On polyhedral formulas for Kirillov-Reshetikhin modules

TL;DR

This paper introduces a new method to verify polyhedral formulas for Kirillov-Reshetikhin modules, especially in exceptional Lie types, using rational function identities and residue comparisons, with applications to types F4 and G2.

Contribution

It develops a novel, computer-assisted approach to prove polyhedral formulas for KR modules, simplifying verification through rational function residue analysis.

Findings

Verified polyhedral formulas for KR modules in types F4 and G2.

Provided a uniform, computational framework for checking these formulas.

Extended the method to cases where formulas are conjectural, aiding future proofs.

Abstract

We propose a method to prove a polyhedral branching formula for Kirillov-Reshetikhin (KR) modules over a quantum affine algebra. When the underlying simple Lie algebra is of exceptional type, such a formula remains conjectural in many cases. Using a linear recurrence relation satisfied by the characters of KR modules, we convert the verification of a polyhedral formula into an identity between two rational functions of a single variable with only simple poles at known locations. It is then sufficient to compare the residues at those poles, which are explicitly computable quantities. By applying this strategy, we obtain new, computer-assisted and easily verifiable proofs of known polyhedral formulas in types $F_4$ and $G_2$ within a uniform framework.

Equations

$$\mathrm{res}{W}_m^{(2)}=\underset{\begin{array}{c}{j}_1+2j_2+j_3+j_4\le m\\ j_1,j_2,j_3,j_4\in {\mathbb{Z}}_{\ge 0}\end{array}}{⨁}p(j_1,j_2,j_3,j_4)L\left(j_1{\lambda }_1+j_2{\lambda }_2+j_3{\lambda }_3+j_4{\lambda }_4\right),$$

$$p(j_1,j_2,j_3,j_4)=\min \left(1+j_3,1+m-j_1-2j_2-j_3-j_4\right)(j_4+1),$$

$$(Q^{(a)}_{m})^{2}=Q^{(a)}_{m+1}Q^{(a)}_{m-1}+\prod_{b:C_{ab}<0}\prod_{k=0}^{-C_{ab}-1}Q^{(b)}_{\bigl{\lfloor}\frac{C_{ba}m-k}{C_{ab}}\bigr{\rfloor}},\,\quad a\in I,m\geq 1$$

$$N^{(a)}(t):={\mathcal{Q}}^{(a)}(t)D^{(a)}(t)$$

$$Q_m^{(a)}=\sum _{(\lambda ,\zeta ,l)}C({\mathcal{Q}}^{(a)},\lambda ,\zeta ,l){\zeta }^m{e}^{m\lambda /l},m\in {\mathbb{Z}}_{\ge 0},$$

$$w\left(C({\mathcal{Q}}^{(a)},\lambda ,\zeta ,l)\right)=C({\mathcal{Q}}^{(a)},w(\lambda ),\zeta ,l),w\in W.$$

$$Q_m^{(a)}=\sum _{\lambda \in {\Lambda }_a}{C}_{\lambda }^{(a)}{e}^{m\lambda }+\sum _{\lambda \in {\Lambda }_a^{\prime }}\sum _{\zeta :{\zeta }^{t_a}=1}{C}_{\lambda ,\zeta }^{(a)}{\zeta }^m{e}^{m\lambda /t_a}.$$

$$C_{{\omega }_a}^{(a)}=\frac{1}{{\prod }_{\alpha \in {\Delta }^{+}}(1-e^{-\alpha }{)}^{[\alpha {]}_a}}.$$

$$W=\bigotimes_{(a,m)}\bigl{(}\operatorname{res}W^{(a)}_{m}\bigr{)}^{\otimes\nu^{(a)}_{m}},$$

$$W\cong \underset{\lambda \in P^{+}}{⨁}m(W,\lambda )L(\lambda ),m(W,\lambda )\in {\mathbb{Z}}_{\ge 0},$$

$$\mathrm{res}{W}_m^{(a)}=\underset{\mathbf{x}\in F_m^{(a)}}{⨁}L({\lambda }_{\mathbf{x}})$$

$$\mathrm{res}{W}_m^{(a)}\stackrel{?}{=}\underset{\mathbf{x}\in F_m^{(a)}}{⨁}p(\mathbf{x})L({\lambda }_{\mathbf{x}}),$$

$$P_m^{(a)}=\sum _{\mathbf{x}\in F_m^{(a)}}p(\mathbf{x})\chi (L({\lambda }_{\mathbf{x}})).$$

$${\mathcal{P}}^{(a)}(t):=\sum _{m=0}^{\infty }{P}_m^{(a)}{t}^m,$$

$${\mathcal{P}}^{(a)}(t)={\mathcal{Q}}^{(a)}(t),$$

$$C({\mathcal{P}}^{(a)},\lambda ,\zeta ,l)=C({\mathcal{Q}}^{(a)},\lambda ,\zeta ,l)$$

$$P_m^{(a)}=\sum _{(\lambda ,\zeta ,l)}C({\mathcal{P}}^{(a)},\lambda ,\zeta ,l){\zeta }^m{e}^{m\lambda /l}$$

