#  Combinatorics of irreducible characters for Lie superalgebra
$\mathfrak{gl}(m,n)$

A.N. Sergeev  Department of Mathematics, Saratov State University,
Astrakhanskaya 83, Saratov 410012 Russian Federation<EMAIL_ADDRESS>
###### Abstract.

In this paper we give a new formula for characters of finite dimensional
irreducible $\mathfrak{gl}(m,n)$ modules. We use two main ingredients: Su-
Zhang formula and Brion’s theorem.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Su-Zhang formula
  4. 4 Graphs and polyhedra
  5. 5 Character formula

##  1\. Introduction

In this paper we prove a new formula for characters of irreducible finite
dimensional $\mathfrak{gl}(m,n)$ modules. Our formula is the same type as Su-
Zhang formula ([10]). Namely Su and Zhang proved that for irreducible module
$L(\lambda)$ the following equality holds true

$ch\,L(\lambda)=\sum_{\mu\in\mathcal{C}^{\text{Trun}}_{\lambda}}(-1)^{|\lambda-\mu|}ch\,K(\mu)$

where $|\lambda-\mu|$ is some integer number,
$\mathcal{C}^{\text{Trun}}_{\lambda}$ is the subset in $\mathbb{Z}^{m+n}$ and
$K(\mu)$ is Kac module. Using this formula they proved a Weyl type character
formula. First they represent $\mathcal{C}_{\lambda}^{\text{Trun}}$ as the
union of fundamental domains under the symmetric group. Then for every
fundamental domain they (implicitly) apply Brion’s theorem.

In our approach instead of using weights we use the language of weight
diagrams [4]. Then we apply Brion’s theorem to the polyhedron for which
$\mathcal{C}^{\text{Trun}}_{\lambda}$ is the set of integer points. The Su-
Zhang formula contains up to $r!2^{r}$ summands where $r$ is the degree of
atypicality. Our formula always contains $2^{r-s}$ summands where s is the
number of connected components of the weight diagram.

Our main result can be formulated in the following way (see for details
section 5).

######  Theorem 1.1.

Let $L(\chi)$ be a finite dimensional irreducible module over Lie superalgebra
$\mathfrak{gl}(m,n)$ with the highest weight $\chi$. Then the following
equality holds true

$D\,chL(\chi)=\sum_{w\in
W_{0}}\varepsilon(w)w\left(\frac{e^{\chi+\rho}\theta(\Gamma_{f},-e^{\alpha_{1}},\dots,-e^{\alpha_{r}})}{\prod_{\alpha\in
S_{\chi}}(1+e^{-\alpha})}\right)$

where $\Gamma_{f}$ is the graph which is explicitly constructed from the
weight diagram $f$ of $\chi$, $S_{\chi}=\\{\alpha_{1}<\dots<\alpha_{r}\\}$ is
a maximal orthogonal set of atypical roots and
$\theta(\Gamma_{f},t_{1},\dots,t_{r})$ is a Laurent polynomial given by
explicit formula.

We also give one more formula (see the end of section 5) of the same type with
the less number of summands. We were motivated mainly by papers [2], [8],
[10].

##  2\. Preliminaries

Let us remind that the Lie superalgebra $\mathfrak{gl}(m,n)$ is the Lie
superalgebra of the linear transformations of a $\mathbb{Z}_{2}$ graded vector
space $V=V_{0}\oplus V_{1}$ ($V$ is also called the standard representation of
$\mathfrak{g}$). We have

$\mathfrak{g}_{\bar{0}}=\mathfrak{gl}(m)\oplus\mathfrak{gl}(n),\quad\mathfrak{g}_{\bar{1}}=V_{0}\otimes
V_{1}^{*}\oplus V_{1}\otimes V_{0}^{*}.$

We also have $\mathbb{Z}$ graded decomposition
$\mathfrak{g}=\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$
where

$\mathfrak{g}_{-1}=V_{1}\otimes V^{*}_{0},\,\,\mathfrak{g}_{1}=V_{0}\otimes
V^{*}_{1}\,.$

Let us fix bases in $V_{0}=<e_{1},\dots,e_{m}>$ and
$V_{1}=<f_{1},\dots,f_{n}>$ respectively. Let $\mathfrak{b}$ be the subalgebra
of upper triangular matrix in $\mathfrak{gl}(m,n)$ and $\mathfrak{k}$ be the
subalgebra of diagonal matrix in $\mathfrak{gl}(m,n)$ in the above basis. By
$\varepsilon_{1},\dots,\varepsilon_{m},\delta_{1},\dots,\delta_{n}$ we will
denote the weights of standard representation with respect to $\mathfrak{k}$.
The corresponding system of positive roots $R^{+}=R^{+}_{0}\cup R^{+}_{1}$ of
$\mathfrak{gl}(m,n)$ can be described in the following way

$R^{+}_{0}=\\{\varepsilon_{i}-\varepsilon_{j}\,:\,1\leq i<j\leq
m\,:\,\delta_{k}-\delta_{l},\,1\leq k<l\leq n\\}$
$R_{1}^{+}=\\{\varepsilon_{i}-\delta_{k},\,1\leq i\leq m,\,1\leq k\leq n\\}.$

Let also

$P=\\{\chi=\lambda_{1}\varepsilon_{1}+\dots+\lambda_{m}\varepsilon_{m}+\mu_{1}\delta_{1}+\dots+\mu_{n}\delta_{n},\mid
n_{i},m_{j}\in\mathbb{Z}\\}$

be the weight lattice and

$P^{+}=\\{\chi\in P\mid\lambda_{i}-\lambda_{j}\geq
0,\,i<j\,:\mu_{k}-\mu_{l}\geq 0,\,k<l\\}$

be the set of highest weights.

We will use the following parity on the weight lattice due to C. Gruson and V.
Serganova [7] and Brundan and Stroppel [4] by saying that $\varepsilon_{i}$
(resp. $\delta_{j}$) is even (resp. odd). It is easy to check that every
finite dimensional module $L$ can be represented in the form

$L=L^{+}\oplus L^{-}$

where $L^{+}$ is the submodule of $L$ in which weight space has the same
parity as the corresponding weight and $L^{-}$ is the submodule in which the
parities differ. We should note that this construction is a particular case of
Deligne construction category $Rep(G,z)$ from the paper [6] for $G=GL(m,n)$
and $z=diag(\underbrace{1,\dots,1}_{m},\underbrace{-1,\dots,-1}_{n}).$

Let us denote by $\mathcal{F}$ the category of finite dimensional modules over
$\mathfrak{gl}(n,m)$ such that every module in $\mathcal{F}$ is semisimple
over Cartan subalgebra $\mathfrak{k}$ and and all its weights are in $P$. By
$K(\mathcal{F})$ we will denote the quotient of the Grothendieck ring of
$\mathcal{F}$ by the relation $[L]-[\Pi(L)]=0$ where $\Pi(L)$ is the module
with the shifted parity $\Pi(L)_{0}=L_{1},\Pi(L)_{1}=L_{0}$ and
$x*v=(-1)^{p(x)}xv,x\in\mathfrak{gl}(m,n).$ For every $L\in\mathcal{F}$ we can
define

$ch\,L=\sum_{\chi}\dim L_{\chi}e^{\chi}$

where the sum is taken over all weights of $L$. It is easy to see that $ch\,L$
is well defined function on $K(\mathcal{F})$.

The ring $K(\mathcal{F})$ can be describe explicitly in the following way. Let

$P_{m,n}=\mathbb{Z}[x_{1}^{\pm 1},\dots,x_{m}^{\pm 1},\,y_{1}^{\pm
1},\dots,y_{n}^{\pm 1}]$

be the ring of Laurent polynomials in variables $x_{1},\dots,x_{m}$ and
$y_{1},\dots,y_{n}.$

If we set $x_{i}=e^{\varepsilon_{i}},\,y_{j}=e^{\delta_{j}}$ then we get a
character map

$ch:K(\mathcal{F})\longrightarrow P_{m,n}.$

Let also

$\Lambda^{\pm}_{m,n}=\\{f\in P_{m,n}^{S_{m}\times S_{n}}\mid
x_{i}\frac{\partial f}{\partial x_{i}}+y_{j}\frac{\partial f}{\partial
y_{j}}\in(x_{i}+y_{j})\\}$

be the subring of $P_{m.n}$ of supersymmetric Laurent polynomials.

