An extension of Macdonald's identity for $\mathfrak{sl}_n$

TL;DR

This paper extends Macdonald's identity for rak{sl}_n to a two-variable form using Wronskians of vector-valued ta-functions, connecting to modular forms and denominator identities.

Contribution

It introduces a new two-variable extension of Macdonald's identity for rak{sl}_n, employing Wronskians of vector-valued ta-functions.

Findings

Provides a novel two-variable identity for rak{sl}_n

Uses Wronskians of vector-valued ta-functions in proof

Connects to modular Wronskians and denominator identities

Abstract

Let $n$ be an odd positive integer. In this short elementary note, we slightly extend Macdonald's identity for $\mathfrak{sl}_{n}$ into a two-variables identity in the spirit of Jacobi forms. The peculiarity of this work lies in its proof which uses Wronskians of vector-valued $\theta$-functions. This complements the work of A. Milas towards modular Wronskians and denominator identities.

Equations

$$\eta (\tau )\stackrel{\text{def}}{=}{q}^{\frac{1}{24}}\prod _{n=1}^{\infty }(1-q^n)$$

$$\eta (\tau {)}^{\dim \mathfrak{g}}=\sum _{l\in \Lambda }\prod _{\alpha >0}\frac{(l+\rho ,\alpha )}{(l,\alpha )}{q}^{\frac{\Vert l+\rho {\Vert }^2}{2g}}(\tau \in \mathcal{H}).$$

$$\eta (\tau {)}^{n^2-1}=\frac{1}{1!2!\cdots (n-1)!}\sum _{\begin{array}{c}(x_1,...,x_n)\in {\mathbb{Z}}^n\\ x_i\equiv i(\text{mod}n)\\ x_1+...+x_n=0\end{array}}\prod _{i<j}(x_i-x_j)q^{\frac{1}{2n}(x_1^2+...+x_n^2)}.$$

$$\theta (\tau ,z)\stackrel{\text{def}}{=}\sum _{r\in \mathbb{Z}}{q}^{\frac{r^2}{2}}{\zeta }^r=\prod _{m=1}^{\infty }(1-q^m)(1+q^{\frac{2m-1}{2}}\zeta )(1+q^{\frac{2m-1}{2}}{\zeta }^{-1})$$

$$\theta (\tau ,z)\eta (\tau {)}^{n^2-1}=\frac{1}{1!2!\cdots (n-1)!}\sum _{(x_1,...,x_n)}\prod _{i<j}(x_i-x_j)q^{\frac{(x_1^2+...+x_n^2)}{2n}}{\zeta }^{\frac{(x_1+...+x_n)}{n}}$$

$$\Gamma =\left\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{SL}}_2(\mathbb{Z})|\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\equiv \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\text{or}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)(\mathrm{mod}2)\right\}.$$

$$(f|\left[\sigma ,\phi \right])(\tau )\stackrel{\text{def}}{=}\phi (\tau {)}^{-1}f\left(\frac{a\tau +b}{c\tau +d}\right).$$

$$(\varphi {|}_m\left[\sigma ,\phi \right])(\tau ,z)\stackrel{\text{def}}{=}\phi (\tau {)}^{-1}\exp \left(-\frac{2i\pi mc{z}^2}{c\tau +d}\right)\varphi \left(\frac{a\tau +b}{c\tau +d},\frac{z}{c\tau +d}\right).$$

$${\mathrm{Wron}}_z\Phi \stackrel{\text{def}}{=}\det (\Phi ,{\partial }_z\Phi ,...,{\partial }_z^{n-1}\Phi ).$$

$${\mathrm{Wron}}_z(\Phi {|}_m\left[\sigma ,\phi \right])=({\mathrm{Wron}}_z\Phi ){|}_{mn}\left[\sigma ,j^{\frac{n(n-1)}{2}}{\phi }^n\right].$$

$${\theta }_{n,i}(\tau ,z)\stackrel{\text{def}}{=}\sum _{\begin{array}{c}x\in \mathbb{Z}\\ x\equiv i(\text{mod}n)\end{array}}{q}^{\frac{x^2}{2n}}{\zeta }^x(z\in \mathbb{C},\tau \in \mathcal{H}).$$

$$\left({\theta }_n{|}_{\frac{n}{2}}[\sigma ,\phi ]\right)={\mathfrak{u}}_n(\sigma ){\theta }_n.$$

$$({\mathrm{Wron}}_z{\theta }_n){|}_{\frac{n^2}{2}}\left[\sigma ,j^{\frac{n(n-1)}{2}}{\phi }^n\right]=(\det {\mathfrak{u}}_n(\sigma ))({\mathrm{Wron}}_z{\theta }_n).$$