$$Q_m^{(2)}=P_m^{(2)},$$

$${\mathcal{Q}}^{(2)}(t)={\mathcal{P}}^{(2)}(t)$$

$$\begin{array}[]{c|c|c}a&\Lambda_{a}\cap P^{+}&\Lambda^{\prime}_{a}\cap P^{+}\\ \hline\cr 1&0,\omega_{1}&\cdot\\ 2&0,\omega_{1},\omega_{2},2\omega_{4}&\cdot\\ 3&\omega_{3}&0,\omega_{1},2\omega_{1},\omega_{2},2\omega_{4},\omega_{1}+2\omega_{4}\\ 4&\omega_{4}&0,\omega_{1}\\ \end{array}.$$

$$a_m=\sum _{\begin{array}{c}{j}_1+2j_2+j_3+j_4\le m\\ j_1,j_2,j_3,j_4\in {\mathbb{Z}}_{\ge 0}\end{array}}p(j_1,j_2,j_3,j_4)x_1^{j_1}{x}_2^{j_2}{x}_3^{j_3}{x}_4^{j_4},$$

$$\sum _{m=0}^{\infty }{a}_m{t}^m=\frac{1}{(1-t)\left(1-t{x}_1\right)\left(1-t^2{x}_2\right)\left(1-t{x}_3\right)\left(1-t^2{x}_3\right)\left(1-t{x}_4\right){}^2}.$$

$$\frac{1}{\Delta (1-t)}\sum _{w\in W}\frac{(-1{)}^{\ell (w)}{e}^{w(\rho )}}{\left(1-e^{w({\lambda }_1)}t\right)\left(1-e^{w({\lambda }_2)}{t}^2\right)\left(1-e^{w({\lambda }_3)}t\right)\left(1-e^{w({\lambda }_3)}{t}^2\right)\left(1-e^{w({\lambda }_4)}t\right){}^2}.$$

$$\begin{gathered} \frac{1}{(1-t)\left(1-e^{w({\lambda }_1)}t\right)\left(1-e^{w({\lambda }_2)}{t}^2\right)\left(1-e^{w({\lambda }_3)}t\right)\left(1-e^{w({\lambda }_3)}{t}^2\right)\left(1-e^{w({\lambda }_4)}t\right){}^2} \\ =\frac{D_{w;0}^{(2)}}{1-t}+\frac{D_{w;{\lambda }_1}^{(2)}}{1-e^{w({\lambda }_1)}t}+\frac{D_{w;{\lambda }_3}^{(2)}}{1-e^{w({\lambda }_3)}t}+\frac{D_{w;{\lambda }_4}^{(2)}}{1-e^{w({\lambda }_4)}t} \\ +\frac{E_{w;{\lambda }_2}^{(2)}}{1-e^{w({\lambda }_2)}{t}^2}+\frac{E_{w;{\lambda }_3}^{(2)}}{1-e^{w({\lambda }_3)}{t}^2}+\frac{E_{w;{\lambda }_4}^{(2)}}{{\left(1-e^{w({\lambda }_4)}t\right)}^2}, \end{gathered}$$

$$\begin{array}{r}{D}_{w;0}^{(2)}=\frac{1}{\left(1-e^{w\left({\lambda }_1\right)}\right)\left(1-e^{w\left({\lambda }_2\right)}\right)\left(1-e^{w\left({\lambda }_3\right)}\right){}^2\left(1-e^{w\left({\lambda }_4\right)}\right){}^2}\end{array}.$$

$${\mathcal{P}}^{(2)}(t)=\frac{1}{\Delta }\sum _{w\in W}(-1{)}^{\ell (w)}{e}^{w(\rho )}\left(\frac{D_{w;0}^{(2)}}{1-t}+\frac{D_{w;{\lambda }_1}^{(2)}}{1-e^{w({\lambda }_1)}t}+\frac{D_{w;{\lambda }_3}^{(2)}}{1-e^{w({\lambda }_3)}t}+\frac{D_{w;{\lambda }_4}^{(2)}}{1-e^{w({\lambda }_4)}t}\right).$$

$$\sum _{w\in W_{\lambda }}(-1{)}^{\ell (w)}{e}^{w(\rho )}{E}_{w;\lambda }^{(2)}=0.$$

$$\sum _{w\in W_{\{1,2\}}}(-1{)}^{\ell (w)}{e}^{w(\rho )}{E}_{w;{\lambda }_2}^{(2)}=0$$

$$\sum _{w^{\prime }\in W^{\prime }}(-1{)}^{\ell (w^{\prime })}{w}^{\prime }\left(\sum _{w\in W_{\{1,2\}}}(-1{)}^{\ell (w)}{e}^{w(\rho )}{E}_{w;{\lambda }_2}^{(2)}\right)$$

$$\sum _{w\in W_{\{1,3\}}}(-1{)}^{\ell (w)}{e}^{w(\rho )}{E}_{w;{\lambda }_3}^{(2)}=0.$$

Full text

1. Introduction

Let $\mathfrak{g}$ be a complex simple Lie algebra. The Kirillov-Reshetikhin (KR) modules constitutes an important family of finite-dimensional irreducible representations of the quantum affine algebra $U_{q}(\widehat{\mathfrak{g}})$. An interesting problem is to understand how a KR module or their tensor product decomposes into irreducible $U_{q}(\mathfrak{g})$-modules. The fermionic formula by Kirillov and Reshetikhin [KR87], proven through a series of works [HKO*+*99, Nak03, Her06, DFK08], gives an answer to this question by expressing the multiplicity of each irreducible summand as a certain combinatorial rule. However, it is difficult to use in practice, and it is often desirable to have a more explicit and computationally cheaper way of decomposing a single KR module.

