ON THE LAW OF LARGE NUMBERS FOR DEMAZURE MODULES OF $\widehat{\mathfrak{sl}}_2$

Abstract

We determine the covariance of the weight distribution in level 1 Demazure modules of sl 2 . This allows us to prove a weak law of large numbers for these weight distributions, and leads to a conjecture about the asymptotic concentration of weights for arbitrary Demazure modules.2010 Mathematics Subject Classification. Primary 06B15 Representation theory, Secondary 60B99 Probability theory on algebraic and topological structures.

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

“…Proposition 3.5 by translations and rotations (averaging over the random variable S Lw ) as follows Equations (3.11) and (3.12) read as: (3.17)X w = N 2 m + N (N + 1)n + (−2N − 1) · S Lw + T Lw .Equations (3.13) and (3.14) read as: (3.18)X w = N 2 m + N (N − 1)n − 2N · S Lw + T Lw .The cases covered here correspond to the cases found in [3, Theorem 4.1]. Let us restrict for simplicity reasons to(3.18) for w = (s 1 s 0 ) N , and compare our findings to[3, Theorem 4.1], where the corresponding case is [3, (4.1)] for even N .…”