$$({\mathrm{Wron}}_z{\theta }_n)(\tau ,z)=\sum _{\begin{array}{c}(x_1,...,x_n)\in {\mathbb{Z}}^n\\ x_i\equiv i(\mathrm{mod}n)\end{array}}\prod _{i<j}(x_i-x_j)q^{\frac{1}{2n}(x_1^2+...+x_n^2)}{\zeta }^{x_1+...+x_n}.$$

$${\mathrm{Wron}}_z{\theta }_n$$

$$=\sum _{x_1,x_2,...\equiv 1,2,...(\text{mod}n)}\det (1,(x_i{)}_i,...,(x_i^{n-1}{)}_i)q^{\frac{1}{2n}(x_1^2+...+x_n^2)}{\zeta }^{x_1+...+x_n}.$$

$$h_r(\tau )\stackrel{\text{def}}{=}\sum _{\begin{array}{c}(x_1,...,x_n)\in {\mathbb{Z}}^n\\ x_i\equiv i(\text{mod}n)\\ x_1+...+x_n=r\end{array}}\prod _{i<j}(x_i-x_j)q^{\frac{1}{2n}(x_1^2+...+x_n^2)-\frac{r^2}{2n^2}}.$$

$$({\mathrm{Wron}}_z{\theta }_n)(\tau ,z)=\sum _{r\in \mathbb{Z}}{h}_r(\tau )q^{\frac{r^2}{2n^2}}{\zeta }^r.$$

$$({\mathrm{Wron}}_z{\theta }_n)(\tau ,z)=h_0(\tau )\left(\sum _{r\in \mathbb{Z}}{q}^{\frac{r^2}{2}}{\zeta }^{rn}\right)=h_0(\tau )\theta (\tau ,nz).$$

$$({\mathrm{Wron}}_z{\theta }_n)(\tau ,z)$$

$$=(\det {\mathfrak{u}}_n(\sigma ){)}^{-1}(h_0|\left[\sigma ,j^{\frac{n(n-1)}{2}}{\phi }^{n-1}\right])(\tau )(\theta {|}_{1/2}[\sigma ,\phi ])(\tau ,nz)$$

$$=\frac{\mathfrak{u}(\sigma )}{\det {\mathfrak{u}}_n(\sigma ))}(h_0|\left[\sigma ,j^{\frac{n(n-1)}{2}}{\phi }^{n-1}\right])(\tau )\theta (\tau ,nz),$$

$$h_0|\left[\sigma ,j^{\frac{n(n-1)}{2}}{\phi }^{n-1}\right]=\frac{\det {\mathfrak{u}}_n(\sigma )}{\mathfrak{u}(\sigma )}{h}_0.$$

$$\frac{1}{2n}\left(1^2+...+{\left(\frac{n-1}{2}\right)}^2+{\left(\frac{-n+1}{2}\right)}^2+...+(-1{)}^2\right)=\frac{n^2-1}{24}.$$

Full text

Let $\eta$ denote the Dedekind $\eta$-function, given by the infinite product

[TABLE]

where $\tau\in\mathcal{H}=\{z\in\mathbb{C}|\operatorname{im}(z)>0\}$ is the Poincaré upper-half plane, and $q$ is the local parameter $e^{2i\pi\tau}$ at infinity. On the other hand, let $R$ be a reduced root system in a real vector space $V$ canonically attached to a semi-simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. Let $(\cdot,\cdot)$ be a scalar product on $V$ invariant under the action of the Weyl group of $R$. To settle notations, we let $\|x\|^{2}=(x,x)$ for $x\in V$, we fix $\Phi$ the highest root of $R$ and set $g=(\Phi,\rho)$ for $\rho$ half the sum of the positive roots in $R$ (positive being defined with respect to the early choice of a Weyl chamber). We let $\Lambda$ be the lattice in $V$ generated by the set $\{2g\alpha/\|\alpha\|~{}|~{}\alpha\in R\}$. In the celebrated paper [3], I. G. Macdonald proved a remarkable identity for the $\dim\mathfrak{g}$-power of the Dedekind $\eta$-function:

[TABLE]

For $n$ an odd positive integer and $\mathfrak{g}=\mathfrak{sl}_{n}$, we draw from [3, App. 1(6)(a)] that equation (1) reduces to

[TABLE]

For $n=5$, (2) is known as Dyson’s identity and is equivalent to an astounding formula for the Ramanujan function $\tau$ (presented in [1]).

For $z\in\mathbb{C}$, we let $\zeta=e^{2i\pi z}$. Let $\theta$ be the usual theta function:

[TABLE]

where we wrote the Jacobi triple product on the right. In this short note, we extend Equation (2) into a two-variables identity involving $\theta$.

