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Recent documents in UST Research Onlineen-usSun, 16 Jan 2022 01:31:34 PST3600Dual filtered graphs
https://ir.stthomas.edu/cas_mathpub/166
https://ir.stthomas.edu/cas_mathpub/166Tue, 21 Dec 2021 14:26:29 PST
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, 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. 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 Mobius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra, as described by Bergeron-Lam-Li, Nzeutchap, and Lam-Shimizono. The Mobius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms.
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Rebecca Patrias et al.Promotion on generalized oscillating tableaux and web rotation
https://ir.stthomas.edu/cas_mathpub/164
https://ir.stthomas.edu/cas_mathpub/164Tue, 21 Dec 2021 14:26:28 PST
We introduce the notion of a generalized oscillating tableau and define a pro- motion operation on such tableaux that generalizes the classical promotion operation on standard Young tableaux. As our main application, we show that this promotion corresponds to rotation of the irreducible A2-webs of G. Kuperberg.
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Rebecca PatriasShifted Hecke insertion and K-theory of OG(n,2n+ 1)
https://ir.stthomas.edu/cas_mathpub/165
https://ir.stthomas.edu/cas_mathpub/165Tue, 21 Dec 2021 14:26:28 PST
Patrias and Pylyavskyy introduced shifted Hecke insertion as an application of their theory of dual filtered graphs. We study shifted Hecke insertion, showing it preserves descent sets and relating it the K-theoretic jeu de taquin of Buch–Samuel and Clifford–Thomas–Yong. As a consequence, we construct symmetric functions that are closely related to Ikeda–Naruse's representatives for the K-theory of the orthogonal Grassmannian. Exploiting this relationship and introducing a shifted K-theoretic Poirier–Reutenauer algebra, we derive a Littlewood–Richardson rule for the K-theory of the orthogonal Grassmannian equivalent to the rules of Clifford–Thomas–Yong and Buch–Samuel.
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Zachary Hamaker et al.Minuscule reverse plane partitions via quiver representation
https://ir.stthomas.edu/cas_mathpub/163
https://ir.stthomas.edu/cas_mathpub/163Tue, 21 Dec 2021 14:26:27 PST
The Hillman–Grassl correspondence is a well-known bijection between multisets of rim hooks of a partition shape λ and reverse plane partitions of λ. We use the tools of quiver representations to generalize Hillman–Grassl in type A and to define an analogue in all minuscule types.
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Alexander Garver et al.Combinatorics of K-theory via aK-theoretic Poirier–Reutenauer bialgebra
https://ir.stthomas.edu/cas_mathpub/162
https://ir.stthomas.edu/cas_mathpub/162Tue, 21 Dec 2021 14:26:26 PST
We use the -Knuth equivalence of Buch and Samuel (2015) to define a -theoretic analogue of the Poirier–Reutenauer Hopf algebra. As an application, we rederive the -theoretic Littlewood–Richardson rules of Thomas and Yong (2009, 2011) and of Buch and Samuel (2015).
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Rebecca Patrias et al.K-Knuth equivalence for increasing tableau
https://ir.stthomas.edu/cas_mathpub/161
https://ir.stthomas.edu/cas_mathpub/161Tue, 21 Dec 2021 14:26:26 PST
A K-theoretic analogue of RSK insertion and Knuth equivalence relations was first introduced in 2006 by Buch, Kresch, Shimozono, Tamvakis, and Yong. The resulting K-Knuth equivalence relations on words and increasing tableaux on [n] has prompted investigation into the equivalence classes of tableaux arising from these relations. Of particular interest are the tableaux that are unique in their class, which we refer to as unique rectification targets (URTs). In this paper we give several new families of URTs and a bound on the length of intermediate words connecting two K-Knuth equivalent words. In addition, we describe an algorithm to determine if two words are K-Knuth equivalent and to compute all K-Knuth equivalence classes of tableaux on [n].
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Christian Gaetz et al.Antipode formulas for some combinatorial Hopf algebras
https://ir.stthomas.edu/cas_mathpub/160
https://ir.stthomas.edu/cas_mathpub/160Tue, 21 Dec 2021 14:26:25 PST
Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.
