Geodesic
Bijecting hidden symmetries for skew staircase shapes
Hamaker, Zachary1 ; Morales, Alejandro H.2 ; Pak, Igor3 ; Serrano, Luis4 ; Williams, Nathan5
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, Tome 6 (2023) no. 4, pp. 1095-1118

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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).

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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

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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Bijecting hidden symmetries for skew staircase shapes},
     journal = {Algebraic Combinatorics},
     pages = {1095--1118},
     publisher = {The Combinatorics Consortium},
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Hamaker, Zachary; Morales, Alejandro H.; Pak, Igor; Serrano, Luis; Williams, Nathan. Bijecting hidden symmetries for skew staircase shapes. Algebraic Combinatorics, Tome 6 (2023) no. 4, pp. 1095-1118. doi: 10.5802/alco.285

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