[TABLE]

where the sum is over $n$-tuples $(x_{1},...,x_{n})$ in $\mathbb{Z}^{n}$ such that $x_{i}\equiv i\ (\text{mod}\ n)$ for all $i$.

In fact, Theorem 1 is equivalent to (2), namely Macdonald’s identity for $\mathfrak{sl}_{n}$ for odd $n$. One deduces (2) simply by comparing the constant coefficients once considered as an equality of Fourier series in $z$, and Theorem 1 follows by taking the sum over $r\in\mathbb{Z}$ of the right-hand side of (2) whose summand is shifted by $(x_{1},...,x_{n})\mapsto(x_{1}+r,...,x_{n}+r)$.

Our proof of Theorem 1, however, is entirely modular and mainly self-contained. In particular, it avoids the use of root system of Lie algebras. Essentially, the method presented here fits naturally in the framework of Jacobi forms: it consists of computing the Eichler-Zagier decomposition of the Wronskian of a family of theta functions. As such, the method employed here is in many ways reminiscent of the work of A. Milas in [4], [5] where Wronskians are used to give alternative proofs of Macdonald’s identities for root systems of type $B_{n}$, $C_{n}$, $BC_{n}$ and $D_{n}$.

This note is designed to be elementary and the reader does not need any background in the theory of Jacobi forms.

1 Proof of Theorem 1

We now turn to the proof of Theorem 1. For $n=1$, the result is clear, so we can assume that $n>1$. Let $\Gamma$ denote the following congruence subgroup of $\operatorname{SL}_{2}(\mathbb{Z})$ of index $3$:

[TABLE]

It is generated by the two elements $S=\left(\begin{smallmatrix}0&-1\\ 1&0\end{smallmatrix}\right)$ and $T^{2}=\left(\begin{smallmatrix}1&2\\ 0&1\end{smallmatrix}\right)$.

As we will use them extensively, we introduce the classical slash operators for modular and Jacobi forms. For all integer $k$, we define the set $G_{k|2}(\Gamma)$ consisting of pairs $\left[\sigma,\varphi\right]$ where $\sigma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma$ and where $\varphi:\mathcal{H}\to\mathbb{C}$ is a holomorphic function such that there exists a root of unity $t$ for which $\varphi(\tau)^{2}=t^{k}(c\tau+d)^{k}$ (for all $\tau\in\mathcal{H}$). The function $\varphi$ will be referred as the automorphy factor (of weight $k/2$) of the pair. As usual, $j$ will be the automorphy factor $(\sigma,\tau)\mapsto c\tau+d$. For legibility, the dependency on $\sigma$ in the automorphy factor shall not appear. The set $G_{k|2}(\Gamma)$ is again a group according to the operation $\left[\sigma_{1},\varphi_{1}\right]\left[\sigma_{2},\varphi_{2}\right]=\left[\sigma_{1}\sigma_{2},\tau\mapsto\varphi_{1}(\sigma_{2}\tau)\varphi_{2}(\tau)\right]$. Note that for $\sigma\in\Gamma$, if $\left[\sigma,\varphi_{1}\right]\in G_{k_{1}|2}(\Gamma)$ and $\left[\sigma,\varphi_{2}\right]\in G_{k_{2}|2}(\Gamma)$, then $\left[\sigma,\varphi_{1}\varphi_{2}\right]\in G_{k_{1}+k_{2}|2}(\Gamma)$. For a holomorphic function $f:\mathcal{H}\to\mathbb{C}$ and $[\sigma,\varphi]=\left[\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right),\varphi\right]\in G_{k|2}(\Gamma)$, let $(f|\left[\sigma,\varphi\right])$ be the $\mathbb{C}$-valued holomorphic function given for all $\tau\in\mathcal{H}$ by

[TABLE]

Let $m$ be a rational number. For a function $\phi:\mathcal{H}\times\mathbb{C}\to\mathbb{C}$, holomorphic in its two variables, we also let

[TABLE]

We extend coordinate-wise the slash operators on vectors of functions. The slash operator (3) was introduce to define Jacobi forms by Eichler and Zagier in [2].

As for the classical slash operator, (3) is not stable by derivative in the variable $z$. There is, however, a relation to the Wronskian of a vector of functions. For $\Phi:\mathcal{H}\times\mathbb{C}\longrightarrow\mathbb{C}^{n}$, we define

[TABLE]

From Leibniz’ derivation rule, we have that $\partial_{z}^{r}(\phi|_{m}[\sigma,\varphi])=(\partial_{z}^{r}\phi)|_{m}[\sigma,j^{r}\varphi]+$ a linear combination of $(\partial_{z}^{i}\phi)|_{m}[\sigma,j^{i}\varphi]$ ($i<r$) with complex coefficients independent of $\phi$. By multilinearity of $\det$, it follows that

[TABLE]

If $[\sigma,\varphi]$ is in $G_{k|2}(\Gamma)$, $\left[\sigma,j^{\frac{n(n-1)}{2}}\varphi^{n}\right]$ belongs to $G_{K|2}(\Gamma)$ where $K=n(n+k-1)$.

