On a conjecture about an analogue of Tokuyama’s theorem for $G_2$

Abstract

We prove the conjecture of Friedlander et al. (J Algebr Comb 41:1089, 2015) about sums over Littelmann patterns for the root system of type $G_2$, which is an analogue of Tokuyama’s theorem Tokuyama (J Math Soc Jpn 40(4):671–685, 1988) for root systems of type $A_r$. We use elementary means to show that the conjecture is implied by a finite set of polynomial identities.

1 Introduction

For integer $r\ge 2$, let $\mathfrak {s}\mathfrak {l}_{r}(\mathbb {C})$ be the Lie algebra of ${\displaystyle r\times r}$ complex matrices with trace zero equipped with the Lie bracket $[X,Y] = XY - YX$. Let $\theta$ be a dominant weight for $\mathfrak {s}\mathfrak {l}_{r}(\mathbb {C})$, and let $V(\theta )$ be the irreducible representation of highest weight $\theta$. A basis for $V(\theta )$ can be given in terms of a set of Gel’fand–Cetlin patterns $GC(\theta )$; this is a set of ordered tuples of integers satisfying certain inequalities depending on $\theta$ and is closely connected to the canonical bases of [6].

Now let $\chi (\theta )$ be the character of $V(\theta )$, and let $\rho$ be the sum of the fundamental weights of $\mathfrak {s}\mathfrak {l}_{r}(\mathbb {C})$. Then Tokuyama’s theorem gives a formula for $\chi (\theta )$ in the form

$$\begin{aligned} \sum _{P \in GC(\theta + \rho )} H(P) = D(\mathbf x ,q; A_{r-1}) \chi (\theta ) \end{aligned}$$

(1)

where $D(\mathbf x ,q; A_{r-1})$, defined in Definition 2, is a q-deformation of the usual Weyl denominator, and where each summand H(P) is a polynomial in $q^{-1}$ computed using statistics taken from the pattern P. Therefore, Tokuyama’s theorem connects two fundamental objects in the representation theory of $\mathfrak {s}\mathfrak {l}_r(\mathbb {C})$.

We note that since a semisimple Lie algebra over an algebraically closed field is determined by its root system, Tokuyama’s theorem can be expressed solely in terms of the root system $A_{r-1}$ of $\mathfrak {s}\mathfrak {l}_{r}(\mathbb {C})$.

A major question is how Eq. (1) generalizes to other root systems. Now, the polynomial $D(\mathbf x ,q; A_{r-1})$ has a natural generalization $D(\mathbf x ,q; R)$ to arbitrary root systems R, as does $\chi (\theta )$ by construction, so the right side of Eq. (1) is easily generalized. A series of papers mentioned below seek combinatorial structures that generalize the left side of (1) for various root systems R. This paper proves a conjecture of Friedlander et al. [5], called Theorem 2 in this paper, for such a generalization to $G_2$ using an elementary bijective proof.

Generalizing Tokuyama’s theorem is a major question in part because the right side of Eq. (1) is the same formula that provides a closed-form evaluation of certain functions called matrix coefficients. These matrix coefficients arise from the principal series representation of an algebraic group over a non-Archimedean field. More specifically, such a matrix coefficient is defined by a certain integral of a canonical function on the algebraic group, where that function is viewed as a vector in the principal series representation of the algebraic group. Shintani [10] and Casselman–Shalika [2] evaluate the integral for one such matrix coefficient, known the spherical-Whittaker function, to be the right side of (1). Generalizations of Tokuyama’s theorem thus would provide insight into evaluating integrals over algebraic groups and the structure of algebraic groups in general.

In addition, Tokuyama’s theorem has played an important part in the development of Weyl group multiple Dirichlet series. These are Dirichlet series in multiple variables and are defined in terms of a given root system. A Weyl group multiple Dirichlet series possesses a set of functional equations isomorphic to the Weyl group of that root system. These series are also conjectured to be the Fourier–Whittaker coefficients of certain Eisenstein series on metaplectic groups. This relationship is described in more detail in [1].

We now mention generalizations of Tokuyama’s theorem to other root systems. Tokuyama’s theorem has been generalized to type C by Hamel and King [7] in which they use symplectic shifted tableaux in place of Gel’fand–Cetlin patterns; and to type B by Friedberg and Zhang [4] in which they use metaplectic double covers in their proof. A conjectural generalization to type D was given by Chinta and Gunnells [3]. For arbitrary root systems, McNamara [9] shows how to compute p-adic Whittaker functions as sums over crystal graphs. For type A, these sums may be interpreted as sums over Gel’fand–Cetlin patterns mentioned above, but have not been computed explicitly for other root systems. The conjecture of [5] (Theorem 2 here) is the first conjecture about an analogue of Tokuyama’s theorem for an exceptional root system.

We now describe our proof strategy and compare it to the work mentioned above. In [11], Tokuyama proves his original theorem using the Pieri rule applied to Schur polynomials. Our proof of Theorem 2 is different from the proof in [11] and other proofs for the analogues to other root systems in that the proof in this paper is purely algebraic, using elementary combinatorics to establish a bijection. Specifically, we express both sides of Theorem 2 as polynomials in four indeterminates whose coefficients are rational functions. We then show that the coefficients are equal. We demonstrate these techniques for the root system $A_2$ in Sect. 3.

2 Background material for Littelmann patterns and root systems

The authors of [5] replace the Gel’fand–Cetlin patterns of (1) with a set of inequalities called Littelmann patterns [8]. Littelmann patterns can be defined for any root system R and yield the Gel’fand–Cetlin patterns when $R=A_r$. We now briefly describe these Littelmann patterns following the description in [5].

Let $\mathfrak {g}$ be a simple complex Lie algebra with root system R. Let $w_L$ be a reduced expression for the longest element in the Weyl group of R. Let $\theta$ be a dominant weight, and let $V_\theta$ the irreducible representation of $\mathfrak {g}$ with lowest weight $-\theta$. As noted in [5], for some root systems, such as $G_2$, $V_\theta$ coincides with the representation with highest weight $\theta$, but differs for other root systems, such as $A_r$. Ultimately what matters for us is the normalized definition of $\chi _\theta$ in Definition 1 and the Littelmann inequalities discussed next.

