Distributions defined by q-supernomials, fusion products, and Demazure modules

TL;DR

This paper proves that certain distributions related to q-supernomials, fusion modules, and Demazure modules become normally distributed as the fusion power scales, revealing a universal asymptotic behavior in these algebraic structures.

Contribution

It establishes the asymptotic normality of distributions associated with q-supernomials and fusion modules, including Demazure modules, as the fusion power grows large.

Findings

Distributions defined by q-supernomials are asymptotically normal.

Fusion modules and Demazure modules exhibit universal normal behavior asymptotically.

Results serve as a central limit theorem for a broad class of graded tensor representations.

Abstract

We prove asymptotic normality of the distributions defined by q-supernomials, which implies asymptotic normality of the distributions given by the central string functions and the basic specialization of fusion modules of the current algebra of sl2. The limit is taken over linearly scaled fusion powers of a fixed collection of irreducible representations. This includes as special instances all Demazure modules of the affine Kac-Moody algebra associated to sl2. Along with an available complementary result on the asymptotic normality of the basic specialization of graded tensors of the type A standard representation, our result is a central limit theorem for a serious class of graded tensors. It therefore serves as an indication towards universal behavior: The central string functions and the basic specialization of fusion and, in particular, Demazure modules behave asymptotically normal, as the number of fusions scale linearly in an asymptotic parameter, N say.