Let $n$ be an odd positive integer, and consider

[TABLE]

Let $\theta_{n}$ be the transpose of $(\theta_{n,1},...,\theta_{n,n})$. The following Lemma is deduced from Poisson’s formula:

Lemma 1**.**

There exists a unitary representation $\mathfrak{u}_{n}:\Gamma\to\operatorname{GL}_{n}(\mathbb{C})$ such that, for all $\sigma\in\Gamma$, there exists $\varphi:\mathcal{H}\to\mathbb{C}$ for which $[\sigma,\varphi]\in G_{1|2}(\Gamma)$ and

[TABLE]

As Lemma 1 is classical, we leave it without proof (see [2, Section II.5]). The representation $\mathfrak{u}_{n}$ and the automorphy factor $\varphi$ can be made explicit, but for our purpose we simply need to know that $\mathfrak{u}_{n}$ is unitary. Note that the function $\theta$ of the introduction is here denoted $\theta_{1,0}$.

By (4) and for $[\sigma,\varphi]$ as in Lemma 1, we find that

[TABLE]

Vandermonde’s identity enables us to compute its Fourier expansion:

Lemma 2**.**

For all $\tau\in\mathcal{H}$ and $z\in\mathbb{C}$, we have

[TABLE]

Proof.

It is enough to prove the formula formally. We have

[TABLE]

By Vandermonde’s identity, $\det(1,(x_{i})_{i},...,(x_{i}^{n-1})_{i})=\prod_{i<j}{(x_{i}-x_{j})}$. ∎

This Lemma implies that $(\operatorname{Wron}_{z}\theta_{n})(\tau,z/n)$ is, up to the factor $1!2!\cdots(n-1)!$, the member on the right-hand side in Theorem 1. The function appearing in right-hand side of (2) is rather related to $h_{r}$ that we now define. For $r\in\mathbb{Z}$, $\tau\in\mathcal{H}$, let

[TABLE]

Note that, for all $\tau\in\mathcal{H}$ and $z\in\mathbb{C}$,

[TABLE]

On one hand, note that for $(x_{1},...,x_{n})$ as in the summation indices of (6), the sum $x_{1}+....+x_{n}$ is always a multiple of $n$, as $n$ is odd. In particular, $h_{r}=0$ if $r$ is not a multiple of $n$.

On the other hand, we note that $h_{r}$ depends only on the class of $r\ (\text{mod}\ n^{2})$ by the change of indices $(x_{1},...,x_{n})\mapsto(x_{1}+n,...,x_{n}+n)$. As $n$ is odd, the change of indices $(x_{2},...,x_{n},x_{1})\mapsto(x_{1}+1,...,x_{n-1}+1,x_{n}+1)$ implies $h_{r}=h_{r+n}$. Consequently, (7) becomes

[TABLE]

By (5) and Lemma 1, we find that $h_{0}$ satisfies a modular invariance property:

[TABLE]

from which one deduces

[TABLE]

In particular, $h_{0}$ behaves like a modular form of weight $(n^{2}-1)/2$ for $\Gamma$ with some character $\sigma\mapsto(\det\mathfrak{u}_{n}(\sigma))\mathfrak{u}(\sigma)^{-1}$ of norm $1$. From the Fourier expansion (6) of $h_{0}$, the latter still holds if we replace $\Gamma$ by $\operatorname{SL}_{2}(\mathbb{Z})$ as $h_{0}$ is invariant by $\tau\mapsto\tau+1$ up to the multiplication by a root of unity. Besides, its order of vanishing at the cusp infinity of $\operatorname{SL}_{2}(\mathbb{Z})$ is at least

[TABLE]

Consequently, $h_{0}/\eta^{n^{2}-1}$ is invariant of weight [math] for $\operatorname{SL}_{2}(\mathbb{Z})$ (for some character of norm $1$) and bounded on $\mathcal{H}$. Therefore, it is a constant function on $\mathcal{H}$. Identifying the constant to be $1!2!\cdots(n-1)!$ as the first nonzero Fourier coefficient of $h_{0}$ finishes the proof of Theorem 1.

Acknowledgments:

The author wishes to thank Ken Ono and Antun Milas for their support and interest in this note.