######  Theorem 2.1.

[11] The ring $K(\mathcal{F})$ is isomorphic to the ring $\Lambda^{\pm}_{m,n}$
under the character map.

######  Remark 2.2.

Actually in the paper [11] a slightly different versions of Grothen-dieck ring
and the algebra $\Lambda^{\pm}_{m,n}$ were considered. But it is easy to check
that they are isomorphic to our ones. We prefer to use characters instead of
supercharacters in this paper in order to avoid some unnecessary signs.

It will be needed later an explicit description of the projective covers of
the irreducible finite dimensional modules due to Brundan [3]. We give the
description here in a slightly different way.

First let us for any $\chi\in P^{+}$ define a pair of sets

$A=\\{(\chi+\rho,\varepsilon_{1}),\dots,(\chi+\rho,\varepsilon_{m})\\},\quad
B=\\{(\chi+\rho,\delta_{1}),\dots,(\chi+\rho,\delta_{n})\\}$

where

$\rho=\frac{1}{2}\sum_{\alpha\in R_{0}^{+}}\alpha-\frac{1}{2}\sum_{\alpha\in
R_{1}^{+}}\alpha+\frac{1}{2}(n-m+1)(\sum_{i=1}^{m}\varepsilon_{i}-\sum_{j=1}^{n}\delta_{j})$
$=\sum_{i=1}^{m}(1-i)\varepsilon_{i}+\sum_{j=1}^{n}(m-j)\delta_{j}.$

Our $\rho$ is slightly different from the standard one but it is more
convenient since the elements of $A$ and $B$ are integers. So instead of
highest weights we will use the set of pairs $(A,B)$ such that
$A,B\subset\mathbb{Z}$ and $|A|=m,\,|B|=n$. We will also use the language of
diagrams which is due to Brundun and Stroppel [4] but we will use it here in a
form due to I. Musson and V. Serganova [9].

######  Definition 2.3.

Let $(A,B)$ be a pair of subsets in $\mathbb{Z}$ such that $|A|=m,\,|B|=n$.
Then the corresponding weight diagram is the following function on
$\mathbb{Z}$

$f(x)=\begin{cases}\times,\,\,x\in A\cap B\\\ \circ,\,\,x\notin A\cup B\\\
>,\,\,x\in A\setminus B\\\ <,\,\,x\in B\setminus A\end{cases}.$

Geometrically we can picture a diagram $f$ as $\mathbb{Z}$-line with $f(x)$
above $x$.

In the category $K(\mathcal{F})$ we have three important classes of modules:
irreducible modules $\\{L(f)\\}$; Kac modules $\\{K(f)\\}$; projective
indecomposable modules $\\{P(f)\\}$. Characters of Kac modules can be easily
described.

######  Definition 2.4.

Let $\chi\in P$ then we define the alternation operation $J$ by the formula

$J(e^{\chi})=\sum_{\sigma\in S_{m}\times
S_{n}}\varepsilon(\sigma)e^{\sigma(\chi)}$

Let

$D=\frac{\prod_{\alpha\in
R_{0}^{+}}\left(e^{\alpha/2}-e^{-\alpha/2}\right)}{\prod_{\alpha\in
R_{1}^{+}}(e^{\alpha/2}+e^{-\alpha/2})}$

and $f$ be a diagram such that

$f^{-1}(\times,>)=\\{a_{1}>a_{2}>\dots>a_{m}\\},\quad
f^{-1}(\times,<)=\\{b_{1}<b_{2}<\dots<b_{n}\\}$

then we have

$Dch\,K(f)=J(e^{\omega(f)}),\quad\omega(f)=\sum_{i=1}^{m}a_{i}\varepsilon_{i}-\sum_{j=1}^{n}b_{j}\delta_{j}=\chi+\rho$

######  Definition 2.5.

Let $f$ be a weight diagram. The corresponding cap diagram can be obtained in
the following way. Take the rightmost $\times$ and make the cap by joining it
to the first $\circ$ on the right. Then take the next $\times$ to the left and
make the cap by joining it to the first $\circ$ on the right which is not the
end of a cap, and so on.

We can also define the cap diagram in another way. Let $f(a)=\times$. Let us
take the first $c$ on the right such that $f(c)=\circ$ and the numbers of
$\times$-s and $\circ$-s in the interval $(a,c)$ are the same. Then draw the
cap joining $a$ and $c$.

Let $f(a)=\times$ then the right end of the corresponding cap will be denoted
by $\tilde{a}$ and by $\sigma_{a}$ we will denote the transposition permuting
$a$ and $\tilde{a}$

######  Definition 2.6.

Let us denote by $\mathcal{P}(f)$ the set $\sigma_{C}(f)$ where $C\subset
f^{-1}(\times)$ and $\sigma_{C}=\prod_{c\in C}\sigma_{c}$.

######  Theorem 2.7.

(Brundan [3]) Let $P(f)$ be the projective cover of an irreducible module
$L(f)$ then in the character ring we have

$ch\,P(f)=\sum_{g\in\mathcal{P}(f)}ch\,K(g)$

We need a topology on the character ring $K(\mathcal{F})$. Let us define

$n(f)=\sum_{f(a)\neq\circ,\,>}a$

and $K(\mathcal{F})_{d},\,d\in\mathbb{Z}$ be the subgroup generated by the
irreducible modules $L(f)$ such that $n(f)\leq d$. Then it is easy to see that
$\\{K(\mathcal{F})_{d},\,d\in\mathbb{Z}\\}$ is an ascending filtration on the
group $K(\mathcal{F})$. So we can consider the infinite series consisting of
Kac modules. It is easy to see that every irreducible module can be
represented in a unique way as the sum of Kac modules:

$L(f)=\sum_{g}a_{gf}K(g)$

Let $f$ be a diagram and $\varphi:f^{-1}(\times\,,<,\,>)\rightarrow\mathbb{Z}$
be an injection such that $\varphi(a)=a$ for any $a\in f^{-1}(<,\,>)$. Then we
can define the following new diagram $g$

$g^{-1}(>)=f^{-1}(>),\,\,\,\,\,g^{-1}(<)=f^{-1}(<),\,\,\,\,\,g^{-1}(\times)=\varphi(f^{-1}(\times)).$

We will denote the above diagram by $\varphi(f)$. We can also define a sign of
$\varphi$ by the following formula

$\varepsilon(\varphi)=\sum_{a\in f^{-1}(\times)}(a-\varphi(a)-n(a,b)),\quad
n(a,b)=|(a,\varphi(a))\cap\\{f^{-1}(<,>\\}|$

We should note, that $(a-\varphi(a)-n(a,b))$ is the number of $\circ$-s inside
the interval $(a,\varphi(a))$.

######  Definition 2.8.

Diagram $f$ is called core free if $f^{-1}(<)=f^{-1}(>)=\emptyset$. Let $f$ be
any diagram. We will denote by $f^{\sharp}$ the diagram obtaining from $f$ by
deleting all symbols $<,\,>$. If $\varphi$ is as above then we have the
corresponding map

$\varphi^{\sharp}:(f^{\sharp})^{-1}(\times)\rightarrow(g^{\sharp})^{-1}(\times)$

.

