n \otimes v) = \sum_{k=1}^n
u_1 \otimes \dots \otimes g(u_k) \otimes \dots \otimes u_n \otimes \iota_k (P) v,
\end{equation}
where $\iota_k$ is the inclusion of $A$ into $A^{\otimes n}$ as the $k$-th factor
$1 \otimes \dots \otimes A \otimes \dots \otimes 1$.
\end{prop}

\begin{proof}
First note that \eqref{swact} defines a Lie algebra action on $V^{\otimes n}\otimes \pi$. To complete the
proof it is enough to observe that this action commutes with the action of $\Sigma_n$.
\end{proof}

\begin{rem}
There are quantum analogs of this functor for a small number of variables. In the classical settings
representations of the type $A$ Hecke algebra corresponds to representations of $U_q(\gl_r)$.
In the one-variable settings this happens with the type $A$ affine Hecke algebra and the affine quantum group.
Finally for two variables there is a correspondence between double affine objects (see \cite{VV}, \cite{Gu}).
In these cases our correspondence  is the classical limit of these constructions.
\end{rem}

\begin{defn}
Let as above $A$ be a commutative finitely-generated algebra with an augmentation $\epsilon$.
Note that $\epsilon$ can be extended as an augmentation of $S^n(A)$ in
the natural way
$$\epsilon(a_1\circ\dots \circ a_n) = \epsilon(a_1) \dots \epsilon(a_n).$$
Introduce the space of
{\em diagonal harmonics}
$$DH_n(A) = \left. A^{\otimes n}\right/ S^n(A)_\epsilon \cdot A^{\otimes n}$$
as a representation of $\sdp{n}{A}$.
\end{defn}

This representation of $\Sigma_n$ is known for $A= \CC[x]$ due to the classical result of Chevalley,
for $A=\CC[x,y]$ due to \cite{H}, for $A = \CC[x,y]/xy$ due to \cite{Ku}.
Note that Conjecture~\ref{triag} below proposes a description of this space for
$A=\CC[x,y,z]$.

\begin{thm}\label{tdual}
For $\g = sl_r$ we have
$$W^A_\epsilon (n\omega_1) \cong  \nabla_n^r\left(DH_n(A)\right).$$
\end{thm}

\begin{proof}
It is shown in \cite{FL2} that $W^A_\epsilon (n\omega_1)$ is the quotient of $S^n(V \otimes A)$ by
the image of the action of $S^n(A)_\epsilon$. As we have  the natural isomorphism
$\nabla_n^r\left(A^{\otimes n} \right) \cong S^n(V \otimes A)$, it is enough to note that
the action of $S^n(A)_\epsilon$ on $S^n(V \otimes A)$ appears from its action on
$A^{\otimes n}$ by multiplication in the dual picture.
\end{proof}

Note that by taking a big enough $r$ we can reconstruct the action of $\Sigma_n$ on $DH_n(A)$
from the action of $\gl_r \otimes 1$ on $W^A_\epsilon (n\omega_1)$. And for $A = \CC[x^1, \dots, x^d]$,
$d < 4$,
the character of $W^A_\epsilon (n\omega_1)$ can be obtained from the following table.

\medskip\noindent
\begin{tabular}{|c|c|c|c|}
\hline
$d$ & $\dim W_{n\omega_1}$ & $\dim W_{n\omega_1}^{k_1\ve_1+ \dots +k_r\ve_r}$ & Reference \cr
\hline\hline
$0$ & $\bin{n+r}{r}$ & $1$   & well known \cr
\hline
$1$ &  &
{\bf Multinomial Coefficients}  & Corollary~\ref{gl} \cr
& $r^n$

&
$\bin{n}{k_1, \dots k_r} = \frac{n!}{k_1!\dots k_r!}$ &  \cr
\hline
$2$ & {\bf Fuss-Catalan Number}  & {\bf ``Fuss-Narayana'' Numbers} &  Theorem~\ref{fsnr}\cr
& $ C_{n+1}^{(r)} = \left.\bin{r(n+1)}{n}\right/ (n+1)$ &
$\left.\bin{n+1}{k_1}\bin{n+1}{k_2}\dots \bin{n+1}{k_r}\right/ (n+1)$
 &  \cr
\hline
$3$

& $\left.r\bin{(2r-1)(n+1)}{n-1}\right/\bin{n+1}{2}$ & $2^{r} (n+1)^{r-2}
\prod\limits_{i=1}^r \frac{\bin{2(n+1)-k_i}{k_i}}{2(n+1)-k_i}$ & Conjecture~\ref{triag} \cr
\hline
\end{tabular}

\section{Explicit results}

\subsection{Binomial case $d=1$}

The following theorem first appeared as a conjecture and was proved for ${\mathfrak sl}_2$ in \cite{CP2},
then the proof was completed for ${\mathfrak sl}_r$ in \cite{CL}, for simply-laced Lie algebra
in \cite{FoLi} and recently H.Nakajima noticed that the general case can be obtained by
combining the results of \cite{BN}, \cite{K1}, \cite{K2}, \cite{N1}, \cite{N2}.