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Rebecca PatriasCoincidences among skew Grothendieck polynomials
https://ir.stthomas.edu/cas_mathpub/158
https://ir.stthomas.edu/cas_mathpub/158Tue, 21 Dec 2021 14:26:24 PST
The question of when two skew Young diagrams produce the same skew Schur function has been well-studied. We investigate the same question in the case of stable Grothendieck polynomials, which are the K-theoretic analogues of the Schur functions. We prove a necessary condition for two skew shapes to give rise to the same dual stable Grothendieck polynomial. We also provide a necessary and sufficient condition in the case where the two skew shapes are ribbons.
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Ethan Alwaise et al.Combinatorial constructions motivated by K-theory of the Grassmannian
https://ir.stthomas.edu/cas_mathpub/159
https://ir.stthomas.edu/cas_mathpub/159Tue, 21 Dec 2021 14:26:24 PST
Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought to as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map in these cases. Next, using the Hecke insertion of Buch-Kresch-Shimozono-Tamvakis-Yong and the K-Knuth equivalence of Buch-Samuel in place of the Robinson-Schensted and Knuth equivalence, we introduce a K-theoretic analogue of the Poirier-Reutenauer Hopf algebra of standard Young tableaux. As an application, we rederive the K-theoretic Littlewood-Richardson rules of Thomas-Yong and of Buch-Samuel. Lastly, we define a K-theoretic analogue of Fomin's dual graded graphs, which we call dual filtered graphs. They key formula in this definition is DU-UD=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. 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 Mobius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra as described by Bergeron-Lam-Li, Lam-Shimozono, and Nzeutchap. The Mobius construction is more mysterious but also potentially more important as it corresponds to natural insertion algorithms.
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Rebecca PatriasShifted Hecke insertion andK-theory of OG(n,2n+ 1)
https://ir.stthomas.edu/cas_mathpub/157
https://ir.stthomas.edu/cas_mathpub/157Tue, 21 Dec 2021 14:26:23 PST
Patrias and Pylyavskyy introduced shifted Hecke insertion as an application of their theory of dual filtered graphs. We study shifted Hecke insertion, showing it preserves descent sets and relating it the K-theoretic jeu de taquin of Buch–Samuel and Clifford–Thomas–Yong. As a consequence, we construct symmetric functions that are closely related to Ikeda–Naruse's representatives for the K-theory of the orthogonal Grassmannian. Exploiting this relationship and introducing a shifted K-theoretic Poirier–Reutenauer algebra, we derive a Littlewood–Richardson rule for the K-theory of the orthogonal Grassmannian equivalent to the rules of Clifford–Thomas–Yong and Buch–Samuel. Our methods are independent of the Buch–Ravikumar Pieri rule.
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Zachary Hamaker et al.Dual filtered graphs
https://ir.stthomas.edu/cas_mathpub/156
https://ir.stthomas.edu/cas_mathpub/156Tue, 21 Dec 2021 14:26:22 PST
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, 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. 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 Mobius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra, as described by Bergeron-Lam-Li, Nzeutchap, and Lam-Shimizono. The Mobius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms.
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Rebecca Patrias et al.Jacobi–Trudi determinants over finite field
https://ir.stthomas.edu/cas_mathpub/154
https://ir.stthomas.edu/cas_mathpub/154Tue, 21 Dec 2021 14:26:21 PST
In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over ℤ to a finite field ����, what is the probability that the Schur function ���� maps to zero? We show that this probability is always at least 1/q and is asymptotically 1/q. Moreover, we give a complete classification of all shapes that can achieve probability 1/q. In addition, we identify certain families of shapes for which the events that the corresponding Schur functions are sent to zero are independent. We also look into the probability that Schur functions are mapped to nonzero values in ����.
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Ben Anzis et al.Doppelgängers: bijections of plane partitions
https://ir.stthomas.edu/cas_mathpub/155
https://ir.stthomas.edu/cas_mathpub/155Tue, 21 Dec 2021 14:26:21 PST
We say two posets are "doppelgängers" if they have the same number of P-partitions of each height k. We give a uniform framework for bijective proofs that posets are doppelgängers by synthesizing K-theoretic Schubert calculus techniques of H. Thomas and A. Yong with M. Haiman's rectification bijection and an observation of R. Proctor. Geometrically, these bijections reflect the rational equivalence of certain subvarieties of minuscule flag manifolds. As a special case, we provide the first bijective proof of a 1983 theorem of R. Proctor---that plane partitions of height k in a rectangle are equinumerous with plane partitions of height k in a trapezoid.