If $\mathfrak{g}$ is of classical type, there is a well-known explicit formula called the domino removal rule. It is a polyhedral formula in the sense that the highest weight of an irreducible summand with non-zero multiplicity is characterized as a lattice point in a suitable bounded polyhedron. Even when $\mathfrak{g}$ is of exceptional type, a polyhedral formula still seems to exist, but an irreducible summand with multiplicity greater than one may appear. Such a formula with multiplicity remains largely conjectural [HKO*+*99], and furthermore, even a conjectural formula has not been written completely (for example, in type $E_{7}$, $E_{8}$, or $F_{4}$). In fact, the only known polyhedral formula with multiplicity is when $\mathfrak{g}$ is of type $G_{2}$ [CM07] by Chari and Moura. Since their method is rather specific to type $G_{2}$, it seems difficult to adapt it to other cases in general.

In this paper, we propose a method to prove a polyhedral formula. The key objects in our approach are the coefficients that appear when the characters of KR modules are written in some exponential form. They are essentially the residues at the poles of the generating function of the characters of KR modules. It turns out that it is possible to decompose a polyhedral formula into a finite list of identities involving these coefficients; see (3.2). Our method seems quite appropriate for a computer-aided mechanical approach. The difficulty of the actual implementation, of course, varies according to the type of $\mathfrak{g}$. As our main objective in mind is of exceptional type, such a mechanical approach could be justified.

After presenting the general strategy, we consider a special case when $\mathfrak{g}$ is of type $F_{4}$. For each node $a$ of the Dynkin diagram of $\mathfrak{g}$ and $m\in\mathbb{Z}_{\geq 0}$, let us denote the corresponding Kirillov-Reshetikhin module, as a $U_{q}(\mathfrak{g})$-module, by $\operatorname{res}W^{(a)}_{m}$. We obtain a computer-aided proof of the following polyhedral formula conjectured in [HKO*+*99] :

Theorem 1.1**.**

Let $\mathfrak{g}$ be of type $F_{4}$. For every $m\in\mathbb{Z}_{\geq 0}$, the following holds :

[TABLE]

where

[TABLE]

and $(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=(2\omega_{4},2\omega_{3},\omega_{2},\omega_{1})$.

Here we have used the same convention for enumerating the nodes of the Dynkin diagram as in [HKO*+*99].

This paper is organized as follows. In Section 2 we review the necessary background for our approach such as the $Q$-system, and linear recurrence relations satisfied by the characters of KR modules. In Section 3, we explain the steps for proving a polyhedral formula for KR modules. In Section 4, we follow the procedures described in Section 3 to give a proof of Theorem (1.1).

2. Background

notation

We will use the following notation throughout the paper.

• •

$\mathfrak{g}$ : simple Lie algebra over $\mathbb{C}$ of rank $r$

• •

$\mathfrak{h}$ : Cartan subalgebra of $\mathfrak{g}$

• •

$I=\{1,\dots,r\}$ : index set for the Dynkin diagram of $\mathfrak{g}$

• •

$\alpha_{a},\,a\in I$ : simple root

• •

$h_{a},\,a\in I$ : simple coroot

• •

$\omega_{a},\,a\in I$ : fundamental weight

• •

$C=(C_{ab})_{a,b\in I}$ : Cartan matrix with $C_{ab}=\alpha_{b}(h_{a})$

• •

$P=\oplus_{a\in I}\mathbb{Z}\omega_{a}$ : weight lattice

• •

$P^{+}\subseteq P$ : set of dominant integral weights

• •

$Q$ : root lattice

• •

$\rho=\sum_{a\in I}\omega_{a}$ : Weyl vector

• •

$\theta\in Q$ : highest root

• •

$\Delta^{+}$ : set of positive roots

• •

$\mathfrak{h}_{\mathbb{R}}^{*}:=\oplus_{a\in I}\mathbb{R}\omega_{a}$

• •

$(\cdot,\cdot):\mathfrak{h}_{\mathbb{R}}^{*}\times\mathfrak{h}_{\mathbb{R}}^{*}\to\mathbb{R}$ : $\mathbb{R}$-bilinear form induced from the Killing form with $(\theta,\theta)=2$

• •

$\mathbb{Z}[P]$ : integral group ring of $P$ (which is the same as the ring $\mathbb{Z}[e^{\pm\omega_{j}}]_{j\in I}$ of Laurent polynomials in $e^{\omega_{j}}$)