\begin{thm}\label{tns}\cite{CP2}\cite{CL}\cite{FoLi}\cite{N2}
We have
$$W^1 (\lambda) \cong \bigotimes_{i=1}^\rkg W^1(\omega_i)^{\otimes \lambda_i}$$
as $\g\otimes 1$-modules.
\end{thm}

\begin{rem}
Note that the structure of $\g \otimes \CC[x]$-module can be obtained by
replacing the tensor product by the {\em fusion product} from \cite{FL1}.
\end{rem}

\begin{rem}
In the one-variable case Weyl modules can be defined in the quantum group settings (see \cite{CP}) and
the analog of Theorem~\ref{tns} follows for them.
\end{rem}

\begin{crl}\label{pold1}
We have $\dim W^1(\lambda)^{\lambda-\mu}$ is a polynomial in $\lambda_1, \dots, \lambda_\rkg$
of degree $\mu^i$ in the variable $\lambda_i$, $i=1\dots \rkg$.
\end{crl}

\begin{proof}
If $\mu \notin Q^+$ then this dimension is constantly zero. If $\mu=0$ then it is constantly one.

We say that $\mu>\nu$ if $\mu - \nu \in Q^+$. Note that it is a partial order and
that for each $\mu \in Q^+$ there is a finite number of $\nu \in Q^+$ such that
$\mu >\nu$.
Now suppose we know the statement for all
$\nu < \mu$, and let us deduce it for the given $\mu$.

Choose $k$ such that $\lambda_k \ne 0$. Then we have
$W^1(\lambda) \cong W^1(\lambda - \omega_k)\otimes W^1(\omega_k)$.
Note that due to the highest weight condition for each weight $\eta$ of $W^1(\omega_k)$
we have either $\eta = \omega_k$ and $\dim W^1(\omega_k)^\eta = 1$, or $\eta \le \omega_k - \alpha_k$.
So
$$\dim W^1(\lambda)^{\lambda-\mu} =
\sum_{\nu\in Q_+}
\left(\dim W^1(\lambda-\omega_k) ^{\lambda-\omega_k-\nu}\right)
\left(\dim W^1(\omega_k) ^{\omega_k-\mu+\nu}\right)=$$
$$=
\dim W^1(\lambda-\omega_k)^{\lambda-\omega_k-\mu} +
\sum_{{\nu\in Q_+}\atop{\nu + \alpha_k \le \mu}}
\left(\dim W^1(\lambda-\omega_k) ^{\lambda-\omega_k-\nu}\right)
\left(\dim W^1(\omega_k) ^{\omega_k-\mu+\nu}\right).
$$
Therefore by the induction hypothesis, $\dim W^1(\lambda)^{\lambda-\mu} - \dim W^1(\lambda-\omega_k)^{\lambda-\omega_k-\mu}$
is a polynomial of degree $\mu^k -1$ in $\lambda_k$ and degree
$\mu^i$ in $\lambda_i$ for $i \ne k$. So the difference
derivatives of $\dim W^d(\lambda)^{\lambda-\mu}$ are polynomials of desired degree and the statement follows.
\end{proof}

\begin{crl}\label{gl}
For $\g = \gl_r$ we have $\dim W^1(n\omega_1) = r^n$ and
$$\dim W^1(n\omega_1)^{k_1\ve_1+ \dots+ k_r\ve_r} = \frac{n!}{k_1! \dots k_r!}$$
when $k_1 + \dots + k_r =n$, and zero otherwise.
\end{crl}

\begin{proof}
It follows from Theorem~\ref{tns} together with the observation that
$\dim W^1(\omega_1)^\mu = 1$ for $\mu = \ve_i$, $i=1\dots r$, and zero otherwise.
\end{proof}

\subsection{Catalan case $d=2$}

From here fix $\g = \gl_r$. Let us show that the character combinatorics of
two-variable Weyl modules
is pretty similar to the Catalan number combinatorics.

\subsubsection{Parking representations}

Consider the following elementary problem `from a real life'. Imagine we have $n$ cars and
exactly $n$ parking spaces along
the road. Suppose that they are distributed into a sequence of $N$ parking lots with capacities
$m_1, \dots, m_N$. Now let each car has its own preferred lot, so we have a set-theoretical
function $f: \{1,\dots, n\} \to \{1,\dots,N\}$. Now the spaces are distributed as follows.
When $i$-th car appears on the road,
it goes directly to $f(i)$-th lot, then parks there, if it is not full, or at the first available lot
after $f(i)$ otherwise.

\begin{prop}
Each car find its space if and only if we have
$$\left|f^{-1}(\{1,\dots, s\})\right| \ge m_1+ \dots + m_s, \qquad s=1\dots N.$$
\end{prop}

\begin{proof}
Follows by induction on the number of cars. After the first car find the space at the $k$-th lot we consider all the
other cars with the same preference function, and capacities $m_1, \dots, m_k-1, \dots, m_N$.
\end{proof}

Note that the answer does not depend on the order of the cars. It motivates the following
definition.

