Weyl Modules for Toroidal Lie Algebras

Abstract

In this paper, we study Weyl modules for a toroidal Lie algebra $\mathcal {T}$ with arbitrary n variables. Using the work of Rao (Pac. J. Math. 171(2), 511–528 1995), we prove that the level one global Weyl modules of $\mathcal {T}$ are isomorphic to suitable submodules of a Fock space representation of $\mathcal {T}$ up to a twist. As an application, we compute the graded character of the level one local Weyl module of $\mathcal {T}$, thereby generalising the work of Kodera (Lett. Math. Phys. 110(11) 3053–3080 2020).

Appendix: on Symmetric Functions

The aim of this appendix is to review some of the basic notions about symmetric functions which we require in the proof of Proposition 4.4. Most of the material covered here can be found in [20]. Let x₁,x₂,⋯ ,x_n be a finite set of indeterminates. The symmetric group on n letters S_n acts on $\mathbb Z[x_{1}, x_{2}, \cdots , x_{n}]$, the ring of polynomials in n variables by permuting the variables. The ring of symmetric polynomials is defined by ${\Lambda }(n) := \{ f \in \mathbb {Z}[x_{1}, \cdots , x_{n}]: \sigma .f = f \forall \sigma \in S_{n}\}$. Let Λ(n)_k be the space of homogeneous symmetric polynomials of degree k. Then Λ(n) is a graded ring with ${\Lambda } (n) = \bigoplus _{k} {\Lambda } (n)_{k}$.

If we have a countable infinite set of indeterminates, say x₁,x₂,⋯, we consider the ring R of power series in x₁,x₂,⋯ of bounded degree. Hence, elements of R can be infinite sums, but only in a finite number of degrees. Let $S_{\infty }$ be the group of all permutations of {1, 2,⋯}. Then $S_{\infty }$ acts on R, and we define the ring of symmetric functions ${\Lambda } :=\{f \in R: \sigma . f=f for all \sigma \in S_{\infty }\}$. This is a subring of R. For each n there is a surjective homomorphism ${\Lambda } \rightarrow {\Lambda }(n)$ obtained by setting x_n+ 1 = x_n+ 2 = ⋯ = 0. We also have a surjective homomorphism of rings ${\Lambda } (n+1) \rightarrow {\Lambda } (n)$ by setting x_n+ 1 = 0 and that restricts to a surjective map ${\Lambda } (n+1)_{k} \rightarrow {\Lambda }(n)_{k}$ for all k and the map is bijective if and only if k ≤ n. We set $\displaystyle {{\Lambda }_{k}=\varprojlim _{k} {\Lambda }(n)_{k}}$. Then ${\Lambda }=\displaystyle {\bigoplus _{k \geq 0} {\Lambda }_{k}}$. If A is any commutative ring, we write ${\Lambda }_{A}:={\Lambda } \otimes _{\mathbb Z} A$ for the ring of symmetric functions with coefficients in A.

There are various $\mathbb {Z}$-bases of the ring, some of which we shall review. They all are indexed by partitions. A partition λ is a (finite or infinite) weakly decreasing sequence λ = (λ₁,λ₂,⋯) of non-negative integers with finitely many non-zero terms. Let $\mathcal P$ denote the set of all partitions. Given two partitions λ and μ we say that λ ≥ μ if and only if |λ| = |μ| and λ₁ + λ₂ + ⋯ + λ_r ≥ μ₁ + μ₂ + ⋯ + μ_r for all r ≥ 1. For a partition λ = (λ₁,λ₂,⋯), the conjugate partition is defined by $\lambda ^{\prime }=(\lambda _{1}^{\prime }, \lambda _{2}^{\prime }, {\cdots } )$, where $\lambda _{i}^{\prime }$ is the number of j’s such that λ_j ≥ i.

For a partition λ = (λ₁,λ₂,⋯) we define $x^{\lambda }:=x_{1}^{\lambda _{1}}x_{2}^{\lambda _{2}}{\cdots }$. The monomial symmetric function m_λ is the sum of all distinct monomials obtainable from x^λ by permutations of the x’s. In particular, when $\lambda =(1^{r})=: (\underbrace {1,1,1,{\cdots } ,1}_{r}, 0,0, {\cdots } )$ we have $\displaystyle {m_{(1^{r})}=e_{r}=\sum \limits _{i_{1} <i_{2}< {\cdots } <i_{r}} x_{i_{1}}x_{i_{2}}{\cdots } x_{i_{r}}}$, the r-th elementary symmetric polynomial. The e_r’s are algebraically independent over $\mathbb Z$. At the other extreme when λ = (r) := (r, 0, 0,⋯) we have $m_{(r)}=p_{r}=\sum {x_{i}^{r}}$, the r-th power sum. It is clear that every f ∈Λ is uniquely expressible as a finite linear combination of the m_λ’s, so that $(m_{\lambda })_{\lambda \in \mathcal P}$ is a $\mathbb Z$-basis of Λ. For each r ≥ 0, the r-th complete symmetric polynomial h_r, is the sum of all monomials of total degree r in the variables x₁,x₂,⋯ so that $h_{r}={\sum }_{|\lambda |=r} m_{\lambda }$. The map $\omega : {\Lambda } \rightarrow {\Lambda }$ defined by e_λ↦h_λ is an involution of graded rings and hence h_r’s are also algebraically independent over $\mathbb Z$.