• •

$\mathbb{K}:=\mathbb{C}(e^{\omega_{j}})_{j\in I}$ : field of rational functions in $e^{\omega_{j}}$ with coefficients in $\mathbb{C}$

• •

$t_{a}:=(\theta,\theta)/(\alpha_{a},\alpha_{a})\in\{1,2,3\}$

• •

$[\alpha]_{a}\in\mathbb{Z}\,(\alpha\in Q,\,a\in I)$ : coefficients in the expansion $\alpha=\sum_{a\in I}[\alpha]_{a}\alpha_{a}$

• •

$s_{a}$ : simple reflection acting on $\mathfrak{h}_{\mathbb{R}}^{*}$ by $s_{a}(\lambda)=\lambda-\lambda(h_{a})\alpha_{a}$

• •

$W$ : Weyl group generated by $\{s_{a}:a\in I\}$

• •

$W_{\lambda},\,\lambda\in P$ : isotropy subgroup of $W$ fixing $\lambda$

• •

$W_{J},\,J\subseteq I$ : standard parabolic subgroup of $W$ generated by $\{s_{a}:a\in J\}$

• •

$L(\lambda),\,\lambda\in P^{+}$ : irreducible highest weight representation of $U_{q}(\mathfrak{g})$

• •

$\chi(V)\in\mathbb{Z}[P]$ : character of a finite-dimensional $U_{q}(\mathfrak{g})$-module $V=\oplus V_{\lambda}$ with weight spaces $V_{\lambda}$, i.e. $\chi(V)=\sum_{\lambda\in P}(\dim V_{\lambda})e^{\lambda}$

• •

$\mathcal{O}(\lambda),\,\lambda\in P$ : $W$-orbit of $\lambda$

2.1. Some properties of characters of KR modules

KR modules form a family of irreducible finite-dimensional representations of the quantum affine algebra $U_{q}(\widehat{\mathfrak{g}})$, where $q\in\mathbb{C}$ is not a root of unity. For every $(a,m,u)\in I\times\mathbb{Z}_{\geq 0}\times\mathbb{C}^{\times}$, there exists a corresponding KR module $W^{(a)}_{m}(u)$. By restriction, we obtain a finite-dimensional $U_{q}(\mathfrak{g})$-module $\operatorname{res}\,W^{(a)}_{m}(u)$, which can be denoted by $\operatorname{res}W^{(a)}_{m}$ since its isomorphism class does not depend on $u$, the spectral parameter, as a $U_{q}(\mathfrak{g})$-module.

Let $Q^{(a)}_{m}:=\chi(\operatorname{res}W^{(a)}_{m})$. The $Q$-system

[TABLE]

is a difference equation that the characters of the KR modules satisfy. Nakajima and Hernandez proved the $q$-characters of KR modules satisfy the $T$-system from which we obtain the $Q$-system (2.1) by ignoring the spectral parameter.

In [Lee18a], we studied a linear recurrence relation with constant coefficients that the sequence $(Q^{(a)}_{m})_{m=0}^{\infty}$ satisfies. We can summarize its main properties in terms of its generating function $\mathcal{Q}^{(a)}(t):=\sum_{m=0}^{\infty}Q^{(a)}_{m}t^{m}$ as follows :

Theorem 2.1** ([Lee18a, Theorem 1.1]).**

Let $\mathfrak{g}$ be a simple Lie algebra which is not of type $E_{7}$ or $E_{8}$. For each $a\in I$, there exist $W$-invariant finite subsets $\Lambda_{a}$ and $\Lambda^{\prime}_{a}$ of $P$ with the following properties :

1. (i)

If we set $D^{(a)}(t):=\prod_{\lambda\in\Lambda_{a}}(1-e^{\lambda}t)\prod_{\lambda\in\Lambda_{a}^{\prime}}(1-e^{\lambda}t^{t_{a}})$, then

[TABLE]

is a polynomial in $t$ with coefficients in $\mathbb{Z}[P]$ and $\deg N^{(a)}<\deg D^{(a)}$. 2. (ii)

$t_{a}\Lambda_{a}\cap\Lambda^{\prime}_{a}=\emptyset$, where $t_{a}\Lambda_{a}=\{t_{a}\lambda\mid\lambda\in\Lambda_{a}\}$. 3. (iii)

$\omega_{a}\in\Lambda_{a}$.

Let us fix $\Lambda_{a}$ and $\Lambda^{\prime}_{a}$ as in the Appendix of [Lee18a]. When $\mathfrak{g}$ is simply-laced, $\Lambda_{a}$ is simply the set of weights of the fundamental representation $L(\omega_{a})$. Note that (ii) shows that $D^{(a)}(t)$ has only simple roots.

From the partial fraction decomposition of $\mathcal{Q}^{(a)}(t)=N^{(a)}(t)/D^{(a)}(t)$, we deduce that for each $(\lambda,\zeta,l)\in P\times\mathbb{C}^{\times}\times\mathbb{Z}_{>0}$ there exists $C(\mathcal{Q}^{(a)},\lambda,\zeta,l)\in\mathbb{K}(e^{\omega_{j}/{t_{a}}})_{j\in I}$ such that

[TABLE]

and it vanishes unless either

• •

$(\lambda,\zeta,l)=(\lambda,1,1)$ with $\lambda\in\Lambda_{a}$; or,

• •

$(\lambda,\zeta,l)=(\lambda,\zeta,t_{a})$ with $\lambda\in\Lambda_{a}^{\prime}$ and $\zeta^{t_{a}}=1$.