######  Lemma 2.9.

The following equality holds true
$\varepsilon(\varphi)=\varepsilon(\varphi^{\sharp})$

###### Proof.

Let $f(a)=\times$ and $a^{\sharp}$ be the corresponding number for
$f^{\sharp}$ then we have $a^{\sharp}=a-n(a)$ where $n(a)$ is the number of
symbols $<,\,>$ on left on $a$. So we have

$\varepsilon(\varphi^{\sharp})=\sum_{a^{\sharp}\in(f^{\sharp})^{-1}(\times)}(a^{\sharp}-\varphi^{\sharp}(a^{\sharp}))=\sum_{a\in
f^{-1}(\times)}(a-n(a)-\varphi(a)+n(\varphi(a))=\varepsilon(\varphi)$

∎

##  3\. Su-Zhang formula

In this section we give a different proof of Su-Zhang formula for the
decomposition irreducible module in terms of Kac modules. We also formulate
their result in a different terms.

######  Definition 3.1.

Let $f$ be a diagram. Let us introduce a partial order on the set
$f^{-1}(\times)$ by using its cap diagram. Let $a,b\in f^{-1}(\times)$ and
$C_{a},C_{b}$ are the corresponding caps. We say that $a\dashv b$ if $C_{b}$
is located under $C_{a}$. It is easy to see that this is indeed a partial
order on the set $f^{-1}(\times)$.

######  Definition 3.2.

Let us denote by $W(f,\mathbb{Z})$ the set of
$\varphi:f^{-1}(\times)\rightarrow\mathbb{Z}$ such that:

$1)$ $\varphi$ is injection

$2)$ $\varphi$ is a morphism of the ordered sets, where $f^{-1}(\times)$ is
ordered by means of $\dashv$ and $\mathbb{Z}$ is ordered in standard way

$3)$ $\varphi(a)\leq a$ for any $a\in f^{-1}(\times)$

$4)$ $\varphi(a)=a$ if $a\in f^{-1}(<,\,>)$.

We also denote by $W(f,g)$ the subset of $\varphi\in W(f,\mathbb{Z})$ such
that $\varphi(f)=g$.

######  Theorem 3.3.

The following equality holds true

$ch\,L(f)]=\sum_{\varphi\in
W(f,\mathbb{Z})}(-1)^{\varepsilon(\varphi)}ch\,K(\varphi(f))$

Let us fix two nonnegative integers $m,n$. We will denote by $F(m,n)$ the set
of diagrams $f$ such that

$|f^{-1}(>)|+|f^{-1}(\times)|=m,\,\,|f^{-1}(<)|+|f^{-1}(\times)|=n$

Now let $\mathcal{A},\mathcal{B}$ be two matrixes enumerated by elements of
$F(m,n)$ and defined by the following rules

$a_{f,g}=\delta_{\\{g\in\mathcal{P}(f)\\}},\quad
b_{f,g}=\sum_{\varphi(f)=g}(-1)^{\varepsilon(\varphi)}$

The statement of the Theorem is equivalent the following equality

$(\mathcal{A}\mathcal{B})_{fg}=\sum_{h}a_{f,h}b_{h,g}=\sum_{h,\,\varphi(g)=h\in\mathcal{P}(f)}(-1)^{\varepsilon(\varphi)}=$
$\sum_{\varphi(g)\in\mathcal{P}(f)}(-1)^{\varepsilon(\varphi)}=(P(f),L(g))=\delta_{f,g}$

where $(P(f),L(g))$ means the canonical bilinear form.

It not difficult to see by using Lemma 2.9 that we only need to prove core
free case. In this case instead of diagram $f$ we will use the corresponding
set $f^{-1}(\times)$ and for $a\in f^{-1}(\times)$ we will denote by
$\tilde{a}$ the right end of the corresponding cap. Let us introduce the
following set

$W(A,\mathcal{P}(B))=\\{f\in W(A,\mathbb{Z})\mid f(A)\in\mathcal{P}(B)\\}.$

######  Definition 3.4.

Let $A\subset\mathbb{Z}$. We will denote by $i(A)$ the following subset

$i(A)=\\{a\in A\mid\tilde{a}-a=1\\}$

It is clear that for any $A$ the set $i(A)$ is not empty. It is also easy to
see that $A$ is totally disconnected if and only if $i(A)=A$ and $A$ is
totally connected if and only if $|i(A)|=1$.

######  Definition 3.5.

Let $A\subset\mathbb{Z},$ and $a\in A$. Then by $C_{a}$ we will denote the
upper half of a circle joining two integers $a$ and $\tilde{a}$ and call it a
cap. The set $\\{C_{a}\mid a\in A\\}$ we will call a cap diagram.

Now we are ready to formulate a theorem that gives an explicit expression for
$L(A)$.

######  Theorem 3.6.

The following equality holds true

$ch\,L(A)=\sum_{f}(-1)^{\varepsilon(f)}ch\,K(f(A))$

where sum is taken over all $f\in W(A,\mathbb{Z})$ and
$\varepsilon(f)=\sum_{x\in A}(x+f(x))$.

Before proving the Theorem let us prove several technical Lemmas.

######  Lemma 3.7.

Let $b\in i(B)$ and $C\in\mathcal{P}(B)$. If $f:A\rightarrow C$ is a bijection
and morphism of ordered sets (as above) then $\tau_{a}\circ f$ is a bijection
and a morphism of ordered sets too.

###### Proof.

Clearly $\tau_{b}\circ f$ is a bijection $A$ on $\tau_{b}(C)$. Besides
$\tau_{b}:C\rightarrow\tau_{b}(C)$ is a morphism of ordered sets in the
standard sense since $C$ and $\tau_{b}(C)$ contain only one of the elements
$b,b+1$.Therefore $\tau_{b}\circ f:A\rightarrow\tau_{b}(C)$ is a bijection and
a morphism of ordered sets too. ∎

######  Lemma 3.8.

If $b\in i(B)$ and $b\notin A$ then $W(A,\mathcal{P}(B))$ is invariant with
respect to $\tau_{b}$.

###### Proof.

Let $f\in W(A,\mathcal{P}(B))$ then by Lemma 3.7 $\tau_{b}\circ f$ is a
morphism of ordered sets. Let us prove that $(\tau_{b}\circ f)(a)\leq a$. If
$f(a)\neq b,b+1$ then $(\tau_{b}\circ f)(a)=f(a)\leq a$. If $f(a)=b+1$ then
$(\tau_{b}\circ f)(a)=b<b+1\leq a$. If $f(a)=b$ then $b\leq a$ besides
$b\notin A$. Therefore $b<a$ and $(\tau_{b}\circ f)(a)=b+1\leq a$. ∎

######  Lemma 3.9.

Let $b\in i(B)$ and $b\in A$. Then the following set

$W_{<b}(A,\mathcal{P}(B))=\\{f\in W(A,\mathcal{P}(B))\mid f(b)<b\\}$

is invariant with respect to $\tau_{b}$.

###### Proof.

Let $f\in W_{<b}(A,\mathcal{P}(B))$ then $(\tau_{b}\circ f)(b)=f(b)<b$.
Besides by Lemma 3.7 $\tau_{b}\circ f$ is a morphism of ordered sets. Let us
prove, that $\tau_{b}(f(a))\leq a$ for any $a\in A$. If $f(a)\neq b,b+1$ then
$(\tau_{b}\circ f)(a)=f(a)\leq a.$ If $f(a)=b+1$ then
$\tau_{b}(f(a))=b<b+1\leq a$. If $f(a)=b$ then $b\leq a$ and $b\neq a$ since
$f(b)<b$. Therefore $b<a$ and $\tau_{b}(f(a))=b+1\leq a$. ∎

Let us introduce the following set

$W_{b}(A,\mathcal{P}(B))=\\{f\in W(A,\mathcal{P}(B)\mid f(b)=b\\}$

Then we have

$W(A,\mathcal{P}(B))=W_{b}(A,\mathcal{P}(B))\cup W_{<b}(A,\mathcal{P}(B))$

######  Proposition 3.10.

Let $b\in i(B),\,b,b+1\in A$. Then $W_{b}(A,\mathcal{P}(B))=\emptyset$.

