ALGEBRAIC COMBINATORICS

Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 213-241.

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 a sign insertion algorithm and conjectured that if one fixes the two-sided cell, the recording tableau of the 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 Schützenberger.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.233
Classification: 05E10, 05E15, 20C32
Keywords: affine symmetric group, affine matrix-ball construction, Kazhdan–Lusztig cells, Lascoux–Schützenberger standardization, sign insertion
Kim, Dongkwan 1; Pylyavskyy, Pavlo 1

1 School of Mathematics University of Minnesota Twin Cities Minneapolis MN 55455 U.S.A.
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Kim, Dongkwan; Pylyavskyy, Pavlo. Sign insertion and Kazhdan–Lusztig cells of affine symmetric groups. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 213-241. doi : 10.5802/alco.233. https://alco.centre-mersenne.org/articles/10.5802/alco.233/

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