Due to the $W$-symmetry of $Q^{(a)}_{m}$, we have

[TABLE]

Let $C^{(a)}_{\lambda}:=C(\mathcal{Q}^{(a)},\lambda,1,1)$ for $\lambda\in\Lambda_{a}$ and $C^{(a)}_{\lambda,\zeta}:=C(\mathcal{Q}^{(a)},\lambda,\zeta,t_{a})$ for $\lambda\in\Lambda_{a}^{\prime}$ and $\zeta^{t_{a}}=1$. We can rewrite (2.3) as

[TABLE]

For $C^{(a)}_{\omega_{a}}$ we have an explicit product formula.

Theorem 2.2**.**

[Lee18b]** For each $a\in I$,

[TABLE]

Here, $[\alpha]_{a}\in\mathbb{Z}_{\geq 0}$ denotes the coefficient in the expansion $\alpha=\sum_{a\in I}[\alpha]_{a}\alpha_{a}$.

We call (2.6) the Mukhin-Young formula, which is originally conjectured in [MY14]. We note that, in general, coefficients other than $C(\mathcal{Q}^{(a)},\lambda,1,1)$ do not seem to admit an expression as compact as (2.6).

2.2. fermionic formula

The fermionic formula, proposed by Kirillov and Reshetikhin [KR87], concerns the decomposition of a tensor product of Kirillov-Reshetikhin modules into irreducible $U_{q}(\mathfrak{g})$-modules. Let $(\nu^{(a)}_{m})_{a\in I,m\geq 1}$ be a family of non-negative integers such that $\nu^{(a)}_{m}$ is zero for all but finitely many $(a,m)$. Consider

[TABLE]

a tensor product of Kirillov-Reshetikhin modules, and its decomposition into irreducible $U_{q}(\mathfrak{g})$-representations

[TABLE]

where $L(\lambda)$ denotes an irreducible $U_{q}(\mathfrak{g})$-representation with highest weight $\lambda$. The fermionic formula provides an explicit combinatorial description of the multiplicity $m(W,\lambda)$ in terms of $(\nu^{(a)}_{m})_{a\in I,m\geq 1}$ and $\lambda$. Since this formula is somewhat complicated and not essentially used in this paper, we refer the reader, for example, to [HKO*+*99] for its precise statement.

2.3. polyhedral formula

Let $\theta=\sum_{a\in I}c_{a}\alpha_{a},\,c_{a}\in\mathbb{Z}_{>0}$ be the highest root of $\mathfrak{g}$. Fix $a\in I$ such that $c_{a}\leq 2$. It is shown in [Cha01] (see also [CM06]) that there exist positive integers $(b_{j})_{j\in J_{a}}$, and dominant integral weights $(\lambda_{j})_{j\in J_{a}}$ for some finite set $J_{a}$ such that

[TABLE]

where $F_{m}^{(a)}=\{(x_{j})_{j\in J_{a}}\mid\sum_{j\in J_{a}}b_{j}x_{j}=m,\,x_{j}\in\mathbb{Z}_{\geq 0}\}$, and $\lambda_{\mathbf{x}}=\sum_{j\in J_{a}}x_{j}\lambda_{j}$ for each $\mathbf{x}\in F_{m}^{(a)}$. When $c_{a}>2$, we still expect to have a similar formula, but now with multiplicity, of the form

[TABLE]

where $p$ is a piecewise step-polynomial. A polyhedral formula for the decomposition of KR modules will mean a formula of the form (2.9).

The fermionic formula can be used to decompose $\operatorname{res}W^{(a)}_{m}$ for small individual $m$’s, from which we can observe patterns and guess the form of (2.9). And then, we need a separate argument to prove (2.9) since it is now a formula which is supposed to be true for all $m\in\mathbb{Z}_{\geq 0}$. In the next section, we explain an approach for a proof of (2.9).

3. framework for proving polyhedral formula

Let $P^{(a)}_{m}$ denote the character of the right-hand side of (2.9), i.e.,

[TABLE]

Here the letter $P$ is chosen from the word ‘polyhedron’. When there is such a polyhedral formula, consider its generating function

[TABLE]

which is expected to be a rational function in $t$ in general. Then, $(\ref{KRdecpolyform})$ is equivalent to

[TABLE]

an identity between two rational functions.

We can state the steps necessary to prove (3.1) as follows :

1. (i)

Prove that $\mathcal{P}^{(a)}$ has at most simple poles, and the set of poles of $\mathcal{P}^{(a)}$ is a subset of the set of poles of $\mathcal{Q}^{(a)}$. Also make sure that the degree of the denominator of $\mathcal{P}^{(a)}$ is greater than that of its numerator. 2. (ii)

Prove the equality of the coefficients

[TABLE]

when $(\lambda,\zeta,l)$ belongs to one of the following cases :

• •

$(\lambda,\zeta,l)=(\lambda,1,1)$ with $\lambda\in\Lambda_{a}\cap P^{+}$;

• •

$(\lambda,\zeta,l)=(\lambda,\zeta,t_{a})$ with $\lambda\in\Lambda_{a}^{\prime}\cap P^{+}$ and $\zeta^{t_{a}}=1$.