###### Proof.

Suppose that $f\in W_{b}(A,\mathcal{P}(B))$ and $f:A\rightarrow C$. Since
$b,b+1\in A$ then $b\dashv b+1$. Therefore $b=f(b)<f(b+1)\leq b+1$. But
$f(b+1)\neq b+1$. This is a contradiction. Proposition is proved. ∎

######  Definition 3.11.

Let us define the following function
$\theta:\mathbb{Z}\setminus\\{b,b+1\\}\rightarrow\mathbb{Z}$

$\theta(x)=\begin{cases}x,\,\,x<b\\\ x-2,\,\,x>b+1\end{cases}$

It is easy to see that $\theta$ is isomorphism of ordered set.

######  Lemma 3.12.

Suppose, that $A,C\subset\mathbb{Z}$ and $b\in A\cap C,\,\,\,\,b+1\notin A\cup
C$. Then the correspondence $f\rightarrow\theta\circ f\circ\theta^{-1}$
defines a bijection between $W_{b}(A,C)$ and
$W(\theta(A\setminus\\{b\\}),\theta(C\setminus\\{b\\})).$

###### Proof.

It is easy to check that
$\theta:A\setminus\\{b\\}\rightarrow\theta(A\setminus\\{b\\})$ is an
isomorphism of ordered set if we consider $A\setminus\\{b\\}$ as the ordered
subset of $A$ and the order defined by the set of arks $D_{A}$.

Therefore if $f\in W_{b}(A,C)\,$ then$\,\,\theta\circ f\circ\theta^{-1}\in
W(\theta(A\setminus\\{b\\}),\theta(C\setminus\\{b\\}).$

In order to prove that our map is a bijection we will construct the inverse
map. Let $g\in W(\theta(A\setminus\\{b\\}),\theta(C\setminus\\{b\\})$ then we
define $f:A\rightarrow C$ by the following rule

$f(a)=\begin{cases}b,\,\,a=b\\\ (\theta^{-1}\circ g\circ\theta)(a),\,\,a\neq
b\end{cases}$

So we need to check that $g=\theta\circ f\circ\theta^{-1}$ and that $f\in
W_{b}(A,C)$. The first property is clear. The second property follows from the
fact that $\theta$ is isomorphism of ordered sets $A\setminus\\{b\\}$ (as the
subset of $A$ with respect to the order defined by $D_{A}$) and $\theta(A)$
with respect to the order defined by $D_{\theta(A\setminus\\{b\\})}$. Besides
$\theta$ is isomorphism of ordered set $\mathbb{Z}\setminus\\{b,b+1\\}$ and
$\mathbb{Z}$ with respect to the standard order on integers. The only
additional statement we need to prove is the following : if $a\dashv b$ then
$f(a)<f(b)=b$. But we have $g(\theta(a))\leq\theta(a)$. Therefore
$f(a)=\theta^{-1}(g(\theta(a)\leq a<b$. Lemma is proved. ∎

######  Corollary 3.13.

$(P(B),L(A))=\delta_{A,B}$.

###### Proof.

Let us use induction on $|A|$ \- the number of elements in $A$. If $|A|=1$
then the statement is clear. Let $|A|>1$. If $B=A$ then $(P(A),L(A))=1$. Let
$B\neq A$. If $iA)\cap i(B)=\emptyset$ then by Lemmas 3.8,3.9 and Proposition
3.10 we have $(P(B),L(A))=0.$ If $i(A)\cap i(B)\neq\emptyset$ then let us take
any element $b$ from this intersection and let us apply Lemma 3.12. Since
$A\neq B$ we have $\theta(A\setminus\\{b\\})\neq\theta(B\setminus\\{b\\})$. By
Lemmas 3.12, 3.10 we have

$(P(B),L(A))=(-1)^{b}(P(\theta(B\setminus\\{b\\}),L(A\setminus\\{b\\}))=0$

∎

So we see that we proved Theorem 3.6 and therefore Theorem 3.3.

######  Remark 3.14.

We can reformulate Theorem 3.3 in geometric terms. Let

$\mathbb{Z}\setminus
f^{-1}(\circ)=\\{c_{1}<c_{2}<\dots<c_{N}\\},\,i=1,\dots,N$

######  Definition 3.15.

Let $f$ be a diagram. Let us define the subset $D_{f}\subset\mathbb{R}^{N}\,$:
$(x_{1},\dots,x_{N})\in D_{f}$ if and only if the following conditions are
fulfilled

$1.$ $x_{i}\leq c_{i}$ if $f(c_{i})=\times$

$2$. $x_{i}=c_{i}$ if $f(c_{i})=\,<,\,>$

$3.$ $x_{i}\leq x_{j}$ if $c_{i}\dashv c_{j}$

Any point $(x_{1},\dots,x_{N})\in D_{f}$ with
$x_{i}\in\mathbb{Z},\,i=1,\dots,N$ can be considered as the diagram $g$ such
that $g(c_{i})=f(c_{i})$ if $f(c_{i})=\,<,\,>$ and $g(c_{i})=x_{i}$ if
$f(c_{i})=\times$. We will denote this diagram by $g_{x}$.

Let us also define a linear map $\pi_{f}:\mathbb{R}^{N}\rightarrow R^{m+n}$ by
the following formula

$\pi_{f}(e_{k})=\begin{cases}\varepsilon_{i},\,\,f(c_{k})=\,>,\,c_{k}=a_{i}\\\
-\delta_{j},\,\,f(c_{k})=\,<,\,c_{k}=b_{j}\\\
\varepsilon_{i}-\delta_{j},\,\,f(c_{k})=\times,\,c_{k}=a_{i}=b_{j}\end{cases}$

where $e_{k},\,k=1,\dots,N$ the standard basis in $\mathbb{R}^{N}$.

######  Corollary 3.16.

The following equality holds true

$Dch\,L(f)]=(-1)^{S(f^{-1}(\times))}\sum_{x\in
D_{f}}(-1)^{S(g_{x}^{-1}(\times))}J(e^{\pi_{f}(x)})$

where for $X\subset\mathbb{Z}$ we define $S(X)=\sum_{x\in X}x$.

###### Proof.

We have

$ch\,L(f)]=\sum_{x\in D_{f}}(-1)^{\varphi_{x}}ch\,K(g_{x})$

where $\varphi_{x}(f)=g_{x}$. Therefore

$Dch\,L(f)=\sum_{x\in D_{f}}(-1)^{\varphi_{x}}J(e^{\omega(g_{x})})$

So we see that we only need to prove that

$J(e^{\omega(g_{x})})=(-1)^{\tau}J(e^{\pi_{f}(x)}),\quad\text{where}\,\,\,\tau=\sum_{c\in
f^{-1}(\times)}n(c,\varphi_{x}(c)).$

The above formula follows from the fact that there are $\sigma_{1}\in
S_{m},\sigma_{2}\in S_{n}$ such that
$\sigma_{1}\sigma_{2}(\omega(g_{x}))=\pi_{f}(x)$ and
$sign(\sigma_{1}\sigma_{2})=(-1)^{\tau}$. ∎

##  4\. Graphs and polyhedra

######  Definition 4.1.

Let $f$ be a weight diagram and let $f^{-1}(\times)=\\{c_{1},\dots,c_{r}\\}$
be ordered as in Definition 3.1. Let us denote by $\Gamma_{f}$ the directed
graph with the set of vertexes $\\{1,\dots,r\\}$ and $i\rightarrow j$ if
$c_{i}\dashv c_{j}$ and the interval $(c_{i},c_{j})$ is empty.

Let $\Gamma$ be a directed graph without multiple edges and cycles (if we
ignore the orientation of $\Gamma$) with $r$ vertexes enumerated by integers
$1,\dots,r$. Let also $c_{1}<c_{2}<\dots<c_{r}$ be a sequence of integers such
that if $i\rightarrow j$ is a directed edge then $c_{i}<c_{j}$. Let us also
define $c_{\Gamma(i)}$ to be equal to the $\min\\{c_{k}\\}$ where $c_{k}$ run
over connected component containing $c_{i}$. We denote by $M(\Gamma)$ the set
of subgraphs of $\Gamma$ with the same set of vertexes.

