Iterated Lusztig-Vogan Bijection and Distinguished Weights
George Cao
TL;DR
This work analyzes distinguished weights arising from iterated Lusztig--Vogan bijections in GL_n, revealing an anti-symmetry that halves the determining data and a polynomial-type growth in the number of distinguished weights that become zero after k LV_p steps, with leading coefficients governed by the telephone numbers. It provides a concrete, recursive framework for counting Λ_{n,k}^+ via partitions and gives explicit classifications for n ≤ 4, highlighting how modular reduction interacts with partition refinements. The main methodological contribution is the LV_p iteration approach, linking diagonal structure in weight space to p-adic scaling and revealing deep combinatorial structure tied to p-cells in modular representation theory. The results yield insight into the distribution and asymptotics of distinguished weights, informing the study of modular representations of GL_n and offering tractable enumerative formulas for small ranks. Overall, the paper connects combinatorial LV_p dynamics with representation-theoretic questions about p-cells and dominant weights, providing both exact classifications in low rank and asymptotic growth patterns governed by the telephone numbers.
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 $n \leq 4$.