Bijecting hidden symmetries for skew staircase shapes

Hamaker, Zachary; Morales, Alejandro H.; Pak, Igor; Serrano, Luis; Williams, Nathan

doi:10.5802/alco.285

Hamaker, Zachary ¹; Morales, Alejandro H.²; Pak, Igor ³; Serrano, Luis ⁴; Williams, Nathan ⁵

¹ Department of Mathematics University of Florida Gainesville, FL 32611
² Department of Mathematics and Statistics University of Massachusetts Amherst, MA 01003
³ Department of Mathematics University of California Los Angeles, CA 90095
⁴ Zapata Computing Canada Inc. 325 Front St. W Toronto, ON, M5V 2Y1
⁵ Department of Mathematical Sciences University of Texas at Dallas Richardson, TX 75080

Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 1095-1118.

Abstract

We present a bijection between the set $SYT (λ / μ)$ of standard Young tableaux of staircase minus rectangle shape $λ = δ_{k}$ , $μ = (b^{a})$ , and the set ${ShSYT}^{'} (η)$ of marked shifted standard Young tableaux of a certain shifted shape $η = η (k, a, b)$ . Numerically, this result is due to DeWitt (2012). Combined with other known bijections this gives a bijective proof of the product formula for $| SYT (λ / μ) |$ . This resolves an open problem by Morales, Pak and Panova (2019), and allows an efficient random sampling from $SYT (λ / μ)$ . Other applications include a bijection for semistandard Young tableaux, and a bijective proof of Stembridge’s symmetry of LR–coefficients of the staircase shape. We also extend these results to set-valued standard Young tableaux in the combinatorics of $K$ -theory, leading to new proofs of results by Lewis and Marberg (2019) and Abney-McPeek, An and Ng (2020).

Received: 2021-04-07
Revised: 2022-10-10
Accepted: 2022-12-13
Published online: 2023-08-29

DOI: 10.5802/alco.285

Classification: 05E10, 05A19, 05E05, 17B10
Keywords: tableau, shifted tableau, Schur P-function, Worley–Sagan insertion, mixed shifted insertion, shifted Hecke insertion, Knuth class, shifted Knuth class, K-Knuth class, queer Lie superalgebra

Author's affiliations:

Hamaker, Zachary ¹; Morales, Alejandro H. ²; Pak, Igor ³; Serrano, Luis ⁴; Williams, Nathan ⁵

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Hamaker, Zachary and Morales, Alejandro H. and Pak, Igor and Serrano, Luis and Williams, Nathan},
     title = {Bijecting hidden symmetries for skew staircase shapes},
     journal = {Algebraic Combinatorics},
     pages = {1095--1118},
     publisher = {The Combinatorics Consortium},
     volume = {6},
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     doi = {10.5802/alco.285},
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Hamaker, Zachary; Morales, Alejandro H.; Pak, Igor; Serrano, Luis; Williams, Nathan. Bijecting hidden symmetries for skew staircase shapes. Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 1095-1118. doi : 10.5802/alco.285. https://alco.centre-mersenne.org/articles/10.5802/alco.285/

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