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Zachary Hamaker et al.Enumerations relating braid and commutation classes
https://ir.stthomas.edu/cas_mathpub/153
https://ir.stthomas.edu/cas_mathpub/153Tue, 21 Dec 2021 14:26:20 PST
We obtain an upper and lower bound for the number of reduced words for a permutation in terms of the number of braid classes and the number of commutation classes of the permutation. We classify the permutations that achieve each of these bounds, and enumerate both cases.
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Susanna Fishel et al.What is Schur positivity and how common is it?
https://ir.stthomas.edu/cas_mathpub/151
https://ir.stthomas.edu/cas_mathpub/151Tue, 21 Dec 2021 14:26:19 PSTRebecca PatriasPromotion on generalized oscillating tableaux and web rotation
https://ir.stthomas.edu/cas_mathpub/152
https://ir.stthomas.edu/cas_mathpub/152Tue, 21 Dec 2021 14:26:19 PST
We introduce the notion of a generalized oscillating tableau and define a promotion operation on such tableaux that generalizes the classical promotion operation on standard Young tableaux. As our main application, we show that this promotion corresponds to rotation of the irreducible -webs of G. Kuperberg.
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Rebecca PatriasGreene–Kleiman invariants for Sulzgruber insertion
https://ir.stthomas.edu/cas_mathpub/150
https://ir.stthomas.edu/cas_mathpub/150Tue, 21 Dec 2021 14:26:18 PST
R. Sulzgruber's rim hook insertion and the Hillman–Grassl correspondence are two distinct bijections between the reverse plane partitions of a fixed partition shape and multisets of rim-hooks of the same partition shape. It is known that Hillman–Grassl may be equivalently defined using the Robinson–Schensted–Knuth correspondence, and we show the analogous result for Sulzgruber's insertion. We refer to our description of Sulzgruber's insertion as diagonal RSK. As a consequence of this equivalence, we show that Sulzgruber's map from multisets of rim hooks to reverse plane partitions can be expressed in terms of Greene–Kleitman invariants.
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Alexander Garver et al.Leading Terms of SL3 Web Invariants
https://ir.stthomas.edu/cas_mathpub/148
https://ir.stthomas.edu/cas_mathpub/148Tue, 21 Dec 2021 14:26:17 PST
We use Khovanov and Kuperberg’s web growth rules to identify the leading term in the invariant associated to an SL3 web diagram, with respect to a particular term order.
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Veronique Bazier-Matte et al.The probability of positivity in symmetric and quasisymmetric functions
https://ir.stthomas.edu/cas_mathpub/149
https://ir.stthomas.edu/cas_mathpub/149Tue, 21 Dec 2021 14:26:17 PST
Given an element in a finite-dimensional real vector space, V, that is a nonnegative linear combination of basis vectors for some basis B, we compute the probability that it is furthermore a nonnegative linear combination of basis vectors for a second basis, A. We then apply this general result to combinatorially compute the probability that a symmetric function is Schur-positive (recovering the recent result of Bergeron--Patrias--Reiner), $e$-positive or $h$-positive. Similarly we compute the probability that a quasisymmetric function is quasisymmetric Schur-positive or fundamental-positive. In every case we conclude that the probability tends to zero as the degree of a function tends to infinity.
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Rebecca Patrias et al.Set-valued domino tableaux and shifted set-valued domino tableaux
https://ir.stthomas.edu/cas_mathpub/147
https://ir.stthomas.edu/cas_mathpub/147Tue, 21 Dec 2021 14:26:16 PST
We prove K-theoretic and shifted K-theoretic analogues of the bijection of Stanton and White between domino tableaux and pairs of semistandard tableaux. As a result, we obtain product formulas for pairs of stable Grothendieck polynomials and pairs of K-theoretic Q-Schur functions.
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Florence Maas-Gariepy et al.