Recall that we already know explicitly where the poles of $\mathcal{Q}^{(a)}$ are located, which are always simple. We can deduce from (i) that $P^{(a)}_{m}t^{m}$ can be written in the form

[TABLE]

with $C(\mathcal{P}^{(a)},\lambda,\zeta,l)\in\mathbb{K}(e^{\omega_{j}/{t_{a}}})_{j\in I}$, which vanishes unless the non-vanishing conditions for $C(\mathcal{Q}^{(a)},\lambda,\zeta,l)$, stated after (2.3), are satisfied.

Step (ii) is equivalent to showing that $\mathcal{P}^{(a)}$ and $\mathcal{Q}^{(a)}$ have the same residues at their poles, which are simple at most. Since both $\mathcal{P}^{(a)}$ and $\mathcal{Q}^{(a)}$ are $W$-invariant, the coefficients follow the same $W$-symmetry in (2.4). Hence, it is enough to consider weights in $P^{+}$ because every element of $P$ has a unique element in $P^{+}$ in its $W$-orbit.

Once $P^{(a)}_{m}$ is explicitly given as in (2.9), it is more or less straightforward to compute $\mathcal{P}^{(a)}$ and $C(\mathcal{P}^{(a)},\lambda,\zeta,l)$. For computing $C(\mathcal{Q}^{(a)},\lambda,\zeta,l)$, we can use the $Q$-system (2.1) along with previously known $C(\mathcal{Q}^{(b)},\lambda,\zeta,l)$ with $b\in I$ such that $C_{ab}<0$; when there is a known polyhedral formula for $b\in I$, we can explicitly compute $C(\mathcal{Q}^{(b)},\lambda,\zeta,l)$.

Suppose that we already have proved (i). Then another way to finish the proof of $\mathcal{P}^{(a)}(t)=\mathcal{Q}^{(a)}(t)$ is by showing $Q^{(a)}_{m}=P^{(a)}_{m}$ for $m=0,1,\dots,|\Lambda_{a}|+t_{a}|\Lambda_{a}^{\prime}|-1$ since now we know that both sequences satisfy the same linear recurrence relation of order $|\Lambda_{a}|+t_{a}|\Lambda_{a}^{\prime}|-1$. Although it is a purely mechanical task to check $Q^{(a)}_{m}=P^{(a)}_{m}$ for given $m$ using the fermionic formula, we have found that it is still computationally challenging for $m$ large. By considering $C(\mathcal{P}^{(a)},\lambda,\zeta,l)$ and $C(\mathcal{Q}^{(a)},\lambda,\zeta,l)$, we localize the problem in the sense that we are looking at a single pole at a time, and thus obtain further simplifications.

In a nutshell, it is possible to check both (i) and (ii) algorithmically. In the next section, we follow this strategy to prove a conjectural polyhedral formula in type $F_{4}$, where we discuss some practical issues in our method in detail.

Remark 3.1**.**

We know that $C(\mathcal{Q}^{(a)},\lambda,\zeta,l)$ is invariant under $W_{\lambda}$ from (2.4). When we explicitly compute $C(\mathcal{P}^{(a)},\lambda,\zeta,l)$, it is given as a sum over $W_{\lambda}$; see (4.9) for an example. Thus we can regard (3.2) as a summation formula over $W_{\lambda}$ for $C(\mathcal{Q}^{(a)},\lambda,\zeta,l)$.

4. Proof of Theorem 1.1

Let $\mathfrak{g}$ be a simple Lie algebra of type $F_{4}$. When $a=1$ or $4$, there is a known polyhedral formula for $\operatorname{res}W^{(a)}_{m}$. The formula for $a=1$ is given in (4.8) and will be used later. The main goal of this section is to prove Theorem 1.1, namely, the polyhedral formula for $a=2$.

For $m\in\mathbb{Z}_{\geq 0}$, let $P^{(2)}_{m}$ be the character of the right-hand side of (1.1) and $\mathcal{P}^{(2)}(t)=\sum_{m=0}^{\infty}P^{(2)}_{m}t^{m}$. By following the strategy outlined in Section 3, we will show that

[TABLE]

that is,

[TABLE]

as rational functions in $t$.

Before turning to proofs, we present a table for $\Lambda_{a}$ and $\Lambda_{a}^{\prime}$ from [Lee18a]. Since they are $W$-invariant, they are given as a disjoint union of $W$-orbits of elements of $P^{+}$ :

[TABLE]

Throughout the section, $\Delta=\sum_{w\in W}(-1)^{\ell(w)}e^{w(\rho)}$ denotes the Weyl denominator and $(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=(2\omega_{4},2\omega_{3},\omega_{2},\omega_{1})$. We often need to explicitly deal with $W$ or its subgroups. One may refer to [Sno90] for an algorithm to find the Weyl orbit of a weight or a minimal coset representative for a coset of a standard parabolic subgroup of a Weyl group. The accompanying Mathematica notebook file for some computer calculations is available at https://github.com/chlee-0/KR-polyhedral-formula.