Given the above data, Littelmann describes how to construct a rational polyhedral cone $C_\theta \in \mathbb {R}^N$, where N is the number of positive roots in R. He also constructs a bijection between the lattice points of $C_\theta$ and the vertices of the crystal graph $B(\theta )$. This $B(\theta )$ is a finite directed graph constructed from representation-theoretic properties of $V_\theta$. The Littelmann patterns $\pi$ are then N-tuples of non-negative integers that index the lattice points in $C_\theta$. The pattern $\pi$ satisfies certain inequalities which we call Littelmann inequalities below. When $R=A_r$ and $w_L = s1(s_2 s_1)(s_3 s_2 s_1)...(s_r...s_1)$, the Littelmann patterns are equivalent to Gel’fand–Cetlin patterns. No knowledge of crystal graphs is necessary to understand the proof in this paper; we need to consider only the Littelmann inequalities themselves.

We review the concepts related to root systems necessary to state Theorem 2. Let R be an irreducible root system of rank r in a vector space V with inner product $\langle \cdot ,\cdot \rangle$. Let $R_+$ denote a choice of positive roots in R with simple roots $\alpha _1, \ldots , \alpha _r$. Let $\varpi _1, \ldots , \varpi _r \in V$ denote the fundamental weights defined by

$$\begin{aligned} 2\frac{(\varpi _i, \alpha _j)}{(\alpha _j, \alpha _j)} = \delta _{ij}. \end{aligned}$$

Let $\varLambda _W \subset V$ denote the weight lattice, and $\varLambda \subset V$ the root lattice, and $\mathbb {C}[\varLambda ]$ the associated ring of Laurent polynomials. Identifying $x^{\alpha _i}$ with $x_i$, we express $\mathbb {C}[\varLambda ]$ as $\mathbb {C}[x_1^{\pm }, \ldots , x_r^{\pm }]$. For any linear combination $\theta$ of the $\alpha _i$

$$\begin{aligned} \theta = \sum _{i=1}^r c_i \alpha _i \end{aligned}$$

for some rational numbers $c_i$, we write

$$\begin{aligned} \mathbf x ^\theta = \prod _{i=1}^r x_i^{c_i}. \end{aligned}$$

Let $\rho$ denote the half-sum of positive roots

$$\begin{aligned} \rho =\frac{1}{2}\sum _{\alpha \in R_+} \alpha = \sum _{i=1}^r \varpi _i. \end{aligned}$$

For a dominant weight $\theta \in \varLambda _W$, the weight $\theta + \rho$ is equal to

$$\begin{aligned} \theta + \rho = \sum _{i=1}^r \ell _i \varpi _i \end{aligned}$$

for some positive integers $\ell _i$. Let W be the Weyl group of R generated by the simple reflection $s_i$. For $w\in W$, let $\mathrm {length}(w)$ be the length of a reduced expression for w and let $\mathrm {sgn}(w)$ be

$$\begin{aligned} \mathrm {sgn}(w) = (-1)^{\mathrm {length}(w)}. \end{aligned}$$

Now let $w_L$ denote the long element in W. We define $\chi _\theta$ to be the character of $V_\theta$ normalized to have constant term 1:

Definition 1

$$\begin{aligned} \chi _\theta = \frac{\mathrm {sgn}(w_L)}{ \mathbf x ^{w_L(\theta +\rho )})}\frac{\sum _{w \in W} \mathrm {sgn}(w) \mathbf x ^{w(\theta +\rho )}}{\prod _{\alpha >0}(1-x^\alpha )}. \end{aligned}$$

We next define the notation for factored expressions alluded to in the introduction.

Definition 2

Let q is an indeterminate. For $\alpha \in R$, denote

$$\begin{aligned} D(\mathbf{x}, q;\alpha ) = \left( 1 - \frac{\mathbf{x}^\alpha }{q}\right) \end{aligned}$$

and

$$\begin{aligned} T(\mathbf{x},q; \alpha )= \frac{\left( 1 - \frac{\mathbf{x}^\alpha )}{q}\right) }{(1- \mathbf{x}^\alpha )}. \end{aligned}$$

We also denote

$$\begin{aligned} D(\mathbf x , q;R)&= \prod _{\alpha \in R_+}D(\mathbf x , q;\alpha )\\ T(\mathbf x , q;R)&= \prod _{\alpha \in R_+}T(\mathbf x , q;\alpha ). \end{aligned}$$

3 Demonstration for $A_2$

We demonstrate our proof techniques for Tokuyama’s theorem for $\mathfrak {g}=\mathfrak {s}\mathfrak {l}_3(\mathbb {C})$ with root system $R=A_2$. The result we prove is

Theorem 1

$$\begin{aligned} \sum _{GC(\ell _1, \ell _2)} h(a_{11})h(a_{12})h(a_{21})x^{a_{11}+a_{12}}y^{a_{21}} = D(\mathbf x ,q; A_2) \chi _\theta \end{aligned}$$

where the terms are defined below. Note that $\chi _\theta$ denotes the character normalized to have constant term 1 given in Eq. (2).

For $A_2$, the positive simple roots are

$$\begin{aligned} \alpha _1&= (\sqrt{2},0) \\ \alpha _2&= \left( -\frac{\sqrt{2}}{2},\frac{\sqrt{6}}{2}\right) . \end{aligned}$$

The fundamental weights $\varpi _1$ and $\varpi _2$

$$\begin{aligned} \varpi _1&= \frac{2}{3}\alpha _1 + \frac{1}{3}\alpha _2\\ \varpi _2&= \frac{1}{3}\alpha _1 + \frac{2}{3}\alpha _2. \end{aligned}$$

The simple reflections $s_1$ and $s_2$ act linearly on R according to $s_i(\alpha _i) =-\alpha _i$ and

$$\begin{aligned} s_1(\alpha _2)&= s_2(\alpha _1) =\alpha _1+\alpha _2. \end{aligned}$$

Recall $\theta \in \varLambda _W$ denotes a dominant weight and

$$\begin{aligned} \theta +\rho = \ell _1 \varpi _1+ \ell _2 \varpi _2 \end{aligned}$$

where $\ell _1$ and $\ell _2$ are positive integers. With $x=x_1$ and $y=x_2$, the normalized character $\chi _\theta$ becomes

$$\begin{aligned} \chi _\theta =\frac{1-x^{\ell _2}-y^{\ell _1} - (x y)^{\ell _1+\ell _2}+x^{\ell _1+\ell _2}y^{\ell _1}+x^{\ell _2}y^{\ell _1+\ell _2}}{(1-x)(1-y)(1-xy)}. \end{aligned}$$

(2)

Therefore

$$\begin{aligned} D(\mathbf x ,q; A_2) \chi _\theta = T(\mathbf x ,q; A_2)(1-x^{\ell _2}-y^{\ell _1} - (x y)^{\ell _1+\ell _2}+x^{\ell _1+\ell _2}y^{\ell _1}+x^{\ell _2}y^{\ell _1+\ell _2}) \end{aligned}$$

(3)

We consider (3) as a polynomial $P_{W;A_2}$ in the four indeterminates $x^{\ell _1}, y^{\ell _1}, x^{\ell _2}$ and $y^{\ell _2}$. We say that the term

$$\begin{aligned} (x^{m_1}y^{n_1})^{\ell _1} (x^{m_2}y^{n_2})^{\ell _2} \end{aligned}$$

has multi-degree $((m_1,n_1),(m_2,n_2))$. Therefore the polynomial $P_{W;A_2}$ has six terms with the multi-degrees and coefficients listed in Table 1.

