A window to the Bruhat order on the affine symmetric group
Salim Rostam
TL;DR
The work addresses the problem of deciding comparability in the strong Bruhat order on the affine symmetric group from window data. It blends Lascoux–Deodhar and Jacon–Lecouvey insights with an explicit abacus-based construction to translate window information into a sequence of $e$-core partitions across all residues, then reduces Bruhat comparison to inclusion of the corresponding Young diagrams. The main contributions are (i) a practical criterion that $w \unlhd w'$ iff, for every $c$, the $c$-charged $e$-core $\lambda^{(c)}$ of $w$ is contained in that of $w'$, (ii) an inductive procedure on abaci that computes these cores efficiently from the window, and (iii) a rim-hook interpretation on partitions that makes the induction concrete, with connections to $(e,\underline{\mathsf{c}})$-cores in the Jacon–Lecouvey framework. This provides a concrete, combinatorial method to compare affine permutations without reduced expressions, with implications for affine Grassmannians and modular representation theory.
Abstract
Given two affine permutations, some results of Lascoux and Deodhar, and independently Jacon-Lecouvey, allow to decide if they are comparable for the strong Bruhat order. These permutations are associated with tuples of core partitions, and the preceding problem is equivalent to compare the Young diagrams in each components for the inclusion. Using abaci, we give an easy rule to compute these Young diagrams one another. We deduce a procedure to compare, for the Bruhat order, two affine permutations in the window notation.