\begin{defn}\cite{PP}\cite{PiSt}\cite{Yan}
For a vector $\v= (m_1, \dots, m_N)$
set $|\v| = m_1 + \dots + m_N$ and
introduce
the set of {\em generalized parking functions}
$$PF(\v) = \{ f : \{1,\dots, |\v|\} \to \NN \ | \
\left|f^{-1}(\{1,\dots, s\})\right| \ge m_1 + \dots + m_s,\  1\le s \le N  \}$$
with the action of $\Sigma_{|\v|}$ by permutations of arguments.
\end{defn}

\begin{defn}
Introduce the representation $\cpf(\v)$ of $\Sigma_{|\v|}$ as the space with basis
$PF(\v)$ and the corresponding action of the symmetric group.
\end{defn}

\begin{rem}
It is shown in \cite{H} that  we have
$DH_n(\CC[x,y]) \cong \cpf((n)^t)\otimes \sgr$ as representations of $\Sigma_n$, where $\sgr$ is
the one-dimensional sign representation of $\Sigma_n$.
\end{rem}

Note that the whole set of functions from $\{1, \dots, n\}$ to $\NN$
enumerates the monomial basis of $\CC[t_1,\dots, t_n]\cong \CC[t]^{\otimes n}$,
namely each $f$ corresponds to $t_1^{f(1)-1} \dots t_n^{f(n)-1}$. So
if $A$ acts on $\CC[t]$ not increasing the degree of polynomials
then $\sdp{|\v|}{A}$ acts on $\cpf(\v)$.
Note that for this end $A$ is not necessarily commutative.

Let $\da = \CC\left<X,Y\right>$ be the associative algebra with the relation
$XY-YX = X$. Then $\da$ acts on $\CC[t]$ as follows:
$X$ acts as $\prt$, $Y$ acts as $t\prt$. As this action does not increase the
degree, the wreath product $\sdp{|\v|}{\da}$ acts on $\cpf(\v)$.
Note that $\da$ is a PBW-algebra, namely it is a flat deformation of the commutative
algebra $\CC[x,y]$, and we expect that the Weyl modules can be deformed to
the parking function representations.

\begin{thm} \cite{FL3}
For a partition $\xi$
there is a filtration on $\CC\left<X,Y\right>$-module
$\nabla_n^r(\cpf(\xi^t)\otimes \sgr)$
whose adjoint graded factor
is a quotient of $W^2(\xi)$.
\end{thm}

\begin{proof}
This filtration is obtained as follows: first introduce the filtration on $\CC\left<X,Y\right>$ such that
$F^m\CC\left<X,Y\right>$ is a linear span of $X^iY^j$ with $j \le m$. This filtration is Fourier dual
to the natural degree filtration (and that is why in \cite{FL3} we use a less convenient dual notation).
Then it defines a filtration on $\gl_r \otimes  \CC\left<X,Y\right>$, thus on
$U(\gl_r \otimes  \CC\left<X,Y\right>)$ and therefore on each cyclic module.

Let $v_1, \dots, v_r$ be the standard basis of $V$. Now introduce
$$v_\xi = \bigwedge_{i=1}^r \bigwedge_{j=0}^{\xi_i-1} v_i\otimes t^{j} \in \nabla_n^r(\cpf(\xi^t)\otimes \sgr)$$
and note that this vector is cyclic. Set
$$F^m \nabla_n^r(\cpf(\xi^t)\otimes \sgr) = F^m U(\gl_r \otimes  \CC\left<X,Y\right>) v_\xi,$$
then $\gr \left(\nabla_n^r(\cpf(\xi^t)\otimes \sgr)\right)$
is a representation of $\gr \left(\gl_r \otimes  \CC\left<X,Y\right>\right)
\cong \gl_r \otimes  \CC[x,y]$
and it is shown in \cite{FL3} that the image of $v_\xi$ is the highest weight vector. So this representation
is a quotient of the Weyl module.
\end{proof}

\begin{crl}
There is an inclusion $\nabla_n^r(\cpf(\xi^t)\otimes \sgr)$ to $W^2(\xi)$
as representations of $\gl_r \otimes 1$.
Moreover, for $\xi = (n)$ we have $W^2(\xi) \cong \nabla_n^r(\cpf(\xi^t)\otimes \sgr)$.
\end{crl}

\begin{proof}
First statement is immediate due to semisimplicity of $\gl_r$, second statement follows from \cite{H}
and Theorem~\ref{tdual}.
\end{proof}

\begin{rem}
Note that the main conjecture of \cite{FL3} that these representations are indeed isomorphic.
\end{rem}

\begin{prop}\label{xis}
We have
$\dim \nabla_n^r(\cpf(\v)\otimes \sgr)^{k_1\ve_1+ \dots+ k_r\ve_r}$ is equal to the number of $|\v|$-elements subsets
$H \subset \{1,\dots, r\} \times \NN$ such that
\begin{equation}\label{xigrad}
|H\cap \{i\} \times \NN | = k_i, \ \ i=1\dots r;
\end{equation}
\begin{equation}\label{xiparks}
|H\cap \{1,\dots, r\} \times \{1,\dots, s\}| \ge m_1 + \dots + m_s,\ \ 1 \le s \le N.
\end{equation}
\end{prop}

\begin{proof}
Each subset $H$ corresponds to a vector
\begin{equation}\label{xism}
v_H = \bigwedge_{i\times j \in H} v_i \otimes t^{j-1},
\end{equation}
and it is shown in \cite{FL3} that the vectors $v_H$ for $H$ under condition~\eqref{xiparks}
span the space $\nabla_n^r(\cpf(\v)\otimes \sgr)$. The condition~\eqref{xigrad} fixes the weight of
these vectors.
\end{proof}