Table 1 Terms for $P_{W;A_2}$

We define the set $GC(\ell _1,\ell _2)$ of Gel’fand–Cetlin patterns $\pi$ for $A_2$. Consider the set of triples of integers $(A_{11}, A_{12}, A_{22})$ such that

• 1. $0 \le A_{11} \le \ell _1$

• 2. $\ell _1 \le A_{12} \le \ell _1+\ell _2$

• 3. $A_{11} \le A_{21} \le A_{12}$

Such a triple is called a Gel’fand–Cetlin pattern. These patterns are identified with the Littelmann patterns for $A_2$ after the following change of variables:

$$\begin{aligned} A_{11}= a_{11}, \, A_{12} = \ell _1+a_{12},\, A_{21} =a_{21}. \end{aligned}$$

The above inequalities then become

• 1. $0\le a_{11} \le \ell _1$

• 2. $0 \le a_{12} \le \ell _2$

• 3. $a_{11} \le a_{21} \le \ell _1+a_{12}$.

These are the Littelmann inequalities for the weight $\theta +\rho = \ell _1\varpi _1+\ell _2\varpi _2$ and reduced word $s_1s_2 s_1$. Define $GC(\ell _1,\ell _2)$ to be the set of 3-tuples of integers $\pi = (a_{11}, a_{12}, a_{21})$ that satisfy the Littelmann inequalities. We denote the lower bounds of these inequalities by $L_{ij}$ and the upper bounds by $U_{ij}$:

$$\begin{aligned} L_{11} = 0, L_{12}=0, L_{21}=a_{11} \end{aligned}$$

and

$$\begin{aligned} U_{11} = \ell _1, U_{12}=\ell _2, U_{21}=\ell _1+a_{12}. \end{aligned}$$

Following the terminology of [5], we say that an entry $a_{ij}$ of $\pi$ is “circled”, denoted by $a_{ij}^\circ$, if $a_{ij}$ attains the lower bound in its inequality; i.e., $a_{ij}$ is circled if $a_{ij}=L_{ij}$. We say that $a_{ij}$ is “boxed”, denoted by $\underline{a_{ij}}$, if $a_{ij}$ attains its upper bound in its inequality; i.e., $a_{ij}$ is boxed if $a_{ij} = U_{ij}$.

We then define $h(a_{ij}) \in \mathbb {Z}[q^{-1}]$ by

Definition 3

$$\begin{aligned} h(a_{ij}) = {\left\{ \begin{array}{ll} 1, &{} \quad \text {if } \, a_{ij} \text { is circled and not boxed} \\ 1-q^{-1}, &{} \quad \text {if } a_{ij} \, \text { is neither circled nor boxed} \\ -q^{-1}, &{} \quad \text {if } \, a_{ij} \text { is boxed and not circled} \\ 0, &{} \quad \text {if } \, a_{ij} \text { is both boxed and circled}. \end{array}\right. } \end{aligned}$$

Now we prove Theorem 1. We note that this proof can be generalized to $A_r, r\ge 1$ to provide an elementary proof of Tokuyama’s theorem which we give in another paper.

Proof

We compute the sum

$$\begin{aligned} \sum _{\pi \in GC(\ell _1,\ell _2)} h(a_{11})h(a_{12})h(a_{21})x^{a_{11}+a_{12}}y^{a_{21}}. \end{aligned}$$

We first perform the sum over $a_{21}$ and then over $a_{11}$ and $a_{12}$. By Lemma 1 in Sect. 5,

$$\begin{aligned} \sum _{a_{21}=a_{11} }^{\ell _1+a_{12}} h(a_{21})y^{a_{21}} = \frac{(1-\frac{y}{q})}{1-y}(y^{a_{11}}-y^{\ell _1+a_{12}}) \end{aligned}$$

(4)

We remember the factor $\frac{1-\frac{y}{q}}{1-y}=T(\mathbf x , q; \alpha _2)$ and take the $y^{a_{11}}$ term in (4). We sum this term over $a_{11}$ and $a_{12}$ again using Lemma 1 in Sect. 5 to get

$$\begin{aligned}&\sum _{a_{12}=0}^{\ell _2} \sum _{a_{11}=0}^{\ell _1} (x y)^{a_{11}} x^{a_{12}} = \left( \sum _{a_{11}=0}^{\ell _1} (x y)^{a_{11}}\right) \left( \sum _{a_{12}=0}^{\ell _2} x^{a_{12}}\right) \nonumber \\&\quad =\left( \frac{\left( 1-\frac{ xy }{q}\right) }{1-xy}(1-(xy)^{\ell _1})\right) \left( \frac{\left( 1-\frac{ x }{q}\right) }{1-x}(1-x ^{\ell _2})\right) . \end{aligned}$$

(5)

Similarly we take the $y^{\ell _1+a_{12}}$ term in (4) and sum over $a_{11}$ and $a_{12}$ to get

$$\begin{aligned}&-y^{\ell _1}\sum _{a_{12}=0}^{\ell _2} \sum _{a_{11}=0}^{\ell _1} x^{a_{11}} (xy)^{a_{12}} = -y^{\ell _1}\left( \sum _{a_{11}=0}^{\ell _1} x^{a_{11}}\right) \left( \sum _{a_{12}=0}^{\ell _2} (xy)^{a_{12}}\right) \nonumber \\&\quad =-y^{\ell _1}\left( \frac{\left( 1-\frac{ x }{q}\right) }{1-x }(1-x^{\ell _1})\right) \left( \frac{\left( 1-\frac{ xy }{q}\right) }{1-xy}(1-(xy)^{\ell _2})\right) . \end{aligned}$$

(6)

Adding (5) and (6) and multiplying by $T(\text {x}, q; \alpha _2)$ gives

$$\begin{aligned}&T(\mathbf x ,q; A_2) \left( (1-(xy)^{\ell _1})(1-x^{\ell _2}) - y^{\ell _1}(1-x^{\ell _1})(1-(xy)^{\ell _2})\right) \\&\quad = T(\mathbf x ,q; A_2) (1 - x^{\ell _2}+ x^{\ell _2}(xy)^{\ell _1}-y^{\ell _1}+y^{\ell _1}(xy)^{\ell _2})-(xy)^{\ell _2}(xy)^{\ell _1}). \end{aligned}$$

This proves Tokuyama’s theorem for $R=A_2$. $\square$

4 Statement of conjecture

We define the terms necessary to state Theorem 2.