For any partition λ we define $e_{\lambda }:=e_{\lambda _{1}}e_{\lambda _{2}}{\cdots }$, $p_{\lambda }:=p_{\lambda _{1}}p_{\lambda _{2}} {\cdots }$ and $h_{\lambda }:=h_{\lambda _{1}}h_{\lambda _{2}} {\cdots }$ as the elementary symmetric function, the power sum symmetric function and the complete symmetric function respectively for the partition λ. It is easy to see that

$$e_{\lambda^{\prime}}=m_{\lambda}+\sum\limits_{\mu < \lambda} a_{\lambda, \mu} m_{\mu}$$

for some nonnegative integers a_λ,μ. Hence $(e_{\lambda })_{\lambda \in \mathcal P}$ form another $\mathbb Z$-basis of Λ. Again since ω is an involution, the set $(h_{\lambda })_{\lambda \in \mathcal P}$ is yet another $\mathbb Z$-basis of Λ.

The generating function for the e_r, h_r and p_r are $E(t)={\sum }_{r \geq 0} e_{r}t^{r}=\prod (1+x_{i}t)$, $H(t)={\sum }_{r \geq 0} h_{r}t^{r}=\prod (1-x_{i}t)^{-1}$ and $P(t)={\sum }_{r \geq 1} p_{r}t^{r-1}={\sum }_{r \geq 1} \frac {x_{i}}{(1-x_{i}t)}$ respectively and they satisfy the following identities.

$$E(t)H(-t)=1, P(t)=\frac{H^{\prime}(t)}{H(t)} and P(-t)=\frac{E^{\prime}(t)}{E(t)}.$$

As a result for each n ≥ 1 we get that

$$\sum\limits_{r=0}^{n} (-1)^{r} e_{r}h_{n-r}=0,$$

$$nh_{n}=\sum\limits_{r=1}^{n} p_{r}h_{n-r}$$

and

$$ ne_{n}=\sum\limits_{r=1}^{n} (-1)^{r-1}p_{r}e_{n-r}. $$

(1)

From the above expression it follows that $\mathbb Q[h_{1}, h_{2}, \cdots , h_{n}]=\mathbb Q[p_{1}, p_{2},{\cdots } ,p_{n}]$ for all n ≥ 1. Letting $n \rightarrow \infty$ we get that ${\Lambda }_{\mathbb Q}= \mathbb Q[h_{1},h_{2}, {\cdots } ]=\mathbb Q[p_{1}, p_{2},{\cdots } ]$. So $(p_{\lambda })_{\lambda \in \mathcal P}$ form a $\mathbb Q$-basis of ${\Lambda }_{\mathbb Q}$ but not a $\mathbb Z$-basis of Λ.

We know that $P(t) = \frac {d}{dt} log H(t)$. Integrating both sides imposing the boundary condition log H(0) = 0 and applying the exponential map, we get that,

$$H(t)=\exp (\sum\limits_{n \geq 1} \frac{p_{n}t^{n}}{n})=\prod\limits_{n \geq 1} \exp (\frac{p_{n}t^{n}}{n})=\prod\limits_{n \geq 1}\sum\limits_{d \geq 0} \frac{{p_{n}^{d}}t^{nd}}{n^{d}d!}=\sum\limits_{\lambda} \frac{p_{\lambda}}{z_{\lambda}}t^{|\lambda|},$$

where $z_{\lambda }= {\prod }_{i \geq 1} i^{m_{i}(\lambda )}m_{i}(\lambda )!$ and m_i(λ) is the number of i appears in λ.

Since the involution ω maps H(t) to E(t) and vice versa, we have ω(p_n) = (− 1)^n+ 1p_n and as a result, for a partition λ we get that, ω(p_λ) = 𝜖_λp_λ, where 𝜖_λ = (− 1)^|λ|−l(λ). Now by applying the involution ω to the above identity we get that,

$$E(t)=\exp (\sum\limits_{n \geq 1} (-1)^{n+1}\frac{p_{n}t^{n}}{n})=\sum\limits_{\lambda} \epsilon_{\lambda} \frac{p_{\lambda}}{z_{\lambda}}t^{|\lambda|}.$$

Replacing t by − t in the last identity we get that

$$ E(-t)= \sum\limits_{n \geq 0} (-1)^{n} e_{n}t^{n}= \exp (-\sum\limits_{n \geq 1} \frac{p_{n}t^{n}}{n})=\sum\limits_{\lambda} (-1)^{(2|\lambda|-l(\lambda))} \frac{p_{\lambda}}{z_{\lambda}}t^{|\lambda|}. $$

(2)