We will denote by $V(\Delta)$ the set of vertexes of $\Delta$ and by
$E(\Delta)$ the set of the edges of $\Delta$. We also will denote by
$[x_{i}\leq c_{i}]$ and $[x_{i}\leq x_{j}]$ the characteristic functions of
the corresponding sets.

######  Definition 4.2.

So for every graph as above we can define a polyhedron $D_{\Gamma}$ in the
space $\mathbb{R}^{r}$ by its characteristic function

$[D_{\Gamma}]=\prod_{i\in V(\Gamma)}[x_{i}\leq c_{i}]\prod_{(i\rightarrow
j)\in E(\Gamma)}[x_{i}\leq x_{j}]$

######  Remark 4.3.

We should note the if $\Gamma=\Gamma_{f}$ then $D_{f}=D_{\Gamma_{f}}$.

Our aim in this section is to calculate the generation function of integer
point in the polyhedron $D_{f}$.

######  Definition 4.4.

The generation function of a polyhedron $P\subset\mathbb{R}^{r}$ is

$G(P)(t_{1},\dots,t_{n})=\sum_{x\in\mathbb{Z}^{n}\cap P}t_{1}^{x_{1}}\dots
t_{n}^{x_{r}}$

We also need Brion’s theorem.

######  Theorem 4.5.

([1], section 6)

Let $D$ be a polyhedron and $V$ be the set of its vertexes. Then we have

$G(D)=\sum_{M\in V}G(tcone(M))$

where $G$ is a generating function of integer points and $tcone(M)$ is the
tangent cone at the point $M$.

The next Lemma describes the vertexes of the polyhedron $D_{\Gamma}$.

######  Lemma 4.6.

Let $\Delta\in M(\Gamma)$. Then the point $M_{\Delta}=(x_{1},\dots,x_{r})$
such that $x_{i}=c_{\Delta(i)}$ is a vertex of the polyhedron $D_{\Gamma}$.
This correspondence is a bijection between the set $M(\Gamma)$ and the set of
vertexes of the polyhedron $D_{\Gamma}$.

###### Proof.

We can suppose that $\Gamma$ is a connected graph. Let $\Delta\in M(\Gamma)$
and $\Delta=\cup\Delta_{\alpha}$ be its decomposition into connected
components. Let also $i_{\alpha}\in\Delta_{\alpha}$ be the minimal vertex.
Then $M=(x_{1},\dots,x_{r})$ has the following characteristic function

$[M_{\Delta}]=\prod_{\alpha}[x_{i_{\alpha}}=c_{i_{\alpha}}]\prod_{(i\rightarrow
j)\in E(\Delta)}[x_{i}=x_{j}].$

Since the number of equations is equal to $|V(\Gamma)|$ the point $M$ is a
vertex of $D_{\Gamma}$. Now let us prove that the correspondence
$\Delta\rightarrow M_{\Delta}$ is a bijection.

Let $M\in D_{\Gamma}$ be a vertex and $f_{1}(M)=\dots=f_{r}(M)=0$ for some
$f_{i}=x_{i}-c_{i}$ or $f_{i}=x_{i}-x_{j}$ with linear independent linear
parts. Let us consider the last equation $f_{r}=0$. Since this vertex is
maximal there exists not more than one edge containing this vertex. Therefore
the sequence $f_{1},\dots,f_{n}$ may contains one of the equations
$x_{r}-x_{i},x_{r}-c_{r}$ or two of them.

In the first case the sequence $f_{1},\dots,f_{r-1}$ does not contain $x_{n}$.
Consider the graph $\tilde{G}$ which can be obtained from $\Gamma$ by deleting
the vertex $v_{n}$ with number $n$. By induction there exists
$\tilde{\Delta}\in M(\tilde{\Gamma})$ such that
$(x_{1},\dots,x_{r-1})=M_{\tilde{\Delta}}$. Let $f_{n}=x_{n}-c_{n}$. If the
vertex $v_{n}$ is a connected component of $\Gamma$ or there is an edge
$i\rightarrow n$. Then $c_{n}>c_{i}$ and we set
$\Delta=\tilde{\Delta}\cup\\{n\\}$ as the disjoint union. Therefore $\Delta\in
M(\Gamma)$ and $M=M_{\Delta}$.

Now let us prove that our map is injection. Suppose that
$\Delta\neq\tilde{\Delta}$. Then we can suppose that there exists $e\in
E(\Delta),\,e\notin E(\tilde{\Delta})$. Therefore $e=(i\rightarrow j)$ and
$c_{i}<c_{j}$ and $i,j$ belong to the same connected component
$\Delta_{\alpha}$ of the graph $\Delta$. In the same time they belong to the
different connected components
$\tilde{\Delta}_{\beta},\tilde{\Delta}_{\gamma}$ of the graph $\tilde{\Delta}$
( indeed suppose that $i,j$ belong to same connected component $G$ in
$\tilde{\Delta}$. Then there exists a path from minimal element in $G$ to $i$
and a path to $j$. Adding to the first path $e$ we get a cycle in $\Delta$.
This is a contradiction). If $k$ is the minimal element in
$\tilde{\Delta}_{\gamma}$ then $k\geq j$ (if $k<j$ then the path from the the
minimal element of $\Delta_{\alpha}$ to $k$ and then to $j$ should be the same
as path from the minimal element of $\Delta_{\alpha}$ to $j$ calling at $i$.
Therefore $k=j$. So we see that $x_{\Delta(j)}=x_{\Delta(i)}$ and
$x_{\tilde{\Delta}(j)}=c_{j}>c_{i}\geq x_{\tilde{\Delta}(i)}$. ∎

According to [1] in order to describe the tangent cones we need to choose the
inequalities (from the defining inequalities of $D_{\Gamma}$) which are active
on the corresponding vertex. In other words we need to choose those
inequalities for which $M_{\Delta}$ is a solution of the corresponding
equation.

######  Corollary 4.7.

Let $\Delta=\cup_{\alpha}\Delta_{\alpha}$ be the decomposition into connected
components and $i_{\alpha}$ be the minimal number in $\Delta_{\alpha}$. Then
the vertex $M_{\Delta}$ can be define by the characteristic function

$[M_{\Delta}]=\prod_{\alpha}[x_{i_{\alpha}}=c_{i_{\alpha}}]\prod_{(i\rightarrow
j)\in E(\Delta)}[x_{i}=x_{j}].$

The tangent cone at the point $M_{\Delta}$ has the following characteristic
function

$[tcone(M_{\Delta})]=\prod_{\alpha}[x_{i_{\alpha}}\leq
c_{i_{\alpha}}]\prod_{(i\rightarrow j)\in E(\Delta)}[x_{i}\leq x_{j}]$

###### Proof.

The polyhedron $D_{\Gamma}$ is given by the inequalities

$l_{i}(N)=x_{i}\leq c_{i},\,i\in V(\Gamma),\quad l_{ij}(N)=x_{i}-x_{j}\leq
0\,\,\text{if}\,\,i\rightarrow j\in E(\Gamma).$

Suppose that $l_{i}(M_{\Delta})=c_{i}$. Therefore $c_{i}=c_{i_{\alpha}}$. So
we see that $i=i_{\alpha}$. Now suppose that $l_{ij}(M_{\Delta})=0$ where
$i\rightarrow j\in E(\Gamma)$. We can suppose that $i\rightarrow j\notin
E(\Delta)$. Therefore $c_{\Delta(i)}<c_{\Delta(j)}$ and we came to
contradiction. ∎

We also need some different (open) cone with the same generating function for
integer points. Let us set

$[tcone^{op}(M_{\Delta})]=\prod_{\alpha}[x_{i_{\alpha}}\leq
c_{i_{\alpha}}]\prod_{(i\rightarrow j)\in E(\Delta)}(1-[x_{i}\leq x_{j}])$

We should note that $[tcone^{op}(M_{\Delta})]$ is not a polyhedron.