The following definition helps us further to work with such functions.

\begin{defn}\label{bp}
For a given $\v$
we say that $1\le s < N$ is a {\em boundary point} for $f \in  PF(\v)$ if
$$\left|f^{-1}(\{1,\dots, s\})\right| = m_1 + \dots + m_s.$$

Similarly for a given $\v$ and a subset $H$ under condition~\eqref{xiparks}
we say that an integer $s$ is a {\em boundary point} if
$$|H\cap \{1,\dots, r\} \times \{1,\dots, s\}| = m_1 + \dots + m_s.$$
\end{defn}

\begin{rem}
Let $\CC[x,y]_q$ be the associative algebra of $q$-polynomials, that is the algebra with relation
$xy-qyx=0$. Then $\CC[x,y]_q$ acts on $\CC[t]$ not increasing the degree, namely $x$ acts by $\prt$
and $y$ acts by $q^{t\prt}$ (that is $y\cdot t^k = q^k t^k$).
So  $\sdp{|\v|}{\CC[x,y]_q}$ acts on $\cpf(\v)$
and $\gl_r \otimes \CC[x,y]_q$ acts on $\nabla_n^r(\cpf(\v))$.
\end{rem}

\begin{rem}
For $\xi = (n)$ the space of parking function can be quantized to a representation of the double
affine Hecke algebra (see \cite{Ch}, \cite{G}), so
Weyl modules can be quantized to a representation of the corresponding toroidal quantum group.
Unfortunately, for other $\xi$ there are no representations of these quantum algebras with
suitable characters.
\end{rem}

\subsubsection{Explicit formula for weight spaces}

The following lemma is a parking function analog of the Reni lemma about Catalan numbers (see \cite{S}).

\begin{lem}\label{Reni}
Let $f : \{1, \dots, n\} \to  \{1, \dots, n+1\}$ be an arbitrary function. Then there exists a unique cyclic permutation
$$\sigma_k  = (1\,2\dots n+1)^k = \left(
\begin{array}{cccccccc}
1 & 2   & & \dots & & \dots & & n+1 \cr
k & k+1 & \dots & n+1   & 1 & 2 & \dots & k-1 \cr
\end{array}\right)
$$
such that
$$|(\sigma_k\circ f)^{-1}(\{1, \dots, s\})| \ge s \quad \mbox{for} \ 1 \le s \le n.$$
\end{lem}

\begin{proof}
For a given $f$ introduce $F: \{1, \dots, n+1\} \to \NN$ by
$$F(s) = |f^{-1}(\{1, \dots, s\})| - \frac{n}{n+1}s.$$
Then note that the values of $F$ are distinct, so
it has a unique minimum. Now suppose that  $F(s)$
is minimal at $s=s_0$ and  note that
the only choice to satisfy the condition is $k = n+2 - s_0$,
so the permuted numeration starts precisely after this minimum.
\end{proof}

\begin{thm}\label{fsnr}
For $\g = \gl_r$ we have
$\dim W^2(n\omega_1) = \left.\bin{r(n+1)}{n}\right/ (n+1)$ and
$$\dim W^2(n\omega_1)^{k_1\ve_1+ \dots+ k_r\ve_r} = \frac{\bin{n+1}{k_1} \dots \bin{n+1}{k_r}}{n+1}$$
when $k_1 + \dots + k_r =n$, and zero otherwise.
\end{thm}

\begin{proof}
Note that the permutation group $\Sigma_{n+1}$ acts
on $\{1, \dots, r\} \times \{1,\dots n+1\}$ mapping $i\times j$ to $i \times \sigma(j)$.
So it acts on the subsets, preserving condition~\eqref{xigrad}.

Applying the Schur-Weyl duality to the statement of Lemma~\ref{Reni}, for any subset
$H \subset \{1, \dots, r\} \times \{1,\dots, n+1\}$
we obtain the cyclic permutation
$\sigma_k$
such that
$$\left|\sigma_k(H)\cap \{1, \dots, r\} \times \{1,\dots ,s\}\right|
\ge s \quad \mbox{for} \ 1\le s \le n.$$

So the number of $n$-element subsets under condition~\eqref{xiparks}
is the number of all
$n$-element subsets divided by the number of cyclic permutations $n+1$, and
the number of subsets under
 conditions~\eqref{xigrad} and~\eqref{xiparks} is the number
of all subsets under condition~\eqref{xigrad} divided by the number of cyclic permutations $n+1$.
\end{proof}

\begin{rem}
The same arguments produce a pretty  answer for $\xi = (n,\dots,n,1)$ and $\xi = (n,\dots, n,n-1)$,
but unfortunately for the more natural case $\xi =  (n,\dots,n)$ the corresponding polynomial is not reducible
over $\QQ$.
\end{rem}

