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.
Highlights
The q-supernomial coefficients encode certain integer partitions as polynomials in a variable q of the form q j1+···+jm=a m i=1 ji =1 L + j +1 .j q the electronic journal of combinatorics 22(1) (2015), #P1.56The L1, . . . , Lm and a are nonnegative integers and a b q denotes the well known q-binomial coefficient that enumerates inversions in words
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
It 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
Summary
The q-supernomial coefficients encode certain integer partitions as polynomials in a variable q of the form q j1+···+jm=a m i=1 ji Our initial motivation for the study of q-supernomials is their appearance as Hilbert series of fusion modules of the current algebra slr ⊗ C[t] that were introduced by Feigin and Loktev [11], that is tensor products of irreducible representations endowed with a grading that is encoded by the variable q. The coefficients of those Hilbert series ( called string functions) encode dimensions of certain isotropic components called weight spaces. The material is broadly divided into two parts: the first part is devoted to the probabilistic-combinatorial problem of deriving limit theorems for the q-supernomial distributions, and the second part explains the representation theoretic interpretation of the limit theorems derived in the first part
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