Anders Skovsted Buch

Laboratoire de Mathématiques

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Harry Tamvakis

University of Maryland, College Park

Andrew Kresch

University of Zurich

Leonardo C. Mihalcea

Virginia Tech

Nicolas Perrin

Laboratoire de Mathématiques

Alexander Yong

University of Illinois Urbana-Champaign

Richárd Rimányi

University of North Carolina at Chapel Hill

Pierre–Emmanuel Chaput

Université de Lorraine

Pierre-Emmanuel Chaput

institut Camille-Jordan

Lev Borisov

Rutgers, The State University of New Jersey

Sjuvon Chung

The Ohio State University

Cooperative Institutions

Rutgers, The State University of New Jersey

Centre National de la Recherche Scientifique

Université de Lorraine

Virginia Tech

Laboratoire de Mathématiques

Université Paris-Saclay

Université de Versailles Saint-Quentin-en-Yvelines

The Ohio State University

Loughborough University

University of Michigan–Ann Arbor

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Alternating signs of quiver coefficients

Journal of the American Mathematical Society (2004)

Anders Skovsted Buch

We prove a formula for the Grothendieck class of a quiver variety, which generalizes the cohomological component formulas of Knutson, Miller, and Shimozono. Our formula implies that the K K -theoretic quiver coefficients have alternating signs and gives an explicit combinatorial formula for these coefficients. We also prove some new variants of the factor sequences conjecture and a conjecture of Knutson, Miller, and Shimozono, which states that their double ratio formula agrees with the original quiver formulas of the author and Fulton. For completeness we include a short proof of the ratio formula.

Quiver

10.1090/s0894-0347-04-00473-4

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Citations (18)

Quiver coefficients are Schubert structure constants

Mathematical Research Letters (2005)

Anders Skovsted Buch Frank Sottile Alexander Yong

We give an explicit natural identification between the quiver coefficients of Buch and Fulton, decomposition coefficients for Schubert polynomials, and the Schubert structure constants for flag manifolds. This is also achieved in K-theory where we give a direct argument that the decomposition coefficients have alternating signs, based on a theorem of Brion, which then implies that the quiver coefficients have alternating signs. Our identification shows that known combinatorial formulas for the latter two numbers give formulas for the quiver coefficients.

Quiver

Identification

Argument (complex analysis)

Structure constants

10.4310/mrl.2005.v12.n4.a10

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Citations (12)

Rational connectedness implies finiteness of quantum K-theory

arXiv (Cornell University) (2013)

Anders Skovsted Buch Pierre–Emmanuel Chaput Leonardo C. Mihalcea Nicolas Perrin

Let X be any generalized flag variety with Picard group of rank one. Given a degree d, consider the Gromov-Witten variety of rational curves of degree d in X that meet three general points. We prove that, if this Gromov-Witten variety is rationally connected for all large degrees d, then the structure constants of the small quantum K-theory ring of X vanish for large degrees.

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Rank (graph theory)

Flag (linear algebra)

Degree (music)

K-Theory

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Citations (3)

Quantum cohomology of Grassmannians

arXiv (Cornell University) (2001)

Anders Skovsted Buch

We give elementary proofs of the main theorems about (small) quantum cohomology of Grassmannians, including the quantum Giambelli and quantum Pieri formulas, the rim-hook algorithm, Siebert and Tian's presentation, and a recent theorem of Fulton and Woodward about the minimal q-power which appears in a product of two Schubert classes.

Hook

Gromov–Witten invariant

10.48550/arxiv.math/0106268

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Citations (23)

Positivity of quiver coefficients through Thom polynomials

Advances in Mathematics (2004)

Anders Skovsted Buch L Fehér Richárd Rimányi

Quiver

Component (thermodynamics)

10.1016/j.aim.2004.10.019

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Citations (18)

Journal of Algebraic Combinatorics (2001)

Anders Skovsted Buch

Quiver

Statement (logic)

10.1023/a:1011245531325

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Citations (11)

K-theory of minuscule varieties

arXiv (Cornell University) (2013)

Anders Skovsted Buch Matthew J. Samuel

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.

Structure constants

Constant (computer programming)

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A Giambelli formula for isotropic Grassmannians

Selecta Mathematica (2016)

Anders Skovsted Buch Andrew Kresch Harry Tamvakis

Schubert polynomial

Schubert variety

Schubert Calculus

10.1007/s00029-016-0250-1

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Citations (21)

Quantum cohomology of partial flag manifolds

Transactions of the American Mathematical Society (2004)

Anders Skovsted Buch

We give elementary geometric proofs of the structure theorems for the (small) quantum cohomology of partial flag varieties SL ⁡ ( n ) / P \operatorname {SL}(n)/P , including the quantum Pieri and quantum Giambelli formulas and the presentation.

Flag (linear algebra)

Generalized flag variety

10.1090/s0002-9947-04-03655-4

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Citations (12)

Specializations of Grothendieck polynomials

arXiv (Cornell University) (2003)

Anders Skovsted Buch Richárd Rimányi

We prove a formula for double Schubert and Grothendieck polynomials specialized to two rearrangements of the same set of variables. Our formula generalizes the usual formulas for Schubert and Grothendieck polynomials in terms of RC-graphs, and it gives immediate proofs of many other important properties of these polynomials.

10.48550/arxiv.math/0308103

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Citations (1)