######  Lemma 4.8.

The difference $[tcone(M_{\Delta})]-[tcone^{op}(M_{\Delta})]$ is a linear
combination of characteristic function of rational polyhedra containing a
line.

###### Proof.

We have

$[tcone^{op}(M_{\Delta})]=(-1)^{E(\Delta)|}[tcone(M_{\Delta})]+\prod_{\alpha}[x_{i_{\alpha}}\leq
c_{i_{\alpha}}]\sum_{p=1}^{q-1}(-1)^{p}s_{p},$

where $s_{p}$ is the elementary symmetric polynomial in characteristic
functions of the edges of $\Delta$ and $q=|E(\Delta)|$. So we see that we only
need to prove that for any $p<q$ and any summand in $s_{p}$ the corresponding
polyhedron contains a line. We can suppose that $\Delta$ is a connected graph.
Let $\chi=\chi_{e_{i_{1}}}\dots\chi_{e_{i_{p}}}$ be one of the summands in the
decomposition of $s_{p}(e_{1},\dots,e_{q})$. Consider the graph
$\Delta^{\prime}$ with the same set of vertexes as $\Delta$ and the set of
edges $e_{i_{1}},\dots,e_{i_{p}}$. Since $p<q$ There exist at least one
connected component $\Delta^{\prime}_{1}$ of $\Delta^{\prime}$ which does not
contain the minimal vertex. Therefore this connected component can be
described by the inequalities

$x_{i}\leq x_{j},\,(i\rightarrow j)\in E(\Delta^{\prime}_{1})$

So we see that this connected component contains a line
$x_{i}=x_{j}=t,\,(i\rightarrow j)\in E(\Delta^{\prime}_{1})$. Lemma is proved
∎

######  Corollary 4.9.

The cone $tcone^{op}(M_{\Delta})$ can be described by the following
inequalities: $x_{i}\leq a,\,i=1,\dots,n$; if $e=(i\rightarrow j)\in\Delta$
then $x_{i}>x_{j}$.

######  Definition 4.10.

Le $\Gamma$ be a directed graph without multiple edges and cycles (if we
ignore the orientation). Let us denote by $D_{\Gamma}(a)$ the following cone
in $\mathbb{R}^{r}$

$[D_{\Gamma}(a)]=\prod_{i=1}^{r}[x_{i}\leq a]\prod_{i\rightarrow j\in
E(\Gamma)}[x_{i}\leq x_{j}]$

Let us also denote by $S_{\Gamma}$ the set of the permutations of the vertexes
of $\Gamma$ such that: if $i\rightarrow j$ is an edge of $\Gamma$ then
$\sigma(i)<\sigma(j)$.

######  Lemma 4.11.

The following statements hold true

$1)$

$D_{\Gamma}(a)=\bigcup_{\sigma\in S_{\Gamma}}D_{\Gamma}(a,\sigma)$

where

$D_{\Gamma}(a,\sigma)=\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}\mid
x_{\sigma^{-1}(1)}<\dots<x_{\sigma^{-1}(r)}\leq a\\}$

$2)$ If $\sigma,\tau\in S_{\Gamma}$ then $\tau^{-1}\sigma$ is a bijection
between $D_{\Gamma}(a,\sigma)$ and $D_{\Gamma}(a,\tau)$.

$3)$ If $\Gamma^{op}$ is obtained from $\Gamma$ by changing all arrows on the
opposite ones then $|S_{\Gamma}|=|S_{\Gamma^{op}}|$.

###### Proof.

Let us prove the first statement. Let $\sigma\in S_{\Gamma}$ and suppose that
$M=(x_{1},\dots,x_{n})\in D_{\Gamma}(a,\sigma)$, $i\rightarrow j\in
E_{\Gamma}$. Then $\sigma(i)<\sigma(j)$ and

$x_{i}=\,x_{\sigma^{-1}(\sigma(i))}<x_{\sigma^{-1}(\sigma(j))}=x_{j}$

Therefore $M\in D_{\Gamma}(a)$. Now let $M\in D_{\Gamma}(a)$. There exists a
unique $\sigma$ such that $x_{\sigma^{-1}(1)}<\dots<x_{\sigma^{-1}(n)}$ and we
only need to prove that $\sigma\in S_{\Gamma}$. Let $[i,j]\in E_{\Gamma}$ then
since $M\in D_{\Gamma}(a)$ we have

$x_{\sigma^{-1}(\sigma(i))}=x_{i}<x_{j}=x_{\sigma^{-1}(\sigma(j))}.$

Therefore $\sigma(i)<\sigma(j)$.

Let us prove the second statement. If $M=(x_{1},\dots,x_{r})\in D(a,\sigma)$
and $i<j$ then (since $\tau\in S_{\Gamma}$) $\tau(i)<\tau(j)$. Therefore
$x_{\sigma^{-1}\tau(i)}<x_{\sigma^{-1}\tau(j)}$ and $\sigma^{-1}\tau(M)\in
D_{\Gamma}(a,\tau)$.

Let $w(i)=r-i+1$ be a permutation of the vertexes. Then it is easy to see that
$w$ is a bijection of $S_{\Gamma}$ into $S_{\Gamma^{op}}$ and the third
statement follows. ∎

######  Definition 4.12.

Let $\Gamma$ be a directed graph without multiple edges and cycles (if we
ignore the orientation of $\Gamma$) with $r$ vertexes enumerated by integers
$1,\dots,r$. Let also $c_{1}<c_{2}<\dots<c_{r}$ be a sequence of integers such
that if $i\rightarrow j$ is a directed edge then $c_{i}<c_{j}$. Let us also
define $c_{\Gamma(i)}$ to be equal to the $\min\\{c_{k}\\}$ where $c_{k}$ run
over connected component containing $c_{i}$. Let us define a Laurent
polynomial

$\theta(\Gamma,t_{1},\dots,t_{r})=\frac{1}{r!}\sum_{\Delta\in
M(\Gamma)}(-1)^{|E(\Delta|}|S_{\Delta}|\prod_{i=1}^{r}t_{i}^{c_{\Delta(i)}-c_{i}}$

The following Proposition shows that function $\theta$ is multiplicative.

######  Proposition 4.13.

If $\Gamma=\Gamma_{1}\cup\Gamma_{2}$ is a disjoint union of two graphs, then

$\theta(\Gamma_{1},t)\theta(\Gamma_{2},s)=\theta(\Gamma,t,s)$

###### Proof.

We have

$\theta(\Gamma_{1},t)\theta(\Gamma_{2},s)=\frac{1}{(r_{1})!}\frac{1}{(r_{2})!}\sum_{\Delta_{1}\in
M(\Gamma_{1}),\Delta_{2}\in
M(\Gamma_{2})}(-1)^{|E(\Delta_{1}|+|E(\Delta_{2})|}$
$\times|S_{\Delta_{1}}||S_{\Delta_{2}}|\prod_{i\in
V(\Gamma_{1})}t_{i}^{c_{\Delta(i)}-c_{i}}\prod_{j\in
V(\Gamma_{2})}s_{j}^{c_{\Delta(j)}-c_{j}}$

It is easy to see that any $\Delta\in M(\Gamma)$ can be uniquely represented
in the form $\Delta=\Delta_{1}\cup\Delta_{2}$ where
$(\Delta_{1},\Delta_{2})\in M(\Gamma_{1})\times M(\Gamma_{2})$. Further we see
that

$|S_{\Delta_{1}\cup\Delta_{2}}|=|S_{\Delta_{1}}||S_{\Delta_{2}}|\frac{(n_{1}+n_{2})!}{(n_{1})!(n_{2})!}$

and

$|E(\Delta_{1})|+|E(\Delta_{2})|=|E(\Delta)|$

Proposition is proved. ∎

##  5\. Character formula

######  Definition 5.1.

Let $f$ be a diagram such that

$f^{-1}(>,\times)=\\{a_{1}>\dots>a_{m}\\},\quad
f^{-1}(<,\times)=\\{b_{1}<\dots<b_{n}\\},\,f^{-1}(\times)=r$

