Distributions Defined by $q$-Supernomials, Fusion Products, and Demazure Modules

Stavros Kousidis, Ernst Schulte-Geers

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 $\frak{sl}_2$. 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 $\frak{sl}_2$. 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.


Keywords


$q$-supernomial; current algebra; affine Kac-Moody algebra; fusion product; Demazure module; basic specialization; asymptotic normality; central limit theorem; local central limit theorem; occupancy statistic; mixing distribution

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