Fig. 1

The root system $G_2$

Let $\mathfrak {g}$ be the simple complex Lie algebra of type with root system $R=G_2$ (see Fig. 1). The simple roots of R are

$$\begin{aligned} \alpha _1&= (\sqrt{2},0)\\ \alpha _2&= \left( -\frac{3\sqrt{2}}{2}, \frac{\sqrt{6}}{2}\right) . \end{aligned}$$

The fundamental weights are therefore

$$\begin{aligned} \varpi _1&= 2\alpha _1+\alpha _2\\ \varpi _2&= 3\alpha _1+2\alpha _2. \end{aligned}$$

The Weyl group W has order 12 and is generated by the simple reflections $s_1$ and $s_2$, which act on R via $s_i(\alpha _i) = -\alpha _i$ and

$$\begin{aligned} s_1(\alpha _2)&= 3\alpha _1+\alpha _2\\ s_2(\alpha _1)&= \alpha _1+\alpha _2. \end{aligned}$$

Letting $x_1 = x$ and $x_2=y$, we have

$$\begin{aligned} D(\mathbf x ,q; G_2)=\left( 1-\frac{x}{q}\right) \left( 1-\frac{y}{q}\right) \left( 1-\frac{x y}{q}\right) \left( 1-\frac{x^2 y}{q}\right) \left( 1-\frac{x^3 y}{q}\right) \left( 1-\frac{x^3 y^2}{q}\right) . \end{aligned}$$

Let $\theta$ be a dominant weight. Then $\theta +\rho = \ell _1\varpi _1+\ell _2\varpi _2$. Recall that the set $B(\theta +\rho )$ is the crystal graph whose vertex set is identified with the set of Littelmann patterns; we thus refer to $B(\theta +\rho )$ as the set of Littelmann patterns which are 6-tuples $\pi =(a,b,c,d,e,f)$ such that the entries a, b, c, d, e, f are non-negative integers that satisfy the following Littelmann inequalities.

Littelmann Inequalities:

• 1. $0 \le \, f \le \ell _2 +a - 2b+ c-2d+e$

• 2. $b \le \, a \le \ell _1 +3b - 2c+ 3d-2e$

• 3. $\frac{c}{2} \le \, b \le \ell _2 +c-2d+e$

• 4. $2d \le \, c \le \ell _1 +3d-2e$

• 5. $e \le \, d \le \ell _2 +e$

• 6. $0 \le \, e \le \ell _1$

These are the inequalities that come from choice of long word $s_2 s_1 s_2 s_1 s_2 s_1$. It is straightforward to check that inequalities $i+1$ through 6 imply that the lower bound of inequality i is less than or equal to the upper bound of inequality i. We refer to inequalities 1 through 6 as the inequalities for the entries f, a, b, c, d and e, in that order. As defined for the root system $A_2$ above, we say that an entry u of $\pi$ is “circled”, denoted by $u^\circ$, if u attains the lower bound in its inequality; e.g., f is circled if $f=0$, a is circled if $a=b$, b is circled if $b=\frac{c}{2}$, etc. Likewise, we say that u is “boxed”, denoted by $\underline{u}$, if u attains its upper bound in its inequality; e.g., f is boxed if $f = \ell _2 +a - 2b+ c-2d+e$, a is boxed if $a= \ell _1 +3b - 2c+ 3d-2e$, b is boxed if $b= \ell _2 +c-2d+e$, etc. Then for any Littelmann pattern $\pi$, the authors of [5] define $\hat{H}(\pi )$ to be a certain polynomial in $\mathbb {Z}[q^{-1}]$. We express this $\hat{H}(\pi )$ as a sum of two terms

$$\begin{aligned} \hat{H}(\pi )= H_\mathrm {std}(\pi )+ H_{\mathrm{adj}}(\pi ) \end{aligned}$$

where $H_{\mathrm{std}}(\pi )$ encompasses the “standard” terms that come from Tokuyama’s original definition; and $H_{\mathrm{adj}}(\pi )$ encompasses the special cases, or “adjusted” terms. That is,

$$\begin{aligned} H_\mathrm {std}(\pi ) = h(a)h(b)h(c)h(d)h(e)h(f) \end{aligned}$$

(7)

where h(u) is defined by Definition 3. The $H_{\mathrm{adj}}(\pi )$ is defined below via Eqs. (8) and (9).

Their conjecture is then

Theorem 2

$$\begin{aligned} \sum _{\pi \in B(\theta +\rho )} \hat{H}(\pi ) \, x^{a+c+e} y^{b+d+f} = D(\mathbf x ,q;R_+) \chi _\theta \end{aligned}$$

We note that to define $\hat{H}(\pi )$, the authors of [5] use a definition depending on whether $\pi$ is generic or one of twenty special cases. Considering $H_{\mathrm{adj}}(\pi )$ on its own allows us to consolidate the special cases, to simplify their characterization, and also to simplify the values of $H_{\mathrm{adj}}(\pi )$. We also see that the after expressing

$$\begin{aligned} \sum _{\pi \in B(\theta +\rho )} H_{\mathrm{adj}}(\pi ) \, x^{a+c+e} y^{b+d+f} \end{aligned}$$

as a polynomial, the coefficients have a factored form similar to that of $T(\mathbf x ,q; R_+)$.