Let us define the following evaluation homomorphism

$ev_{f}:\mathbb{R}[t_{1},\dots,t_{N}]\rightarrow\mathbb{R}[e^{\pm\varepsilon_{1}},\dots,e^{\pm\varepsilon_{m}},e^{\pm\delta_{1}},\dots,e^{\pm\delta_{n}}],\,\,N=m+n-r$

by the following rule:

$ev_{f}(t_{k})=\begin{cases}e^{\varepsilon_{i}},\,\text{if}\,\,c_{k}=a_{i},\,f(a_{i})=\,\,>\\\
e^{-\delta_{j}},\,\,\text{if}\,\,c_{k}=b_{j},\,f(b_{j})=\,\,<\\\
-e^{\varepsilon_{i}-\delta_{j}}\,\,\text{if}\,\,c_{k}=a_{i}=b_{j},\,f(a_{i})=f(b_{j})=\times\end{cases}$

######  Lemma 5.2.

The following formula holds true

$J(ev_{f}(G(tcone(M_{\Delta})))=\frac{|S_{\Delta}|}{|V(\Delta)|!}J\left(ev_{f}\left(\prod_{i\not\in
f^{-1}(\times)}t_{i}^{c_{i}}\prod_{i\in
f^{-1}(\times)}\frac{t_{i}^{c_{\Delta(i)}}}{(1-t_{i}^{-1})}\right)\right)$

###### Proof.

By Proposition 4.13 we can suppose that $\Delta$ is a connected graph and set
$a=c_{\Delta_{i}}$. Therefore we can apply Lemma 4.11 to
$tcone^{op}(M_{\Delta})=D_{\Delta^{op}}(a)$. According to this Lemma

$J(ev_{f}G(D_{\Delta^{op}}(a)))=|S_{\Delta^{op}(a)}|J(ev_{f}G(D_{\Delta^{op}}(a)(id)))$

and it is easy to see that

$J(ev_{f}G(D_{\Delta^{op}}(a)(id)))=\frac{1}{|V(\Delta)|!}J\left(ev_{f}\left(\prod_{i\not\in
f^{-1}(\times)}t_{i}^{c_{i}}\prod_{i\in
f^{-1}(\times)}\frac{t_{i}^{c_{\Delta(i)}}}{(1-t_{i}^{-1})}\right)\right)$

∎

Now we are going to give a formula for characters of irreducible finite
dimensional representations of the superalgebra Lie $\mathfrak{gl}(m,n)$. Let
$\chi$ be an integer dominant weight then we can define a pair of the set
$(A,B)$ and the diagram $f$. We also define the $S_{\chi}$ \- a maximal
$\chi+\rho$ isotropic set. The set $S_{\chi}$ is in a bijection with the set
of $f^{-1}(\times)$ by the following rule: if $f(c)=\times$ and
$c=a_{i}=b_{j}$ then $\varepsilon_{i}-\delta_{j}\in S_{\chi}$. So we can order
the set $S_{\chi}=\\{\alpha_{1}<\dots<\alpha_{r}\\}$ using the natural order
on $f^{-1}(\times)$ and we also have the corresponding graph $\Gamma_{f}$. Let
also

$D=\frac{\prod_{\alpha\in
R_{0}^{+}}\left(e^{\alpha/2}-e^{-\alpha/2}\right)}{\prod_{\alpha\in
R_{1}^{+}}(e^{\alpha/2}+e^{-\alpha/2})}.$

######  Theorem 5.3.

The following equality holds true

$Dch\,L(\chi)=\sum_{w\in
W_{0}}\varepsilon(w)w\left(\frac{e^{\chi+\rho}\theta(\Gamma_{f},-e^{\alpha_{1}},\dots,-e^{\alpha_{r}})}{\prod_{\alpha\in
S_{\chi}}(1+e^{-\alpha})}\right)$ (1)

###### Proof.

By Collorary 3.16 we have

$Dch\,L(f)=(-1)^{S(f^{-1}(\times))}\sum_{x\in\mathbb{Z}^{N}\cap
D_{f}}(-1)^{S(g_{x}^{-1}(\times))}J(e^{\pi_{f}(x)})$
$(-1)^{S(f^{-1}(\times))}\sum_{x\in\mathbb{Z}^{N}\cap
D_{f}}J(ev_{f}(t_{1}^{x_{1}}\dots t_{N}^{x_{N}}))=$
$(-1)^{S(f^{-1}(\times))}J(ev_{f}(G(D_{f})))$

So we need to calculate the function $J(G(D_{f}))$. By Brion’s theorem we have

$G(D_{f})=\sum_{M\in V}G(tcone(M)).$

Therefore we need to calculate $J(ev_{f}(G(tcone(M))))$ for any vertex $M$ of
$D_{f}$. By Lemma 4.6 any vertex can be described by means of graph $\Delta$.
So by Lemma 5.2 we have

$Dch\,L(f)=$
$(-1)^{S(f^{-1}(\times))}J\left(ev_{f}\left(\sum_{\Delta}(-1)^{|E(\Delta)|}\frac{|S_{\Delta}|}{|V(\Delta)|!}\frac{\prod
t_{i}^{c_{\Delta(i)}}}{\prod(1-t_{i}^{-1})}\right)ev_{f}(\prod_{f(c_{i})\neq\times}t_{i}^{c_{i}})\right)=$
$(-1)^{S(f^{-1}(\times))}J\left(ev_{f}\left(\prod t_{i}^{c_{i}}\sum_{\Delta\in
M(\Gamma)}(-1)^{|E(\Delta)|}\frac{|S_{\Delta}|}{|V(\Delta)|!}\frac{\prod
t_{i}^{c_{\Delta(i)}-c_{i}}}{\prod(1-t_{i}^{-1})}\right)\right)$

But

$ev_{f}\left(\prod_{i\in f^{-1}(\times)}(1-t_{i}^{-1})\right)=\prod_{\alpha\in
S_{\lambda}}(1+e^{-\alpha})$

and

$\frac{1}{|V(\Gamma)|!}\sum_{\Delta\in
M(\Gamma)}(-1)^{|E(\Delta)|}|S_{\Delta}|\prod
t_{i}^{c_{\Delta(i)}-c_{i}}=\theta(\Gamma,t_{i_{1}},\dots,t_{i_{r}})$

where $i_{1}<i_{2}\dots<i_{r}=f^{-1}(\times)$. ∎

######  Remark 5.4.

From Proposition 4.13 we see that we only need to write down the character
formula only for connected graphs.

######  Example 5.5.

We will consider the most atypical case of $\mathfrak{gl}(3,3)$. Let us
consider the following weight

$\chi=(c+3)\varepsilon_{1}+(c+2)\varepsilon_{2}+(c+2)\varepsilon_{3}-(c+5)\delta_{1}-(c+3)\delta_{2}-(c+3)\delta_{3}$
$\chi+\rho=(c+3)(\varepsilon_{1}-\delta_{3})+(c+1)(\varepsilon_{2}-\delta_{2})+c(\varepsilon_{3}-\delta_{1})$
$S_{\chi}=\\{\alpha_{1}<\alpha_{2}<\alpha_{3}\\}.$

The corresponding weight diagram is
$f^{-1}(\times)=\\{c,c+1,c+3\\},f^{-1}(<)=f^{-1}(>)=\emptyset$. The
corresponding cap diagram is the following

$\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c+1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c+2}$$\textstyle{c+3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c+4}$$\textstyle{c+5}$

In this case $\Gamma_{f}$ is the following graph

$\textstyle{c_{2}}$$\textstyle{c_{3}}$$\textstyle{c_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $c_{1}=c,c_{2}=c+1,c_{3}=c+3$. The corresponding polyhedron is

