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

Stavros Kousidis ; Ernst Schulte-Geers

The electronic journal of combinatorics, Tome 22 (2015) no. 1

Résumé

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.

Détail

DOI : 10.37236/4914

Classification : 05A16, 05A30, 60B99, 06B15
Mots-clés : \(q\)-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