4.1. Step (i)

From (2.2), it is clear that $\mathcal{Q}^{(2)}(t)$ can have only poles of order at most 1 and they can only be found at $t=e^{-\lambda},\,\lambda\in\Lambda_{2}$. Let us find the poles of $\mathcal{P}^{(2)}(t)$. To write $\mathcal{P}^{(2)}(t)$ explicitly, define a sequence $\{a_{m}\}_{m=0}^{\infty}$ by

[TABLE]

whose generating function is

[TABLE]

By combining this with the Weyl character formula, $\mathcal{P}^{(2)}(t)$ can be written as

[TABLE]

At this point, it is not entirely clear whether $\mathcal{P}^{(2)}(t)$ has only simple poles at $t=e^{-\lambda},\,\lambda\in\Lambda_{2}$ or not. For example, $\mathcal{P}^{(2)}(t)$ may have a double pole at $t=e^{-\lambda_{4}}$.

Consider the partial fraction decomposition of a summand in (4.2) :

[TABLE]

where $D^{(2)}_{w;\lambda},\,E^{(2)}_{w;\lambda_{4}}\in\mathbb{K}$, and $E^{(2)}_{w;\lambda_{2}}$ and $E^{(2)}_{w;\lambda_{3}}\in\mathbb{K}[t]$ are polynomials of degree at most 1. They are uniquely determined by this form of decomposition. For example,

[TABLE]

Because these expressions are long but easy to find, we do not write them here; one can refer to the accompanying file for an explicit description.

Proposition 4.1**.**

We have

[TABLE]

Proof.

It is sufficient to show that for $\lambda\in\{\lambda_{2},\lambda_{3},\lambda_{4}\}$,

[TABLE]

We may use computers to verify this directly. Below, we will explain how to reduce the amount of calculation to check (4.5). While this reduction is not essential as long as we focus on type $F_{4}$ whose Weyl group is manageable in size, it might be useful for treating a similar vanishing sum over bigger groups in other types.

For $\lambda=\lambda_{2}=2\omega_{3}$, $W_{\lambda}=W_{\{1,2,4\}}$. The parabolic subgroup $W_{\{1,2\}}$ of $W_{\{1,2,4\}}$ satisfies

[TABLE]

which implies the vanishing of the sum over $W_{\{1,2,4\}}$ since (4.5) can be written as

[TABLE]

where $W^{\prime}$ is the set of minimal coset representatives for cosets in $W_{\{1,2,4\}}/W_{\{1,2\}}$.

Similarly, when $\lambda=\lambda_{3}=\omega_{2}$, we have $W_{\lambda}=W_{\{1,3,4\}}$ and the following sum over the parabolic subgroup $W_{\{1,3\}}$ vanishes :

[TABLE]

When $\lambda=\lambda_{4}$, we have not found any proper parabolic subgroup of $W_{\lambda}=W_{\{2,3,4\}}$, over which the sum vanishes. However, if we let

[TABLE]

then the left-hand side of (4.5) becomes

[TABLE]

where $W^{\prime\prime}$ is the set of minimal coset representatives for cosets in $W_{\{2,3,4\}}/W_{\{2,4\}}$. The size of $W^{\prime\prime}$ is 12, and it is possible to partition this set into 6 pairs of distinct elements so that the contribution from each pair to the above sum is zero. In other words, for each $w_{1}\in W^{\prime\prime}$, there exists $w_{2}\in W^{\prime\prime},\,w_{1}\neq w_{2}$ such that

[TABLE]

∎

This proposition immediately implies the following :

Proposition 4.2**.**

The rational function $\mathcal{P}^{(2)}(t)$ has only simple poles, possibly at $t=e^{-\lambda}$, $\lambda\in\Lambda_{2}$ and no other poles.

Now we know that the poles of $\mathcal{Q}^{(2)}(t)$ and $\mathcal{P}^{(2)}(t)$ can only appear at $t=e^{-\lambda},\,\lambda\in\Lambda_{2}$.

4.2. Step (ii)

It remains to carry out the second step of our strategy in Section 3 to prove (4.1). As $\Lambda^{\prime}_{2}$ is empty, we have to show

[TABLE]

for $\lambda\in\{0,\omega_{1},\omega_{2},2\omega_{4}\}\in\Lambda_{2}\cap P^{+}$. We first explain how to compute both sides, and then check their equality. Let us write $C^{(2)}_{\lambda}=C(\mathcal{Q}^{(2)},\lambda,1,1)$ and $D^{(2)}_{\lambda}=C(\mathcal{P}^{(2)},\lambda,1,1)$.

how to calculate $C^{(2)}_{\lambda}$

Let us explain how to calculate $C^{(2)}_{\lambda}$. Recall the $Q$-system relation (2.1)

[TABLE]

for $a=2$. By rewriting this relation using (2.5), we obtain an expression for $C^{(2)}_{\lambda}$ in terms of $C^{(1)}_{\mu},\,\mu\in\Lambda_{1}$ :

[TABLE]

where $S_{\lambda}:=\{(\nu,\mu)\in\Lambda_{1}\times\Lambda_{1}:\nu\neq\mu,\,\nu+\mu=\lambda\}$.

