Shifted Hecke insertion and the K-theory of OG(n,2n+1)

We define a K-theoretic analogue of Fomin's dual graded graphs, which we call dual filtered graphs. The key formula in the definition is DU − U D = D + I. Our major examples are K-theoretic analogues of Young's lattice, of shifted Young's lattice, and of the Young-Fibonacci lattice. We suggest notions of tableaux, insertion algorithms, and growth rules whenever such objects are not already present in the literature. (See the table below.) We also provide a large number of other examples. Most of our examples arise via two constructions, which we call the Pieri construction and the Möbius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra, as described in [1,19,16]. The Möbius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms. Tableaux Insertion Growth Young Standard Young tableaux [33] RSK insertion [23, 25, 13] [9] Shifted Young Standard shifted Young tableaux with and without circles [24] Shifted Robinson-Schensted insertion [24, 32] [11] Young-Fibonacci Young-Fibonacci tableaux [9, 10] Young-Fibonacci insertion [9] [11] Möbius Young Increasing and set-valued tableaux [29, 5] Hecke insertion [6] Section 4.4

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“…The result of such insertion agrees with that of K-theoretic jeu de taquin rectification described in [8]. See [12].…”

Section: Theorem 53 [18]supporting

confidence: 83%

Dual filtered graphs

Patrias

Pylyavskyy²

2018

Algebraic Combinatorics

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“…There is interest in finding K-analogues of elements of the classical Young tableau theory; see, e.g., [Le00, Bu02, BKSTY08, ThYo09b, BuSa13, GMPPRST16, PaPy14, HKPWZZ15,LiMoSh16]. Although the Grothendieck functions were originally studied for geometric reasons, the combinatorics has been part of a broader conversation in algebraic and enumerative combinatorics, e.g., Hopf algebras [LaPy07,PaPy16,Pa15], cyclic sieving [Pe14,Rh15,PrStVi14], Demazure characters [Mo16+], homomesy [BlPeSa16], longest increasing subsequences of random words [ThYo11], poset edge densities [ReTeYo16], and plane partitions [DiPeSt15,HPPW16].…”

Section: Introductionmentioning

confidence: 99%

Genomic tableaux

Pechenik¹,

Yong²

2016

J Algebr Comb

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We explain how genomic tableaux ] are a semistandard complement to increasing tableaux [Thomas-Yong '09]. From this perspective, one inherits genomic versions of jeu de taquin, Knuth equivalence, infusion and Bender-Knuth involutions, as well as Schur functions from (shifted) semistandard Young tableaux theory. These are applied to obtain new Littlewood-Richardson rules for K-theory Schubert calculus of Grassmannians (after [Buch '02]) and maximal orthogonal Grassmannians (after [Clifford- , ). For the unsolved case of Lagrangian Grassmannians, sharp upper and lower bounds using genomic tableaux are conjectured.Date: March 28, 2016.

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Dual filtered graphs

Patrias¹,

Pylyavskyy²

2015

Discrete Mathematics &Amp; Theoretical Computer Science

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International audience We define a $K$ -theoretic analogue of Fomin’s dual graded graphs, which we call dual filtered graphs. The key formula in the definition is $DU - UD = D + I$. Our major examples are $K$ -theoretic analogues of Young’s lattice, the binary tree, and the graph determined by the Poirier-Reutenauer Hopf algebra. Most of our examples arise via two constructions, which we call the Pieri construction and the Möbius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra, as described in Bergeron-Lam-Li, Nzeutchap, and Lam-Shimozono. The Möbius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms. Nous définissons un analogue $K$ -théorique aux graphes gradués en dualité de Fomin que nous appelons les graphes filtrés en dualité. La formule importante pour la définition est $DU - UD = D + I$. Nos principaux exemples sont un analogue $K$ -théorique aux graphe de Young, l’arbre binaire, et un graphe déterminé par l’algèbre de Hopf de Poirier-Reutenauer. La plupart de nos exemples surviennent de deux constructions que nous appelons la construction de Pieri et la construction de Möbius. La construction de Pieri est étroitement liée à la construction des graphes gradués en dualité d’une algèbre graduée de Hopf à la Bergeron-Lam-Li, Nzeutchap, et Lam-Shimozono. La construction de Möbius est plus mystérieuse, mais aussi peut-être plus importante car cette construction correspond aux algorithmes d’insertion naturelles.

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Shifted Hecke insertion and the K-theory of OG(n,2n+1)

Cited by 3 publications

References 4 publications

Dual filtered graphs

Dual filtered graphs

Genomic tableaux

Dual filtered graphs

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