We now define $H_{\mathrm{adj}}(\pi )$ by consolidating below the twenty special cases used by [5]. The adjusted contribution $H_{\mathrm{adj}}(\pi )$ is defined in general to be 0 unless $\pi =(a,b,c,d,e,f)$ satisfies certain conditions. The first condition is that $\pi$ has what in [5] is called a “bad middle”, which means $b=d+1$ and $c= 2d+1$. Therefore the Littelmann inequalities for $\pi$ with a bad middle become

“Bad Middle” Littelmann Inequalities:

• 1. $0 \le \, f \le \ell _2 +a -2d+e-1$

• 2. $d+1 \le \, a \le \ell _1+ 2d-2e+1$

• 3. $d+.5 \le \, d+1 \le \ell _2 +e+1$

• 4. $2d \le \, 2d+1 \le \ell _1 +3d-2e$

• 5. $e \le \, d \le \ell _2 +e$

• 6. $0 \le \, e \le \ell _1$

Thus such $\pi$ are determined by the values of a, d, e and f. The definitions for circling and boxing the entries of $\pi$ still hold. Let $\pi ''=(a,d,e)$, and we set

$$\begin{aligned} H_{\mathrm{adj}}(\pi ) = H_{\mathrm{adj}}(\pi '') h(f). \end{aligned}$$

(8)

By calculating $\hat{H}(\pi )$ and $H_\mathrm {std}(\pi )$ in each of the twenty cases of [5], we can determine $H_{\mathrm{adj}}(\pi '')$. We see that the values of $H_{\mathrm{adj}}(\pi '')$ become more concise than those given in [5] and that the twenty cases are consolidated to the following definition.

$$\begin{aligned} H_{\mathrm{adj}}(\pi '') = {\left\{ \begin{array}{ll} \frac{(1-q^{-1})}{q}, \, \mathrm {if} \, \pi ''=(a^\circ ,d^\circ ,e^\circ ,) \\ -\frac{(1-q^{-1})}{q^2}, \, \mathrm {if}\,\pi ''= (\underline{a}, d \text { or } d^\circ ,e^\circ ) \\ \frac{(1-q^{-1})}{q^3}, \, \mathrm {if}\,\pi ''=(a = 2d+1-e, \underline{d}, e \text { or } e^\circ )\\ \frac{(1-q^{-1})^2}{q} \, \mathrm {if}\, \pi ''=(a^\circ , d^\circ , e), (a, d^\circ , e^\circ ), (a^\circ , d, e^\circ ), (a, d^\circ , e),(a^\circ , d, e) \\ -\frac{(1-q^{-1})^2}{q^2}, \, \mathrm {if}\, \pi ''=(a, \underline{d},e \text { or } e^\circ ,) \text { such that } a \ne 2d+1-e \\ \frac{(1-q^{-1})^3}{q}, \, \mathrm {if}\,\pi ''=(a,d,e) \\ \frac{(1-q^{-1})^3}{q}, \, \mathrm {if}\,\pi ''=( a,d, e^\circ ) \text { such that } a \ne 2d+1-e \\ \frac{(1-q^{-1})((1-q^{-1})^2 +q^{-1})}{q}, \, \mathrm {if}\, \pi ''=(a = 2d+1-e,d,e^\circ ). \end{array}\right. } \end{aligned}$$

(9)

This means, for example, that if $\pi = (a,b,c,d,e,f)=(1,1, 1, 0,0,0)$, then $\pi$ has a bad middle with a, d and e circled (because we assume $\ell _1, \ell _2>0$), so

$$\begin{aligned} H_{\mathrm{adj}}((1,1,1,0,0,0)) = \frac{(1-q^{-1})}{q}. \end{aligned}$$

We see that the definition of $H_{\mathrm{adj}}(\pi )$ depends only on the circling and boxing of a, d and e and whether $a = 2d+1-e$.

5 Proof of conjecture

Now we can prove Theorem 2.

The strategy of the proof is to interpret

$$\begin{aligned} \sum _{\pi \in B(\theta +\rho )} \hat{H}(\pi ) \, x^{a+c+e} y^{b+d+f} \end{aligned}$$

(10)

as a rational function in x, y and $q^{-1}$. This rational function depends on the numbers $\ell _1$ and $\ell _2$, which appear only as exponents of x and y in the numerator of the rational function. We therefore interpret this rational function as a polynomial, say $P_H$, in the four indeterminates

$$\begin{aligned} x^{\ell _1}, y^{\ell _1}, x^{\ell _2}, y^{\ell _2} \end{aligned}$$

(11)

whose coefficients we prove will be of the form

$$\begin{aligned} \frac{p_1(x,y,q^{-1})}{p_2(x,y)} \end{aligned}$$

(12)

where $p_1$ and $p_2$ are polynomials. Now the right side of Theorem 2 is also a polynomial, say $P_W$, in the four indeterminates (11) with coefficients of the form (12). Therefore Theorem 2 can be proved by equating the coefficients of the two polynomials $P_H$ and $P_W$. Recall

$$\begin{aligned} P_W = D(\mathbf x ,q; G_2) \chi _\theta \end{aligned}$$

where $\chi _\theta$ is the normalized character defined in Definition 1. The polynomial $P_W$ has 12 terms, as there are 12 elements in the Weyl group W, and the coefficients are of the form

$$\begin{aligned} \text {sgn}(w) T(\mathbf x ,q;G_2). \end{aligned}$$

We denote the multi-degree of the term

$$\begin{aligned} (x^{m_1}y^{n_1})^{\ell _1}(x^{m_2} y^{n_2})^{\ell _2} \end{aligned}$$

by

$$\begin{aligned} ( (m_1,n_1), (m_2,n_2)). \end{aligned}$$

The twelve multi-degrees of $P_W$ and the coefficients are given in Table 2.

Table 2 Terms for $P_W$

We show how to express (10) as a polynomial in the indeterminates (11). As $\hat{H}(\pi ) = H_{\mathrm{std}}(\pi )+ H_{\mathrm{adj}}(\pi )$, we compute separately the two sums

$$\begin{aligned} \sum _{\pi \in B(\theta +\rho )} H_{\mathrm{std}}(\pi ) \, x^{a+c+e} y^{b+d+f} \end{aligned}$$

(13)

and

$$\begin{aligned} \sum _{\pi \in B(\theta +\rho )} H_{\mathrm{adj}}(\pi ) \, x^{a+c+e} y^{b+d+f}. \end{aligned}$$

(14)

We first compute (13). We sum over the six entries in the order f, a, b, c, d, e. We write (13) as

$$\begin{aligned} \sum _e h(e)x^e \Bigl ( \sum _d h(d)y^d \Bigl ( \sum _c h(c)x^c \Bigl ( \sum _b h(b)y^b \Bigl ( \sum _a h(a)x^a \Bigl ( \sum _f h(f)y^f \Bigr ) \Bigr ) \Bigr ) \Bigr ) \Bigr ) \end{aligned}$$

(15)

where each entry ranges over integers satisfying its respective Littelmann inequality. Thus f ranges from 0 to $\ell _2 +a - 2b+ c-2d+e$, a ranges from b to $\ell _1 +3b - 2c+ 3d-2e$, b ranges from $\frac{c}{2}$ to $\ell _2 +c-2d+e$, etc.

