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Grothendieck polynomials and quiver formulas

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Anders Skovsted Buch, Andrew Kresch, Harry Tamvakis, Alexander Yong

Combinatorics · Submitted 27 Jun 2003

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove an explicit combinatorial formula for the coefficients, which shows that they have alternating signs.

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arXiv IDmath/0306389 (version 1)
Pith IDN4R4JYHS, Pith's permanent record of this paper and its review
CitationAlexander Yong, Anders Skovsted Buch, Andrew Kresch, Harry Tamvakis. "Grothendieck polynomials and quiver formulas." arXiv:math/0306389 (2003). https://pith.science/paper/N4R4JYHS
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Fulton's universal Schubert polynomials give cohomology formulas for a class of degeneracy loci, which generalize Schubert varieties. The K-theoretic quiver formula of Buch expresses the structure sheaves of these loci as integral linear combinations of products of stable Grothendieck polynomials. We prove an explicit combinatorial formula for the coefficients, which shows that they have alternating signs. Our result is applied to obtain new expansions for the Grothendieck polynomials of Lascoux and Sch\"utzenberger.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Fulton's universal Schubert polynomials give cohomology formulas for a class of degeneracy loci, which generalize Schubert varieties. background 1.00 0.04 0.04 asserted 1.00
s1 The K-theoretic quiver formula of Buch expresses the structure sheaves of these loci as integral linear combinations of products of stable Grothendieck polynomials. background 0.99 0.05 0.10 asserted 1.00
s2 We prove an explicit combinatorial formula for the coefficients, which shows that they have alternating signs. result 0.84 0.96 0.71 asserted 1.00
s3 Our result is applied to obtain new expansions for the Grothendieck polynomials of Lascoux and Sch\"utzenberger. result 0.48 0.88 0.14 asserted 1.00

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Original abstract

Fulton's universal Schubert polynomials give cohomology formulas for a class of degeneracy loci, which generalize Schubert varieties. The K-theoretic quiver formula of Buch expresses the structure sheaves of these loci as integral linear combinations of products of stable Grothendieck polynomials. We prove an explicit combinatorial formula for the coefficients, which shows that they have alternating signs. Our result is applied to obtain new expansions for the Grothendieck polynomials of Lascoux and Sch\"utzenberger.

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Papers whose main result relates to this one's

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States the same result 1 paper
  • We prove K-theoretic generalizations of the component formulas of Knutson, Miller, and Shimozono, and deduce that K-theoretic quiver coefficients have alternating signs.

    Alternating signs of quiver coefficients · 0.81

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