\subsubsection{Recurrence relation}

Let us present a quadratic relation for characters of  Weyl modules
that generalizes the recurrence relation for Calatan numbers.

\begin{defn}
Introduce a symmetric function $c_\xi = \ch \nabla^r_n(\cpf(\xi^t)\otimes \sgr)$, that is the Frobenius
characteristic map of $\cpf(\xi^t)\otimes \sgr$.
\end{defn}

\begin{thm}\label{rec}
\begin{enumerate}
\item
We have
$$c_{\xi} = \sum_{i=\xi_r}^{\xi_1-1}\, c_{\left(\xi_{> i}\right)'}\, \cdot \, c_{\xi_{\le i}},$$
where
$(\xi_{> i})_j = \max(\xi_j-i,0)$, $(\xi_{\le i})_j = \min (\xi_j,i)$,
$$
(\xi')^t_j =
\begin{cases}
\xi^t_j +1 & 1 = j < \xi_1 \cr
\xi^t_j -1 & 1< j= \xi_1 \cr
 \xi^t_j & \mbox{otherwise}
\end{cases}
$$
\nopagebreak\smallskip

\begin{center}
\centerline{\psfig{file=rec.eps,width=7cm}}
\end{center}

\item Suppose for a given $k$ we have $\xi_k - \xi_{k+1}>1$. Then
$$c_{\xi} - c_{\xi-\alpha_k} = c_{\xi_{>(\xi_k-1)}} c_{\xi_{\le (\xi_k-1)}} +
\sum_{i=\xi_{k+1}+1}^{\xi_k-2}  c_{(\xi_{> i}) - \alpha_k} c_{\xi_{\le i}}.$$
\end{enumerate}
\end{thm}

\begin{proof}
For (i) note that $PF((\xi')^t)\subset PF(\xi^t)$ is the subset of parking functions
without boundary points (see Definition~\ref{bp}), so the same is true in the dual picture.

Now for a subset $H$ let $i$ be the biggest boundary point.
Splitting such $H$ into two parts:
$$H_{>i} = H\cap \{1, \dots, r\} \times \{i+1, \dots\}, \qquad
H_{\le i}  = H\cap \{1, \dots, r\} \times \{1, \dots, i\}$$
and multiplying their contributions to character, we identify both sides of the equality.

Proof of (ii) is similar, here we notice only boundary points on the interval
$(\xi_{k+1}, \xi_k)$ and subtracting $\alpha_k$ to avoid them.
\end{proof}

\begin{crl}\label{pol2}
We have $\dim \nabla^r_n(\cpf(\xi^t)\otimes \sgr)^{\xi-\mu}$ is a polynomial in $\xi_1, \dots, \xi_r$
of degree $2\mu^i$ in $\xi_i-\xi_{i+1}$.
\end{crl}

\begin{proof}
Proof is similar to the proof of Corollary~\ref{pold1}, but now we use Theorem~\ref{rec}~(i)
to show that this function is a polynomial and Theorem~\ref{rec}~(ii) to calculate the degree.

Note that the structure of Weyl module depends only on the restriction of $\xi$ to
${\mathfrak sl}_r$.
So for $\lambda = \sum \lambda_i \omega_i$ let $P_\mu(\lambda) = \dim \nabla^r_n(\cpf(\xi^t)\otimes \sgr)^{\xi-\mu}$ with $\xi_i - \xi_{i+1} = \lambda_i$.
Suppose that $\lambda = \lambda_k \omega_k + \dots + \lambda_{r-1}\omega_{r-1}$ with
$\lambda_k > 0$.
Then by Theorem~\ref{rec}~(i) we have
\begin{equation}\label{recwt}
P_\mu(\lambda) =
\sum_{\nu \in Q^+} P_{\mu,\nu}(\lambda) \quad \mbox{with}
\end{equation}
\begin{align*}
P_{\mu,\nu}(\lambda) \qquad = \qquad
P_\nu(\omega_k) &\cdot  P_{\mu-\nu}(\lambda-\omega_k) \quad + \\
+ \sum_{i=2}^{\lambda_k} P_{\nu-\alpha_k}(i\omega_k-\alpha_k)
&\cdot P_{\mu-\nu}(\lambda - i\omega_k)\quad + \\
+ \sum_{j=k+1}^{r-1} \sum_{i=1}^{\lambda_{j}}
P_{\nu-\alpha_{k\dots j}} (\lambda_{k\dots j-1} + i\omega_j - \alpha_{k\dots j})
&\cdot P_{\mu-\nu}(\lambda_{j\dots r-1}-i\omega_j),
\end{align*}
where
$\alpha_{a\dots b} = \alpha_a + \dots + \alpha_b$, $\lambda_{a\dots b} =
\lambda_a \omega_a + \dots + \lambda_b\omega_b$ for $1 \le a \le b <r$.

