Iterated Lusztig-Vogan Bijection and Distinguished Weights

TL;DR

This paper studies distinguished weights defined through iterated Lusztig-Vogan bijections, revealing their anti-symmetry, asymptotic distribution, and explicit classifications for small cases, advancing understanding in representation theory.

Contribution

It establishes anti-symmetry properties, asymptotic behavior, and explicit classifications of distinguished weights, providing new insights into their structure and distribution.

Findings

All distinguished weights are anti-symmetric under reversal and negation.

The distribution of distinguished weights follows a polynomial asymptotic with a leading coefficient related to telephone numbers.

Explicitly computed all distinguished weights for n ≤ 4.

Abstract

The distinguished weights form a subset of the weight lattice and are closely tied to the notion of $p$-cells. These weights are defined via iterations of the Lusztig-Vogan bijection. We prove that all distinguished weights exhibit an anti-symmetry under the composition of reversal and negation. We show that the distribution of these weights follows a polynomial asymptotic, with a leading coefficient relating to the telephone numbers. As an explicit computation, we determine all the distinguished weights for $n \leq 4$.