Expected degree of weights in Demazure modules of $ {\hat{\mathfrak{sl}}_2} $

Abstract

We compute the expected degree of a randomly chosen element in a basis of weight vectors in the Demazure module Vw(Λ) of sl 2 . We obtain en passant a new proof of Sanderson's dimension formula for these Demazure modules.

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

“…Some examples ofμ (k) are shown in Figure 2. Furthermore, based on our present result and [2,Corollary 4.3] we propose the following conjecture:…”

“…Proof. We first consider the case that N ≡ j mod (2). The variance of the finite weight follows by (3.1), and the fact that degree and finite weight are uncorrelated follows from the symmetry of the weight distribution.…”

“…We believe that in order to answer them, one needs to look at weight distributions with a probabilistic eye, starting with the computation of the expected value and the covariance, possibly followed by skewness and kurtosis later. We determined the expected value of the weight distribution of any Demazure module of sl 2 in [2]. The next step is to determine the covariance, which we complete in the present article for the modules in level 1: Theorem 7.1.…”

“…Let j ∈ {0, 1} and w ∈ W aff such that the length l(ws j ) < l(w) = N . Then the covariance matrix Σ of the degree −d, and the finite weight α ∨ 1 , in V w (Λ j ) is given by (2),…”

“…Var µ N (−d) max{ −d, λ : V (s1s0) k (mΛ 0 ) λ = 0}Let us propose a conjecture about such a concentration in general Demazure modules V w (Λ) with analogously normalized and scaled weight distributionμ w,Λ . By[2, Corollary 4.3] we know thatE[μ w,Λ ] → c, Λ + 2 3( c, Λ + 1) as l(w) → ∞. Fix a dominant integral weight Λ and a sequence (w (k) ) in W aff such that l(w (k) ) → ∞.…”