Let us perform an induction on $\mu$: suppose we know the
statement for all $\nu<\mu$.  Note that if $P_1$ and $P_2$ are polynomials
then $\sum_{i=1}^n P_1(i) P_2(n-i)$ is a polynomial in $n$.
So all the sums in the right hand side except the first summand for
$\nu=0$ are polynomials by the induction hypothesis.
Therefore
the difference derivative
$P_\mu(\lambda) -
P_{\mu} (\lambda-\omega_k)$
is a polynomial in $\lambda_k, \dots, \lambda_{r-1}$
so by induction on $r-k$ we have $P_\mu$ is a polynomial.

To calculate the degree of $P_\mu$ in $\lambda_k$ we can now
suppose that  $\xi_k - \xi_{k+1}= \lambda_k > 1$ and use
 Theorem~\ref{rec}~(ii)
Here we have
\begin{equation}\label{recwt2}
P_\mu(\lambda) = P_{\mu-\alpha_k}(\lambda - \alpha_k) + \sum_{\nu \in Q^+} P_{\mu,\nu}^k(\lambda)
 \quad \mbox{with}
\end{equation}
\begin{align*}
P_{\mu,\nu}^k(\lambda) \qquad = \qquad
P_\nu(\lambda_{1\dots k-1} + \omega_k) &\cdot P_{\mu-\nu}(\lambda_{k\dots r-1} - \omega_k) \quad+\\
+ \sum_{i=2}^{\lambda_k-1} P_{\nu-\alpha_k} (\lambda_{1\dots k-1} + i\omega_k-\alpha_k)
&\cdot P_{\mu-\nu}(\lambda_{k\dots r-1} - i\omega_k).
\end{align*}
Note that if $P(i)$ is a polynomial of degree $m$ then $\sum_{i=1}^n P(i)$ is a polynomial
in $n$ of degree $m+1$. So in the case $\lambda_{1\dots k-1} = 0$ we have from~\eqref{recwt2}
by induction of $\mu$ that
$P_\mu(\lambda) - P_\mu (\lambda-\omega_k)$  has degree $2(\mu^k-1)+1$, so the statement follows.

Now general case follows from~\eqref{recwt2} by induction on $\mu$ as
$P_\mu(\lambda) -  P_{\mu}(\lambda_{k\dots r-1} - \omega_k)$ has a smaller degree.
\end{proof}

\subsection{Hyper-Catalan case $d=3$}

\begin{cnj}\label{triag} Let $\g = \gl_r$.
\begin{enumerate}
\item We have
$$\dim W^3(n\omega_1) = r\prod_{i=1}^{n-1} \frac{2(n+1)r - n -i}{n+2-i} =
\left.r\bin{(2r-1)(n+1)}{n-1}\right/\bin{n+1}{2}.$$
\item We have
$$\dim W^3(n\omega_1)^{k_1\ve_1+ \dots+ k_r\ve_r} = 2^{r} (n+1)^{r-2}
\prod_{i=1}^r \frac{\bin{2(n+1)-k_i}{k_i}}{2(n+1)-k_i}$$
when $k_1 + \dots + k_r =n$, and zero otherwise.
\end{enumerate}
\end{cnj}

\begin{rem}
This conjecture supports the conjecture of \cite{H1} that
$$\dim DH_n(\CC[x,y,z]) = 2^n (n+1)^{n-2}.$$
Moreover, it follows that the trace of a permutation  $\sigma \in \Sigma_n$  consisting of cycles of length $k_1+1, \dots, k_m+1$
on $DH_n(\CC[x,y,z])$ is equal to
$$2^{n-k_1- \dots -k_m} (n+1)^{n-2-k_1- \dots -k_m} \prod_{i=1}^m \left[{2k_i+1}\atop{k_i+1}\right].$$

\end{rem}

\subsection{Singular cases}

First let us mention the result of \cite{Ku} on diagonal coinvariants for a double point, that is
$A = \CC[x,y]/xy\CC[x,y]$ and $\epsilon(P) = P(0,0)$.

It is shown there that for $n>0$ we have
$$W^A_\epsilon(n\omega_1) \cong V^{\otimes n} \oplus (n-1)\left(\wedge^2 V \otimes V^{\otimes n-2}\right).$$
So for $n\ge 1$ the dimensions of corresponding weight spaces are values of  a polynomial.

Another observation is related to $A=\CC[x,y]/x^l\CC[x,y]$ and $\epsilon(P) = P(0,0)$.

\begin{defn}
Introduce $PF(\v)^{(l)}$ as the subset of $PF(\v)$ formed by such functions
that on any interval $[s+1,s+l]$,
$0\le s <  N-l$, there is at least one boundary point.
\end{defn}

\begin{prop}
The quotient of $\nabla_n^r\left(\cpf(\v)\otimes \sgr\right)$
by the action of the Lie algebra $\gl_r \otimes X^l\CC\left<X,Y\right>$ is isomorphic to
 $\nabla_n^r\left(\cpf(\v)^{(l)}\otimes \sgr\right)$.
\end{prop}

\begin{proof}
By $\CC[t]^k \subset \CC[t]$ denote the subspace of polynomials of degree less or equal $k$.
Note that $\CC\left<X,Y\right>$ acts on $\CC[t]^k$ and that the image of
$X^l\CC\left<X,Y\right>$ in ${\rm End} \left(\CC[t]^k\right)$ consists of all operators,
decreasing degree by at least $l$, so it is a linear span of
the operators $M_{ab}$ for $a-b \ge l$
sending $t^a$ to $t^b$ and $t^i$ to zero for $i \ne a$.