Note that, depending on the values of a, b, c, d, and e, the bounds in the Littelmann inequalities may or may not be integers. In particular, the lower bound for the entry b is $\frac{c}{2}$. This means that if c is odd, then b, which we require to be an integer, cannot attain the lower bound in its Littelmann equality, and furthermore when $b = \lceil \frac{c}{2} \rceil$ it is not circled. Therefore, in the following lemmas we evaluate sums over Littelmann patterns considering whether or not the bounds may be attained.

Lemma 1

Let u be an entry of $\pi$ and let U and L be the upper and lower bounds, respectively, of the Littelmann inequality for u. Recall the expression h(u) from Definition 3. If U and L are integers and $U \ge L$, then

$$\begin{aligned} \sum _{L \le u \le U} h(u)X^u = \frac{(1-q^{-1} X)}{1-X}(X^L-X^U). \end{aligned}$$

(16)

Proof

Since U and L are integers, u attains the values of U and L. First consider $U>L$. Directly applying the definition of h(u) yields

$$\begin{aligned} \sum _{L \le u \le U} h(u)X^u = X^L + (1-q^{-1})\frac{X^{L+1} -X^{U}}{1-X} -q^{-1} X^{U} \end{aligned}$$

which simplifies to

$$\begin{aligned} \frac{(1-q^{-1} X)}{1-X}(X^L-X^U). \end{aligned}$$

(17)

If $U=L$, then in the left side of (16) u attains only one value which is both boxed and circled. Thus $h(u)=0$ and the sum is zero. But if $U=L$, then (17) is also zero. Therefore the lemma is true in the case $U=L$ as well. $\square$

Lemma 1 suffices to evaluate the sums for the entries f and a in the nested sum (15). Lemma 1 also shows that, in order to evaluate the sum for b, we must consider sums of the form

$$\begin{aligned} \sum _{ c/2 \le b\le U } h(b) X^b \end{aligned}$$

where $X=x^m y^n$ for some integers m and n. As discussed above, this sum will depend on the parity of c. We therefore make use of characteristic function ${{\varvec{1}}}_{0 \mathrm {mod}2}(n)$, where

$$\begin{aligned} {\varvec{1}}_{0 \mathrm {mod}2}(n) = {\left\{ \begin{array}{ll} 1, &{} \quad \text {if} \, n \equiv 0 \mod 2\\ 0, &{} \quad \text {if} \, n \equiv 1 \mod 2. \\ \end{array}\right. } \end{aligned}$$

The next lemma evaluates the sum for b.

Lemma 2

Let $U=\ell _2+c-2d+e$ be the upper bound in the Littelmann inequality for the entry b. If $U\ge \frac{c}{2}$, then

$$\begin{aligned} \sum _{ \frac{c}{2} \le b\le U } h(b) X^b&= \left( X^{\frac{c}{2}}+ \frac{X^{\frac{c}{2}+1}(1-q^{-1})}{1-X}\right) {{\varvec{1}}}_{0 \mathrm {mod}2}(c) \nonumber \\&\quad + \left( \frac{X^{\frac{c+1}{2}}(1-q^{-1})}{1-X}\right) {{\varvec{1}}}_{0 \mathrm {mod}2}(c+1) \nonumber \\&\quad - \frac{X^U(1-q^{-1})}{1-X} - q^{-1}X^U. \end{aligned}$$

Proof

If c is even, then u attains the lower bound in its Littelmann inequality. If $U>\frac{c}{2}$, then directly applying the definition of h(u) as in Lemma 1 yields

$$\begin{aligned} \sum _{ \frac{c}{2} \le b\le U } h(b) X^b = X^{\frac{c}{2}}+ \frac{X^{\frac{c}{2}+1}(1-q^{-1})}{1-X} - \frac{X^U(1-q^{-1})}{1-X} - q^{-1}X^U. \end{aligned}$$

(18)

Equation (18) is also true if $U=\frac{c}{2}$, because both sides are 0 as in Lemma 1.

If c is odd, then the smallest value b attains is $\lceil \frac{c}{2} \rceil = \frac{c+1}{2}$; at this value b is not circled. Therefore in this case for $U> \frac{c+1}{2}$

$$\begin{aligned} \sum _{ \frac{c}{2} \le b\le U } h(b) X^b = \sum _{ b=\frac{c+1}{2} }^U h(b) X^b= \left( \frac{X^{\frac{c+1}{2}}(1-q^{-1})}{1-X}\right) - \frac{X^U(1-q^{-1})}{1-X} - q^{-1}X^U. \end{aligned}$$

(19)

Equation (19) is also true when $U = \frac{c+1}{2}$, as the sum then consists of only one term for which the entry u is boxed.

Combining (18) and (19) using characteristic functions proves the lemma. $\square$

From Lemma 2, we see that the sum over c in (15) is a sum of sums of the form

$$\begin{aligned} \frac{p_1(x,y,q^{-1})}{p_2(x,y)}\sum _{L\le c \le U} h(c)Y^c X^{\frac{C_1+C_2c}{2}} {{\varvec{1}}}_{0 \mathrm {mod}2} (C_1+C_2c) \end{aligned}$$

(20)

where X and Y each are monomials in x and y; $p_1$ and $p_2$ are polynomials with integer coefficients; $C_1$ and $C_2$ are integers; and $U = \ell _1+3d-2e$ and $L=2d$ are integers as well. If $C_2$ is even, this sum may be evaluated by Lemma 1. If $C_2$ is odd, we use Lemma 3.

