Sign insertion and Kazhdan–Lusztig cells of affine symmetric groups

Dongkwan Kim, Pavlo Pylyavskyy

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)213-241
Number of pages29
JournalAlgebraic Combinatorics
Issue number1
StatePublished - 2023

Bibliographical note

Funding Information:
The authors thank an anonymous referee for their useful comments.

Publisher Copyright:
© The author(s), 2023.


  • Kazhdan–Lusztig cells
  • Lascoux–Schützenberger standardization
  • affine matrix-ball construction
  • affine symmetric group
  • sign insertion


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