By $E_{ij} \in \gl_r$ denote the matrix units. Then $E_{ij} \otimes M_{ab}$
acts on monomials~\eqref{xism} sending $v_H$ to
$v_{H'}$ where
$$ H' = (H \setminus i\times a)\cup j \times b,$$
so the result contains no boundary points on the interval $[b,a-1]$.
And vise versa, if $a$ and $b-1$ are two neighbor boundary points of $H$ and $a>b$ then
there exists $i$ and $j$ such that $j \otimes b \in H$, $i \otimes a \not\in H$
and for
$$H^\circ = (H \setminus j\times b)\cup i \times a$$
we have $v_{H^\circ} \in \nabla_n^r\left(\cpf(\v)\otimes \sgr\right)$.
So $v_H = (E_{ij} \otimes M_{ab}) v_{H^\circ}$.
\end{proof}

\begin{thm}
Let  $\g = \gl_r$  $A=\CC[x,y]/x^l\CC[x,y]$, $\epsilon(P) = P(0,0)$.
Then there exists a filtration on
$\nabla_n^r\left(\cpf(\xi^t)^{(l)}\otimes \sgr\right)$
such that
$\gr \nabla_n^r\left(\cpf(\xi^t)^{(l)}\otimes \sgr\right)$
is a quotient of $W^A_\epsilon(\xi)$.

\end{thm}

\begin{proof}
Note that the filtration on $\nabla_n^r\left(\cpf(\xi^t)\otimes \sgr\right)$
produces a filtration on the quotient $\nabla_n^r\left(\cpf(\xi^t)^{(l)}\otimes \sgr\right)$.
Then
$\gr \nabla_n^r\left(\cpf(\xi^t)^{(l)}\otimes \sgr\right)$ is a highest weight representation
of $\gl_r \otimes \CC[x,y]$, where $\gl_r \otimes x^l \CC[x,y]$ acts by zero, so it is
a highest weight representation of $\gl_r \otimes A$, hence a quotient of $W^A_\epsilon(\xi)$.
\end{proof}

\begin{cnj}
We have $W^A_\epsilon(\xi)  \cong  \gr \nabla_n^r\left(\cpf(\xi^t)^{(l)}\otimes \sgr\right)$.
\end{cnj}

\begin{thm}\label{recfin}
Let $c_{\xi}^{(l)} = \ch \nabla_n^r\left(\cpf(\xi^t)^{(l)}\otimes \sgr\right)$. Then we have
$$c_{\xi}^{(l)} = \sum_{i=\max(\xi_1-l,\xi_r)}^{\xi_1-1} c_{\xi_{> i}'}^{(l)} c_{\xi_{\le i}}^{(l)}
=  \sum_{i=\max(\xi_1-l,\xi_r)}^{\xi_1-1} c_{\xi_{> i}'} c_{\xi_{\le i}}^{(l)}.$$
\end{thm}

\begin{proof}
Similar to proof of Theorem~\ref{rec} (i), just note that the last boundary point is not less than $\xi_1-l$.
Here $c_{\xi_{> i}'}^{(l)} = c_{\xi_{> i}'}$ because $\xi_{> i}'$ has $l$ rows or less.
\end{proof}

\begin{rem}
Note that there are no analog of  Theorem~\ref{rec} (ii) for this case.
\end{rem}

\begin{ex}
Set $r=2$. Let $C_n$ be the usual Catalan numbers and
$$C_{n+1}^{(l)} = \dim \nabla_n^2\left(\cpf((n)^t)^{(l)}\otimes \sgr\right).$$
Then we have $C_n^{(l)} = C_n$ for $1\le n \le l+1$ and
$$C_{n+1}^{(l)} = 2C_{n}^{(l)} + C_{n-1}^{(l)} + 2C_{n-2}^{(l)} + 5 C_{n-3}^{(l)}
+ \dots + C_{l-1} C_{n-l+1}^{(l)}.$$
\end{ex}

\begin{crl}
Fix $\mu$ and suppose that $\xi_i - \xi_{i-1} > l\mu^i+l$ for all $i$.
Then the dimension of $\nabla_n^r\left(\cpf(\xi^t)^{(l)}\otimes \sgr\right)^{\xi-\mu}$ is a polynomial in
 $\xi_1, \dots, \xi_r$ of degree $\mu^i$ in $\xi_i-\xi_{i+1}$.
\end{crl}

\begin{proof}
Proof is based on Theorem~\ref{recfin} and is similar to the proof of
Corollary~\ref{pol2}.
Let $P_\mu^{(l)}(\lambda) = \dim \nabla^r_n(\cpf(\xi^t)^{(l)}\otimes \sgr)^{\xi-\mu}$
with $\xi_i - \xi_{i+1} = \lambda_i$.
Then it satisfy the truncated analog of relation~\eqref{recwt}
with only first $l$ summands in the right hand side for $P_{\mu,\nu}$.