To handle (4.6) explicitly, we need a way to compute $C^{(1)}_{0}$ and $C^{(1)}_{\omega_{1}}$. And these are all we need to find $C^{(1)}_{\mu},\,\mu\in\Lambda_{1}$ because $C^{(1)}_{\lambda}$ with non-zero $\lambda\in\mathcal{O}(\omega_{1})$ is given by $C^{(1)}_{w(\omega_{1})}=w(C^{(1)}_{\omega_{1}})$.

Lemma 4.3**.**

We have

[TABLE]

and

[TABLE]

Proof.

Note that $C^{(1)}_{\omega_{1}}$ is given by Theorem 2.2, the Mukhin-Young formula. To find $C^{(1)}_{0}$, we can exploit the known polyhedral formula from [Cha01]

[TABLE]

By the Weyl character formula,

[TABLE]

where

[TABLE]

Therefore,

[TABLE]

∎

how to calculate $D^{(2)}_{\lambda}$

Recall that we have

[TABLE]

By Proposition 4.1, we can write $D^{(2)}_{\lambda}$ as

[TABLE]

For a dominant weight $\lambda$, $W_{\lambda}$ is a standard parabolic subgroup of $W$. Once we enumerate the elements of $W_{\lambda}$, it is straightforward to compute $D^{(2)}_{\lambda}$. Of course, it becomes computationally easier to manipulate (4.9) when $W_{\lambda}$ is a proper subgroup of $W$. In this sense, the most difficult case arises when $\lambda=0$.

Now we can compute both (4.6) and (4.9) and thus, are ready to check $C^{(2)}_{\lambda}=D^{(2)}_{\lambda}$ for $\lambda\in\{0,\omega_{1},\omega_{2},2\omega_{4}\}$. To use (4.6) we need $S_{\lambda}=\{(\nu,\mu)\in\Lambda_{1}\times\Lambda_{1}:\nu\neq\mu,\,\nu+\mu=\lambda\}$, as described below. The cases of $\lambda\in\{\omega_{1},\omega_{2},2\omega_{4}\}$ do not bring much difficulty, and a computer can easily simplify $C^{(2)}_{\lambda}-D^{(2)}_{\lambda}$ and return zero. We give further comments on the $\lambda=0$ case, which is the most difficult one.

4.2.1. $\lambda=\omega_{1}$ case

[TABLE]

4.2.2. $\lambda=\omega_{2}$ case

In this case, $S_{\omega_{2}}=\left\{(\omega_{1},\omega_{2}-\omega_{1}),(\omega_{2}-\omega_{1},\omega_{1})\right\}$. Thus (4.6) gives

[TABLE]

In fact, this identity is a special case of [Lee18b, Corollary 3.5].

4.2.3. $\lambda=2\omega_{4}$ case

[TABLE]

4.2.4. $\lambda=0$ case

Our goal is to check whether $C^{(2)}_{0}-D^{(2)}_{0}$ is actually zero, but this calculation is not quite straightforward as before, since they are quite huge rational functions. (4.6) can be rewritten as

[TABLE]

and $|\mathcal{O}(\omega_{1})|=24$. And $D^{(2)}_{0}$ involves an alternating sum of orbits of (4.4) over the entire Weyl group $W$ and hence, it is obtained by adding $|W|=1152$ rational functions in $e^{\omega_{1}},\dots,e^{\omega_{4}}$.

It is slightly better to work with $\Delta C^{(2)}_{0}$ and $\Delta D^{(2)}_{0}$ to simplify their denominators. Let us consider

[TABLE]

We can rewrite the above as

[TABLE]

where $W^{\{1,3,4\}}$ denotes the set of minimal coset representatives of cosets in $W/W_{\{1,3,4\}}$. Note that the size of $W^{\{1,3,4\}}$ is 96. Let

[TABLE]

In our computer calculation, we further considered a partition of $W^{\{1,3,4\}}$ into 9 disjoint subsets $W^{\{1,3,4\}}_{i},\,i=1,\dots,9$, say, $W^{\{1,3,4\}}=\bigsqcup_{i=1}^{9}W^{\{1,3,4\}}_{i}$. We can write (4.10) as

[TABLE]

Finally, we start with $\Delta C^{(2)}_{0}$, subtract $(\sum_{w^{\prime}\in W^{\{1,3,4\}}_{i}}f_{w^{\prime}})$, and simplify the expression at each step $i=1,\dots,9$. Once we subtract every summand, the result becomes be zero, as we wanted. On our desktop computer with a 3.50GHz CPU and 8GB of RAM, it took about 1450 seconds to complete this calculation. We note that the partition for $W^{\{1,3,4\}}$ we used is simply found through many computer experiments to reduce the time required to complete the calculation, and may be hardly optimal.