Stable Grothendieck polynomials and K-theoretic factor sequences

Buch, Anders Skovsted; Kresch, Andrew; Shimozono, Mark; Tamvakis, Harry; Yong, Alexander

doi:10.1007/s00208-007-0155-6

Cited by 64 publications

(124 citation statements)

References 24 publications

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“…These monoids arise naturally as subgroup monoids in groups. The theory of Coxeter monoids can be traced back to Bourbaki [11], and was developed in [12,26,53,66]. A further generalisation to Jtrivial monoids has been studied by Denton et al [20].…”

Section: Theoremmentioning

confidence: 99%

Fast Distance Multiplication of Unit-Monge Matrices

Tiskin

2013

Algorithmica

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Monge matrices play a fundamental role in optimisation theory, graph and string algorithms. Distance multiplication of two Monge matrices of size n can be performed in time O(n 2 ). Motivated by applications to string algorithms, we introduced in previous works a subclass of Monge matrices, that we call simple unit-Monge matrices. We also gave a distance multiplication algorithm for such matrices, running in time O(n 1.5 ). Landau asked whether this problem can be solved in linear time. In the current work, we give an algorithm running in time O(n log n), thus approaching an answer to Landau's question within a logarithmic factor. The new algorithm implies immediate improvements in running time for a number of algorithms on strings and graphs. In particular, we obtain an algorithm for finding a maximum clique in a circle graph in time O(n log 2 n), and a surprisingly efficient algorithm for comparing compressed strings. We also point to potential applications in group theory, by making a connection between unit-Monge matrices and Coxeter monoids. We conclude that unit-Monge matrices are a fascinating object and a powerful tool, that deserve further study from both the mathematical and the algorithmic viewpoints.

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Section: Theoremmentioning

confidence: 99%

Fast Distance Multiplication of Unit-Monge Matrices

Tiskin

2013

Algorithmica

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“…One computes using Theorem 1 Hecke insertion and factor sequence formulae. In [Buch et al 2008] a generalization of the Robinson-Schensted and Edelman-Greene insertion algorithms was given. In fact, increasing tableaux also play a prominent role there, although in a different, but related way.…”

Section: Thenmentioning

confidence: 99%

“…Indeed, there has been significant interest in the Grothendieck ring of X and of related varieties; see work on, for example, quiver loci [Buch 2002a;2005a;Buch et al 2008], Hilbert series of determinantal ideals [Knutson and Miller 2005;Knutson et al 2008;2009], applications to invariants of matroids [Speyer 2006], and in relation to representation theory [Griffeth and Ram 2004;Lenart and Postnikov 2007;Willems 2006]. See also work of concerning combinatorial Hopf algebras.…”

Section: Introductionmentioning

confidence: 99%

“…Evidence of the efficacy of this approach is provided through our study of minuscule Schubert calculus; other uses are also suggested. In particular, as a non-algebraic geometric application, in a followup paper [Thomas and Yong 2008b], we relate the ideas in this paper to [Buch et al 2008] and the study of longest strictly increasing subsequences in random words.…”

Section: Introductionmentioning

confidence: 99%

“…In [Knutson and Yong 2004], Lascoux's rule was reformulated in terms of diagram marching moves, and it was also extended to compute a wider class of K-theoretic Schubert structure constants. More recently, in [Buch et al 2008], a rule was given for another class of combinatorial numbers generalizing C ν λ,µ . This rule specializes to a new formula for C ν λ,µ and in fact gives an independent proof of Buch's rule.…”

Section: Introductionmentioning

confidence: 99%

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A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus

Thomas

Yong

2009

ANT

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We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of [Sch ützenberger '77] for standard Young tableaux. We apply this to give a new combinatorial rule for the K-theory Schubert calculus of Grassmannians via K-theoretic jeu de taquin, providing an alternative to the rules of [Buch '02] and others. This rule naturally generalizes to give a conjectural root-system uniform rule for any minuscule flag variety G/P, extending [Thomas-Yong '06]. We also present analogues of results of Fomin, Haiman, Schensted and Sch ützenberger.

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