Lemma 3

Let U and L be the upper and lower bounds, respectively, of the Littelmann inequality for the entry u. Let $C_1$ and $C_2$ be integers with $C_2$ odd. If $U\ge L$, then

$$\begin{aligned}&\sum _{L \le u \le U} h(u)Y^u X^{(C_1+C_2u)/2} {{\varvec{1}}}_{0 \mathrm {mod}2} (C_1+C_2u) \\&\quad = \left( Y^L X^{(C_1+C_2L)/2} +\frac{(1-q^{-1})Y^{L+2}X^{(C_1+C_2(L+2))/2} }{1-Y^2X^{C_2}}\right) {\mathbf{1}}_{0 \mathrm {mod}2}(C_1+C_2L) \\&\qquad +\frac{(1-q^{-1})Y^{L+1}X^{(C_1+C_2(L+1))/2}}{1-Y^2X^{C_2}}\mathbf{1 }_{0 \mathrm {mod}2} (C_1+C_2(L+1))\\&\qquad -\left( \frac{(1-q^{-1})Y^{U}X^{(C_1+C_2U)/2}}{1-Y^2X^{C_2}} + q^{-1}Y^UX^{(C_1+C_2U)/2}\right) {\mathbf{1}}_{0 \mathrm {mod}2} (C_1+C_2 U) \\&\qquad - \frac{(1-q^{-1})Y^{U+1}X^{(C_1+C_2(U+1))/2}}{1-Y^2X^{C_2}}{{\varvec{1}}}_{0 \mathrm {mod}2} (C_1+C_2(U+1)). \end{aligned}$$

Proof

The proof is very similar to that of Lemma 2. Since $C_2$ is odd, the entries that are neither boxed nor circled contribute a geometric sum with common ratio $Y^2 X^{C_2}$. Then the parities of U and L determine the highest and lowest values, respectively, of u that can contribute a nonzero term to the sum. The four terms of characteristic functions in the lemma thus come from these cases. $\square$

Lemma 3 allows us to evaluate the sum for c in (15) and shows that the sum for d is a sum of sums of the form

$$\begin{aligned} \sum _{L\le d \le U} h(c)Y^d X^{\frac{C_1+C_2d}{2}} {{\varvec{1}}}_{0 \mathrm {mod}2} (C_1+C_2d) \end{aligned}$$

where X and Y each are some monomials in x and y; where $C_1$ and $C_2$ are integers; and where $U = \ell _2+e$ and $L=e$ are integers as well. But this sum for d can be evaluated using Lemmas 1 or 3, depending on whether $C_2$ is even or odd, respectively. Likewise these lemmas can also evaluate the sum for e.

Lemma 3 motivates the following definition.

Definition 4

A function $f: \mathbb {Z}^2 \rightarrow \mathbb {Q}(x,y)[q^{-1}]$ is called exponential-congruent if

$$\begin{aligned} f(\ell _1, \ell _2) = A X^{\ell _1 }Y^{\ell _2} Z^{(C_1\ell _1+C_2 \ell _2+C_3)/2}{{\varvec{1}}}_{0 \mathrm {mod}2}(C_1\ell _1+C_2 \ell _2+C_3) \end{aligned}$$

where $A \in \mathbb {Q}(x,y)[q^{-1}]$; X, Y and Z are monomials in x and y; and $C_1, C_2$ and $C_3$ are integers. If $C_1$ and $C_2$ are both even, then such a function f is called exponential.

Note that we abuse notation in the following because even though we interpret $P_H$ and $P_W$ as polynomials in the indeterminates (11), we write $P_H(\ell _1, \ell _2)$ and $P_W(\ell _1, \ell _2)$.

Theorem 3

Let $B(\theta +\rho )$ be the set of Littelmann patterns $\pi$ for $\theta +\rho =\ell _1\varpi _1 +\ell _2\varpi _2$, where $\ell _1$ and $\ell _2$ are positive integers. Let $\hat{H}(\pi ) = H_{\mathrm{std}}(\pi )+H_{\mathrm{adj}}(\pi )$ be the coefficient defined previously. Then

$$\begin{aligned} \sum _{\pi \in B(\theta +\rho )} \hat{H}(\pi ) \, x^{a+c+e} y^{b+d+f} = P_H(\ell _1,\ell _2) \end{aligned}$$

where $P_H(\ell _1,\ell _2)$ is a finite sum of exponential-congruent functions.

Proof

Recall from Sect. 4 that

$$\begin{aligned} \hat{H}(\pi ) = H_{\mathrm{std}}(\pi ) + H_{\mathrm{adj}}(\pi ) \end{aligned}$$

where $H_{\mathrm{std}}(\pi )$ is defined by Eq. (7) and $H_{\mathrm{adj}}(\pi )$ by (8) and (9). Let $f_{\mathrm{std}}(\ell _1,\ell _2)$ denote

$$\begin{aligned} f_{\mathrm{std}}(\ell _1,\ell _2)=\sum _{\pi \in B(\theta +\rho )} H_{\mathrm{std}}(\pi ) \, x^{a+c+e} y^{b+d+f}. \end{aligned}$$

The function $f_{\mathrm{std}}(\ell _1,\ell _2)$ is expressed as the nested sum (15), which Lemmas 1, 2, and 3 evaluate as a finite sum of exponential-congruent functions. Now let $f_{\mathrm{adj}}(\ell _1,\ell _2)$ denote

$$\begin{aligned} f_{\mathrm{adj}}(\ell _1,\ell _2)=\sum _{\pi \in B(\theta +\rho )} H_{\mathrm{adj}}(\pi ) \, x^{a+c+e} y^{b+d+f}. \end{aligned}$$

(21)

This is a sum over Littelmann patterns with bad middle which are described by the “bad middle” Littelmann inequalities in Sect. 4. This sum can also be expressed as a nested sum over the entries f, a, d and e whose upper and lower bounds are always integer. Therefore Lemma 1 suffices to evaluate the sum for the entry f. For each of the cases in (9), geometric sums can evaluate the sums for entries a, d and e. This shows that $f_{\mathrm{adj}}(\ell _1,\ell _2)$ is finite sum of exponential functions.

Thus

$$\begin{aligned} f_{\mathrm{std}}(\ell _1,\ell _2)+ f_{\mathrm{adj}}(\ell _1,\ell _2) = P_{H}(\ell _1,\ell _2) \end{aligned}$$

is a finite sum of exponential-congruent functions. $\square$

Let $P_W(\ell _2, \ell _2)$ denote the right side of Theorem 2. Theorem 3 shows that Theorem 2 is equivalent to the statement

$$\begin{aligned} P_H(\ell _1, \ell _2) = P_W(\ell _1, \ell _2) \end{aligned}$$

for all integer $\ell _1, \ell _2>0$. The function $P_W(\ell _1, \ell _2)$ is a sum of twelve exponential functions, while $P_H(\ell _1,\ell _2)$ is a finite sum of congruent-exponential functions. Now a priori an arbitrary finite sum of exponential-congruent functions may not reduce to a finite sum of exponential functions. However, in Theorem 4 we verify that this is the case for $P_H(\ell _1,\ell _2)$ and that it is equivalent to $P_W(\ell _1, \ell _2)$, proving the theorem.