$D_{f}=\\{x_{1}\leq c_{1},\,x_{2}\leq c_{2},\,x_{3}\leq c_{3},\,x_{1}\leq
x_{2},\,x_{1}\leq x_{3}\\}.$

The polyhedron $D_{f}$ has four vertexes

$M_{1}=(c_{1},c_{1},c_{1}),\,M_{2}=(c_{1},c_{2},c_{1}),\,M_{3}=(c_{1},c_{1},c_{3}),\,M_{4}=(c_{1},c_{2},c_{3}),\,$

and these vertexes correspond to the four possibilities for graph $\Delta$

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
30.56378pt\raise-36.10999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{c_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
4.9358pt\raise-4.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
69.31462pt\raise-4.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
71.25513pt\raise-36.10999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\quad\quad|S_{\Delta}|=2,\,\,\prod_{i=1}^{3}t_{i}^{c_{\Delta(i)}-c_{i}}=t^{-1}_{2}t_{3}^{-3}$
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.56378pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 67.69135pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{c_{3}}$}}}}}}}{\hbox{\kern-3.0pt\raise-36.10999pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
30.56378pt\raise-36.10999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{c_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
69.31462pt\raise-4.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
71.25513pt\raise-36.10999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\quad\quad|S_{\Delta}|=3,\,\,\prod_{i=1}^{3}t_{i}^{c_{\Delta(i)}-c_{i}}=t_{3}^{-3}$

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{c_{2}}$}}}}}}}{\hbox{\kern 34.12756pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 67.69135pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{c_{3}}$}}}}}}}{\hbox{\kern-3.0pt\raise-36.10999pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
30.56378pt\raise-36.10999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{c_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
4.9358pt\raise-4.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
71.25513pt\raise-36.10999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\quad\quad|S_{\Delta}|=3,\,\,\prod_{i=1}^{3}t_{i}^{c_{\Delta(i)}-c_{i}}=t_{2}^{-1}$

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.56378pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{c_{2}}$}}}}}}}{\hbox{\kern 34.12756pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 67.69135pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{c_{3}}$}}}}}}}{\hbox{\kern-3.0pt\raise-36.10999pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
30.56378pt\raise-36.10999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{c_{1}}$}}}}}}}{\hbox{\kern
71.25513pt\raise-36.10999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\quad\quad|S_{\Delta}|=6,\,\,\prod_{i=1}^{3}t_{i}^{c_{\Delta(i)}-c_{i}}=1$

Therefore in this case we have

$\theta(\Gamma,t_{1},t_{2},t_{3})=1-\frac{1}{2}t_{2}^{-1}-\frac{1}{2}t_{3}^{-3}+\frac{1}{3}t_{2}^{-1}t_{3}^{-3}$

So we have the following equality

$D\,chL(\chi)=J\left(e^{\chi+\rho}\frac{1+\frac{1}{2}e^{-\alpha_{2}}+\frac{1}{2}e^{-3\alpha_{3}}+\frac{1}{3}e^{-\alpha_{2}-3\alpha_{3}}}{(1+e^{-\alpha_{1}})(1+e^{-\alpha_{2}})(1+e^{-\alpha_{3}})}\right)$

Actually more general result can be proved by the same way.

Let $f$ be a diagram. Then we can represent
$f^{-1}(\times)=\cup[[c_{i},d_{i}]]$ as the disjoin union of the of integer
segments and define $\tilde{c}_{i}$ as the maximum $c_{j}$ where $c_{j}$ run
over the segment containing $c_{i}$. Let $\Gamma_{f}$ be the same graph as
before and edge $i\rightarrow j$ is called special if
$\tilde{c}_{i}<\tilde{c}_{j}$. Let us also define $c_{\Gamma(i)}$ to be equal
to the $\min\\{c_{k}\\}$ where $c_{k}$ run over connected component containing
$c_{i}$. We denote by $M(\Gamma)$ the set of subgraphs of $\Gamma$ with the
same set of vertexes and the same set of non special edges. Let us also denote
by $\Gamma_{0}$ the subgraph of $\Gamma$ consisting of vertexes such that
$\tilde{c}_{i}=\tilde{c}_{\Gamma(i)}$ and the edges of $\Gamma_{0}$ are the
same as in $\Gamma$. It is easy to see that if $\Delta\in M(\Gamma)$ then
$\Delta\supset\Gamma_{0}$. Let us define the following Laurent polynomial

$\tilde{\theta}(\Gamma,t_{1},\dots,t_{n})=\frac{1}{n!}\sum_{\Delta\in
M(\Gamma)}(-1)^{|E(\Delta\setminus\Delta_{0})|}|S_{\Delta^{*}}|\prod_{i=1}^{n}t_{i}^{c_{\Delta(i)}-c_{i}}$

where $\Delta^{*}$ is the graph which can be obtained from $\Delta$ by
replacing all edges in $\Delta\setminus\Delta_{0}$ by the opposite ones.

######  Theorem 5.6.

The following equality holds true

$D\,chL(\chi)=(-1)^{\nu}J\left(\frac{e^{\chi+\rho+\gamma}\tilde{\theta}(\Gamma_{f},-e^{\alpha_{1}},\dots,-e^{\alpha_{r}})}{\prod_{\alpha\in
S_{\chi}}(1+e^{-\alpha})}\right)$ (2)

where $\nu=\sum_{i}(\tilde{c}_{i}-c_{i})$ and
$\gamma=\sum_{i}(\tilde{c}_{i}-c_{i})\alpha_{i}$.

######  Example 5.7.

Let us suppose that $c_{1}=c,\,c_{2}=c+1,c_{3}=c+3$. Then we have
$\tilde{c}_{1}=\tilde{c}_{2}=c+1,\,\tilde{c}_{3}=c+3$. Therefore we have two
possibilities for graph $\Delta$ and the corresponding summands

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.17778pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\tilde{c}_{3}}$}}}}}}}{\hbox{\kern-3.0pt\raise-39.02667pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
31.17778pt\raise-39.02667pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\tilde{c}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
4.71529pt\raise-4.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
71.99113pt\raise-4.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
73.71112pt\raise-39.02667pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\quad\quad|S_{\Delta^{*}}|=1,\,\,\prod_{i=1}^{3}t_{i}^{\tilde{c}_{\Delta(i)}-\tilde{c}_{i}}=t_{3}^{-2}$
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.04294pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{a_{3}}$}}}}}}}{\hbox{\kern-3.0pt\raise-36.10999pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
31.04294pt\raise-36.10999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{a_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
5.06248pt\raise-4.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
73.17175pt\raise-36.10999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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So we see that $S=1$ and $\gamma=\alpha_{1}$. Therefore

$D\,chL(\chi)=-J\left(e^{\chi+\rho+\alpha_{1}}\frac{\frac{1}{2}-\frac{1}{2}e^{-2\alpha_{3}}}{(1+e^{-\alpha_{1}})(1+e^{-\alpha_{2}})(1+e^{-\alpha_{3}})}\right)$

######  Example 5.8.

(See [5]) Suppose that diagram $f$ is PDC. This means that the corresponding
graph $\Gamma=\Gamma_{f}$ is a disjoint union of segments. It is easy to check
that in this case $\Gamma_{0}=\Gamma$. Therefore
$\tilde{\theta}(\Gamma,t_{1},\dots,t_{r})=\frac{|S_{\Gamma}|}{|V(\Gamma)|!}$
and we have the following formula for irreducible character

$D\,chL(f)=(-1)^{\nu}\frac{|S_{\Gamma}|}{|V(\Gamma)|!}J\left(\frac{e^{\chi+\rho+\gamma}}{\prod_{\alpha\in
S_{\chi}}(1+e^{-\alpha})}\right).$

But $|S_{\Gamma}|=\frac{(n_{1}+\dots+n_{r})!}{(n_{1})!\dots(n_{r})!}$. So this
formula coincides with the formula for $PDC$ modules in [5].

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