First step is to show that $P_\mu^{(l)}(\lambda)$
is a polynomial in $\lambda_k$ of degree $\mu^k$
when $\lambda = \lambda_k\omega_k + \dots \lambda_{r-1}\omega_{r-1}$
and $\lambda_k > l\mu^k+l$ for $i \ge k$. Here the relation is simplified
as
$$P_\mu^{(l)}(\lambda) =
\sum_{\nu \in Q^+} P_{\mu,\nu}^{(l)}(\lambda) \quad \mbox{with}
$$
$$
P_{\mu,\nu}^{(l)}(\lambda) =
P_\nu^{(l)}(\omega_k) \cdot  P_{\mu-\nu}^{(l)}(\lambda-\omega_k) +
\sum_{i=2}^{l} P_{\nu-\alpha_k}^{(l)}(i\omega_k-\alpha_k)
\cdot P_{\mu-\nu}^{(l)}(\lambda - i\omega_k),
$$
so the statement follows by induction on $\mu$.

Second step is to
prove the same statement for an arbitrary
$\lambda$ with $\lambda_k > l\mu^k+l$
by induction on $\lambda_1 + \dots + \lambda_{k-1}$ using the truncated version of~\eqref{recwt}.
\end{proof}


TITLE: Learning C* Algebra via Problems
QUESTION [0 upvotes]: I have been reading the book "C*-Algebras and Operator Theory" by G.J.Murphy. I have almost finished reading the first two chapters. I was wondering if someone could suggest me some book/Lecture notes/source with some good problems to go over. Murphy's book has some. But they are at the end of the chapter and very few.
Thanks for the help!!

REPLY [2 votes]: I think the book An Introduction to Operator Algebras written by Kehe Zhu is very well suited to learn about $C^*$-algebras actively. The book consists of 26 (very) short chapters and each of these chapters contains several well-chosen exercises.


TITLE: Proving controllability using eigenvectors and eigenvalues
QUESTION [0 upvotes]: Considering the pair $(A,b)$ where $A∈R^{n×n}$ and $b ∈R^{n×1}$ how can I show that if more than one linearly independent eigenvector can be associated with a single eigenvalue of $A$, then $(A,b)$ is uncontrollable.
My initial thought was to use the PBH test but the proof isn't clear to me.

REPLY [2 votes]: If more than one linearly independent eigenvector can be associated with a single eigenvalue $\lambda$ of $A$ then the rank of $\lambda\,I-A$ drops by more than one, since those eigenvectors span the null space of $\lambda\,I-A$. However, when using the PBH test and that $b$ has only one column it can be noted that $b$ can at most increase the rank of $[\lambda\,I-A, b]$ by one, which implies rank-deficiency and thus uncontrollability.


TITLE: If $f$ is measurable from $E$ to $G$, and if $g=f$ almost everywhere, must $g$ be measurable?
QUESTION [2 upvotes]: Even if this is not generally true, are there special cases of $E$ and $G$ for which this is true?

REPLY [2 votes]: In general, this is true when the measure is complete.


TITLE: divisors on relative projective space
QUESTION [1 upvotes]: Let $X$ be a separated irreducible variety which is regular in codimension $1$ (I want to talk about Weil divisors), and write $\mathbb{P}^n_X = \mathbb{P}^n_k \times X$ for the projective space over $X$. Now if $Y \subset \mathbb{P}^n_X$ is a prime Weil divisor, i.e. an irreducible closed subvariety of codimension $1$, we can consider the generic fiber $Y_K$ (here $K$ is the function field of $X$).
Since closed embeddings are stable under base change, $Y_K$ is a closed subscheme of $\mathbb{P}^n_K$, and clearly $Y_K \neq \varnothing$ if and only if the projection of $Y$ to $X$ is dense. My question: in this case ($Y_K \neq \varnothing$), can we conclude that $Y_K$ is an integral subvariety of $\mathbb{P}^n_K$ of codimension $1$? Hopefully this is true, since then we will have a well-defined degree map $\text{Pic } \mathbb{P}^n_X \to \mathbb{Z}$.
Edit: Maybe I should be asking a more general question: if $Y \to X$ is a dominant morphism of (separated, if necessary) irreducible varieties and $K$ denotes the function field of $X$, is $Y_K = Y \times_X \text{Spec } K$ an irreducible variety over $K$? The codimension $1$ assertion seems easier, since it is local, but I don't know a good criterion for irreducibility.

REPLY [1 votes]: I guess you mean by "irreducible varieties" integral scheme of finite type. Then the anwser is yes.
First the local rings of $Y_K$ are localizations of local rings of $Y$ (can replace $X$ by an affine dense open subvariety, then $K$ is a localization of $O(X)$), so $Y_K$ is locally integral.
Second, the generic point of $Y$ belongs to $Y_K$, so it is also the generic point of $Y_K$. This implies that $Y_K$ is irreducible.
Edit Codimension of $Y_K$. Let $d=\dim X$. Then $\dim Y=n+d-1$, and we have
$$\dim Y_K=\dim Y - \dim X=n-1$$
(dimension formula for irreducible algebraic varieties).
So $Y_K$ is of codimension $1$ in $\mathbb P^n_K$.