Theorem 4

Let $P_H(\ell _1, \ell _2)$ and $P_W(\ell _1, \ell _2)$ be as defined above. For any positive integers $\ell _1$ and $\ell _2$, then

$$\begin{aligned} P_H(\ell _1, \ell _2) = P_W(\ell _1, \ell _2). \end{aligned}$$

Proof

The strategy of the proof is to calculate the coefficients of the exponential terms on either side of the theorem statement and verify that they are equal. Lemmas 1, 2, and 3 allow for automated calculation of $P_H(\ell _1, \ell _2)$ by a computer.

Recall that

$$\begin{aligned} P_H(\ell _1, \ell _2) = f_{\mathrm{std}}(\ell _1,\ell _2)+ f_{\mathrm{adj}}(\ell _1,\ell _2) \end{aligned}$$

in the notation of the proof of Theorem 3. We calculate that $f_{\mathrm{std}}$ is a sum of 544 terms that are exponential-congruent functions and that $f_{\mathrm{adj}}$ is a sum of 106 terms that are exponential functions. We now specify parities for $\ell _1$ and $\ell _2$ so that $P_H$, when restricted to such $\ell _1, \ell _2$, becomes a finite sum of exponential functions. That is, suppose

$$\begin{aligned} \frac{p_1(x,y,q^{-1})}{p_2(x,y)}(x^{m_1}y^{n_1})^{\ell _1 }(x^{m_2}y^{n_2})^{\ell _2} (x^{m_3}y^{n_3})^{\frac{C_1\ell _1+C_2 \ell _2+C_3}{2}}\mathbf{1 }_{0 \mathrm {mod}2} (C_1\ell _1+C_2 \ell _2+C_3) \end{aligned}$$

(22)

is a term in $P_H(\ell _1, \ell _2)$. For a choice of $\epsilon _1, \epsilon _2 \in \{0,1 \}$, we set

$$\begin{aligned} \ell _1 = 2L_1+\epsilon _1, \, \ell _2 = 2L_2+\epsilon _2 \end{aligned}$$

where $L_1$ and $L_2$ range over the set of positive integers. The term (22) then becomes

$$\begin{aligned}&\frac{p_1(x,y,q^{-1})}{p_2(x,y)}(x^{m_1}y^{n_1})^{\epsilon _1} (x^{m_2}y^{n_2})^{\epsilon _2} (x^{m_3}y^{n_3})^{\frac{C_1\epsilon _1+C_2 \epsilon _2+C_3}{2}} \nonumber \\&\quad \times (x^{2m_1+C_1m_3}y^{2n_1+C_1n_3})^{L_1 }(x^{2m_2+C_2 m_3}y^{2n_2+C_2n_3})^{L_2} \end{aligned}$$

(23)

if

$$\begin{aligned} C_1\epsilon _1+C_2 \epsilon _2+C_3 \equiv 0 \mod 2 \end{aligned}$$

and 0 otherwise. If the term (23) is not 0, then we say it has multi-degree

$$\begin{aligned} ((2m_1+C_1 m_3, 2n_2+C_1 n_3),(2m_2+C_2 m_3, 2n_2+C_2n_3)) \end{aligned}$$

and coefficient

$$\begin{aligned} \frac{p_1(x,y,q^{-1})}{p_2(x,y)}(x^{m_1}y^{n_1})^{\epsilon _1} (x^{m_2}y^{n_2})^{\epsilon _2} (x^{m_3}y^{n_3})^{\frac{C_1\epsilon _1+C_2 \epsilon _2+C_3}{2}}. \end{aligned}$$

We then consider the set of multi-degrees that occur in these sums. We choose $\epsilon _1=\epsilon _2=0$. There are 33 distinct multi-degrees that occur from the standard terms, and 14 distinct multi-degrees that come from the adjusted terms. The union of these sets contains 35 distinct multi-degrees. When we combine like terms for the standard terms, there are 18 multi-degrees with nonzero coefficients, and when we combine like terms for the adjusted terms, there are 10 multi-degrees with nonzero coefficients. We present the multi-degrees and coefficients with $\epsilon _1 = \epsilon _2=0$ for the standard terms in Table 3 and for the adjusted terms in Table 4. Note that because we are considering $\ell _i = 2L_i + \epsilon _i$ for $i=1$ and 2, a multi-degree in Table 2 must be multiplied by 2 to obtain the corresponding multi-degree in Tables 3 and 4.

To express these coefficients, we define

$$\begin{aligned} T_1(\mathbf x )&= \frac{(1-q^{-1})(1-q^{-1}x)(1-q^{-1}y)(1-q^{-1}x^3y^2)}{(1-x)(1-y)(1-x^4y^2)(1-x^3y^2)}\\ T_2(\mathbf x )&= \frac{(1-q^{-1})(1-q^{-1}y)(1-q^{-1}xy)(1-q^{-1}x^3y)}{(1-y)(1-xy)(1-x^4y^2)(1-x^3y)} \\ T_3(\mathbf x )&= \frac{(1-q^{-1})^2(1-q^{-1}y)(1-q^{-1}x^4y^2)}{(1-x)(1-xy)(1+x^2y)(1-x^3y)(1-x^3y^2)} \end{aligned}$$

and abbreviate $T(\mathbf x )=T(\mathbf x ,q;G_2)$ defined above.

Table 3 Standard terms for $P_H$ with $\epsilon _1 = \epsilon _2=0$

Table 4 Adjusted terms for $P_H$ with $\epsilon _1 = \epsilon _2=0$

Now we add the coefficients for the standard terms and adjusted terms for each multi-degree and see that they add up to

$$\begin{aligned} \text {sgn}(w) T(\mathbf x ) \end{aligned}$$

which gives us $P_W$. We do the same procedure the three other combinations of $\epsilon _1$ and $\epsilon _2$ and obtain similar results; the coefficient in Tables 3 and 4 for multi-degree $((m_1, n_1),(m_2,n_2))$ gets multiplied by $(x^{m_1}y^{n_1})^{\frac{\epsilon _1}{2}}(x^{m_2}y^{n_2})^{\frac{\epsilon _2}{2}}$. This proves the theorem. $\square$