Crystals for set-valued decomposition tableaux

Marberg, Eric; Tong, Kam Hung

doi:10.5802/alco.437

Marberg, Eric ¹; Tong, Kam Hung ¹

¹ Department of Mathematics HKUST Clear Water Bay, Hong Kong

Algebraic Combinatorics, Volume 8 (2025) no. 4, pp. 857-896.

Abstract

We describe two crystal structures on set-valued decomposition tableaux. These provide the first examples of interesting “$K$-theoretic” crystals on shifted tableaux. Our first crystal is modeled on a similar construction of Monical, Pechenik, and Scrimshaw for semistandard (unshifted) set-valued tableaux. Our second crystal is adapted from the “square root” operators introduced by Yu on the same set. Neither of our shifted crystals is normal, but we conjecture that our second construction is connected with a unique highest weight element. These results lead to partial progress on a conjectural formula of Cho–Ikeda for $K$-theoretic Schur $P$-functions. We also study a new category of “square root crystals” that includes our second construction and Yu’s set-valued tableau crystals as examples. We observe that Buch’s formula for the coefficients expanding products of symmetric Grothendieck functions has a simple description in terms of the tensor product for this category.

Received: 2024-07-29
Revised: 2025-02-21
Accepted: 2025-04-21
Published online: 2025-09-01

DOI: 10.5802/alco.437

Classification: 05E05, 20G42
Keywords: Crystals, $K$-theoretic Schur $P$-functions, queer Lie superalgebras, decomposition tableaux, set-valued tableaux

Author's affiliations:

Marberg, Eric ¹; Tong, Kam Hung ¹

¹ Department of Mathematics HKUST Clear Water Bay, Hong Kong

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Marberg, Eric; Tong, Kam Hung. Crystals for set-valued decomposition tableaux. Algebraic Combinatorics, Volume 8 (2025) no. 4, pp. 857-896. doi : 10.5802/alco.437. https://alco.centre-mersenne.org/articles/10.5802/alco.437/

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