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Uncrowding algorithm for hook-valued tableaux.(English)Zbl 1491.05195

A hook tableau is a semi-standard Young tableau shaped like an ‘L’, in French notation. A hook-valued tableau is a tableau where each box contains a hook tableau, such that

i) if a box \(A\) is to the left of a box \(B\), but in the same row, then \(\max(A) \leqslant \min(B)\), and
ii) if a box \(A\) is below a box \(B\), but in the same column, then \(\max(A) < \min(C)\).
Here, \(\max(A)\) refers to the maximal entry of the hook tableau in box \(A\), and \(\min(A)\) is the minimal entry. Hook-valued tableaux generalise set-valued tableau and multiset-valued tableau: these result from the cases where the hooks consist of single columns and single rows, respectively. Just as with other sorts of tableaux, there is a crystal structure on hook-valued tableaux, introduced byG. Hawkes andT. Scrimshaw [Algebr. Comb. 3, No. 3, 727–755 (2020;Zbl 1441.05236)]. This specialises in the crystal structure of set-valued tableaux and multiset-valued tableaux.
For set-valued tableaux, there exists an uncrowding operator which maps a set-valued tableau to a pair consisting of a semi-standard Young tableau and a flagged increasing tableau. The operator is “uncrowding” in the sense that the output semistandard Young tableau has the same underlying multiset of numerical entries as the original set-valued tableau, except now we have one numerical entry per box, instead of a set of numerical entries per box. The flagged increasing tableau which is part of the output records data on how the tableau was uncrowded, thus allowing the original tableau to be reconstructed from the pair. A flagged increasing tableau is a tableau of skew shape which is increasing in both rows and columns such that entries in row \(i\) are at least \(i - 1\). An important property of the uncrowding operator on set-valued tableaux is that it intertwines with crystal operators.
The heart of the paper is the definition of an uncrowding operator for hook-valued tableaux. The output of such an operator is a set-valued tableaux and a column-flagged increasing tableaux. A column-flagged increasing tableau is the transpose of a flagged increasing tableau. This operator also has the desired property of intertwining with crystal operators. One can then uncrowded completely by uncrowding the output set-valued tableau.
There also exists an uncrowding operator on multiset-valued tableaux. The authors prove that their uncrowding operator on hook-valued tableaux generalises this operator. They also provide an inverse to their uncrowding map, giving a “crowding” map, which reassembles the original hook-valued tableau from a set-valued tableau and a column-flagged increasing tableau. This crowding map can only be applied to pairs that are compatible with each other in a certain way.
The authors also introduce an alternative uncrowding map on hook-valued tableaux which outputs a multiset-valued tableau and flagged increasing tableau. This uncrowding operator uncrowds the legs of the hooks in the hook-valued tableau, rather than the arms, as it were. It likewise intertwines with crystal operators.
In the final section, the authors apply their results to canonical Grothendieck polynomials. They use the uncrowding map to show that canonical Grothendieck polynomials have a tableau Schur expansion. Canonical Grothendieck polynomials are symmetric polynomials that can be expressed as generating functions of hook-valued tableaux. A symmetric function is said to have a tableau Schur expansion if it is the weighted sum of the Schur functions of a particular set of tableaux. A corollary of this result is an expansion of canonical Grothendieck polynomials in terms of stable symmetric Grothendieck polynomials and dual stable symmetric Grothendieck polynomials. Here, stable symmetric Grothendieck polynomials are generating functions of set-valued tableau and dual stable symmetric Grothendieck polynomials are generating functions of reverse plane partitions.

MSC:

05E10 Combinatorial aspects of representation theory
05E05 Symmetric functions and generalizations
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14N15 Classical problems, Schubert calculus
20G42 Quantum groups (quantized function algebras) and their representations

Citations:

Zbl 1441.05236

Cite

References:

[1]Jason Bandlow and Jennifer Morse. Combinatorial expansions in \(K\)-theoretic bases. Electron. J. Combin., 19(4):Paper 39, 27, 2012. ·Zbl 1267.05037
[2]Daniel Bump and Anne Schilling. Crystal bases. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. Representations and combinatorics. ·Zbl 1440.17001
[3]Anders Skovsted Buch. A Littlewood-Richardson rule for the \(K\)-theory of Grassmannians. Acta Math., 189(1):37-78, 2002. ·Zbl 1090.14015
[4]Melody Chan and Nathan Pflueger. Combinatorial relations on skew Schur and skew stable Grothendieck polynomials. Algebr. Comb., 4(1):175-188, 2021. ·Zbl 1460.05193
[5]William Fulton. Young tableaux, volume 35 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. ·Zbl 0878.14034
[6]Ajeeth Gunna and Paul Zinn-Justin. Vertex models for canonical Grothendieck polynomials and their duals. arXiv preprint arXiv:2009.13172, 2020.
[7]Jin Hong and Seok-Jin Kang. Introduction to quantum groups and crystal bases, volume 42 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002. ·Zbl 1134.17007
[8]Graham Hawkes and Travis Scrimshaw. Crystal structures for canonical Grothendieck functions. Algebraic Combinatorics, 3(3):727-755, 2020. ·Zbl 1441.05236
[9]Cristian Lenart. Combinatorial aspects of the \(K\)-theory of Grassmannians. Ann. Comb., 4(1):67-82, 2000. ·Zbl 0958.05128
[10]Thomas Lam and Pavlo Pylyavskyy. Combinatorial Hopf algebras and \(K\)-homology of Grassmannians. Int. Math. Res. Not. IMRN, (24):Art. ID rnm125, 48, 2007. ·Zbl 1134.16017
[11]Alain Lascoux and Marcel-Paul Schützenberger. Symmetry and flag manifolds. In Invariant theory (Montecatini, 1982), volume 996 of Lecture Notes in Math., pages 118-144. Springer, Berlin, 1983. ·Zbl 0542.14031
[12]Jennifer Morse, Jianping Pan, Wencin Poh, and Anne Schilling. A crystal on decreasing factorizations in the \(0\)-Hecke monoid. Electron. J. Combin., 27(2):Paper 2, 29, 2020. ·Zbl 1441.05237
[13]Cara Monical, Oliver Pechenik, and Travis Scrimshaw. Crystal structures for symmetric Grothendieck polynomials. Transform. Groups, 26(3):1025-1075, 2021. ·Zbl 1472.05152
[14]Rebecca Patrias. Antipode formulas for some combinatorial Hopf algebras. Electron. J. Combin., 23(4):Paper 4, 30, 2016. ·Zbl 1351.05234
[15]Vic Reiner, Bridget E. Tenner, and Alexander Yong. Poset edge densities, nearly reduced words, and barely set-valued tableaux. J. Combin. Theory, Ser. A, 158:66-125, 2018. ·Zbl 1391.05269
[16]Damir Yeliussizov. Duality and deformations of stable Grothendieck polynomials. J. Algebraic Combin., 45(1):295-344, 2017. ·Zbl 1355.05263
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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