Sign insertion and Kazhdan-Lusztig cells of affine symmetric groups

TL;DR

This paper proves Blasiak's conjecture linking sign insertion and Kazhdan-Lusztig cells in affine symmetric groups, uniting two approaches and exploring new operations and connections to standard Young tableaux.

Contribution

It establishes the conjecture that the recording tableau of sign insertion uniquely determines the two-sided cell, connecting combinatorial models in affine type A.

Findings

Proof of Blasiak's conjecture.

Introduction of partial rotations connecting cells.

Connection between sign insertion and standardization map.

Abstract

Combinatorics of Kazhdan-Lusztig cells in affine type $A$ was originally developed by Lusztig, Shi, and Xi. Building on their work, Chmutov, Pylyavskyy, and Yudovina introduced the affine matrix-ball construction (abbreviated AMBC) which gives an analog of Robinson-Schensted correspondence for affine symmetric groups. An alternative approach to Kazhdan-Lusztig theory in affine type $A$ was developed by Blasiak in his work on catabolism. He introduced sign insertion algorithm and conjectured that if one fixes the two-sided cell, the recording tableau of sign insertion process determines uniquely and is determined uniquely by the left cell. In this paper we unite these two approaches by proving Blasiak's conjecture. In the process, we show that certain new operations we introduce called partial rotations connect the elements in the intersection of a left cell and a right cell. Lastly, we investigate the connection between Blasiak's sign insertion and the standardization map acting on the set of semi-standard Young tableaux defined by Lascoux and Sch\"utzenberger.