K-theoretic Schubert calculus
for OG(n,2n+1) and jeu de taquin for shifted increasing tableaux

Clifford, Edward; Thomas, Hugh; Yong, Alexander

doi:10.1515/crelle-2012-0071

Cited by 29 publications

(56 citation statements)

References 7 publications

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“…We also recover the result from [9] that row-wise superstandard tableaux of type B are unique rectification targets. We can therefore use Proposition 7.1 to derive the following two corollaries from Theorem 6.6 and Theorem 6.16.…”

Section: K -Knuth Equivalence Of Hook-closed Tableauxsupporting

confidence: 73%

“…Corollary 3.19 was proved in [34] for Grassmannians of type A and in [7,9] for maximal orthogonal Grassmannians. = /.…”

Section: Combinatorial K -Theory Ringsmentioning

confidence: 96%

“…Then the shape D .5; 3; 2/ has the following row-wise and column-wise superstandard tableaux: It was proved in [9,34] that the row-wise superstandard tableau S is a unique rectification target if X is a Grassmannian of type A or a maximal orthogonal Grassmannian; alternative proofs are given in Sections 6 and 7 below. Let X D OG.n; 2n/ be a maximal orthogonal Grassmannian.…”

Section: Rectificationsmentioning

confidence: 99%

“…So far no generalization of this rule has been found for other homogeneous spaces. Thomas and Yong proved their conjecture for Grassmannians of type A in [34], while the conjecture for maximal orthogonal Grassmannians follows from combinatorial results of Clifford, Thomas, and Yong [9] together with a K-theoretic Pieri formula proved by Buch and Ravikumar [7]. Thomas and Yong proved their conjecture for Grassmannians of type A in [34], while the conjecture for maximal orthogonal Grassmannians follows from combinatorial results of Clifford, Thomas, and Yong [9] together with a K-theoretic Pieri formula proved by Buch and Ravikumar [7].…”

Section: Introductionmentioning

confidence: 97%

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K-theory of minuscule varieties

Buch

Samuel

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Based on Thomas and Yong's K-theoretic jeu de taquin algorithm, we prove a uniform Littlewood-Richardson rule for the K-theoretic Schubert structure constants of all minuscule homogeneous spaces. Our formula is new in all types. For the main examples of Grassmannians of type A and maximal orthogonal Grassmannians it has the advantage that the tableaux to be counted can be recognized without reference to the jeu de taquin algorithm.Brought to you by | University of Edinburgh Authenticated Download Date | 5/31/15 10:21 PM 2 `/ , where r is the number of boxes in row r of .One can check that for each box˛2 ƒ X with˛> , there exists a simple rooť 2 X ¹ º such that˛ ˇ2 ƒ X . This implies that height.˛/, defined as the sum of the coefficients obtained when˛is written as a linear combination of simple roots, is equal to the maximal cardinality of a totally ordered subset of ƒ X with˛as its maximal element.Lemma 2.2. Let˛;ˇ2 ƒ X . If˛6 Äˇandˇ6 Ä˛, then .˛;ˇ/ D 0.Proof. The cominuscule condition implies that˛Cˇ… R, and the incomparability gives˛ ˇ… R. The lemma therefore follows from [15, Lemma 9.4].Definition 2.3. Given a straight shape ƒ X , define an element w 2 W as follows. If D ;, then set w D 1. Otherwise set w D w X¹˛º s˛2 W , where˛2 is any maximal box.To see that w is well defined, assume that˛andˇare two distinct maximal boxes of the straight shape . Then Lemma 2.2 implies that .˛;ˇ/ D 0, so s˛commutes with sˇ. By Brought to you by |

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Section: K -Knuth Equivalence Of Hook-closed Tableauxsupporting

confidence: 73%

“…Corollary 3.19 was proved in [34] for Grassmannians of type A and in [7,9] for maximal orthogonal Grassmannians. = /.…”

Section: Combinatorial K -Theory Ringsmentioning

confidence: 96%

Section: Rectificationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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K-theory of minuscule varieties

Buch

Samuel

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…For Grassmannians, various alternatives to Buch's original rule [Buc02] for K w u,v are now known [Vak06,TY09b,PY17b]. However, only the rule of H. Thomas and A. Yong [TY09b] is currently known to extend to all of the minuscule varieties [BR12, CTY14,BS16]. This Thomas-Yong rule is based on a jeu de taquin theory for increasing tableaux.…”

Section: Introductionmentioning

confidence: 99%

Unique Rectification in $d$-Complete Posets: Towards the $K$-Theory of Kac-Moody Flag Varieties

Ilango

Pechenik

Zlatin

2018

Electron. J. Combin.

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The jeu-de-taquin-based Littlewood-Richardson rule of H. for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of 'unique rectification targets' in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to K-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's 'd-complete posets' to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of d-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain K-theoretic Schubert structure constants in the Kac-Moody setting.

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Around 16-dimensional quadratic forms in $$I^3_q$$ I q 3

Karpenko

2016

Math. Z.

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K-theoretic Schubert calculus for OG(n,2n+1) and jeu de taquin for shifted increasing tableaux

Cited by 29 publications

References 7 publications

K-theory of minuscule varieties

K-theory of minuscule varieties

Unique Rectification in $d$-Complete Posets: Towards the $K$-Theory of Kac-Moody Flag Varieties

Around 16-dimensional quadratic forms in $$I^3_q$$ I q 3

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