Algebra & Number Theory

A jeu de taquin theory for increasing tableaux, with applications to {\textsl K}\hskip-2pt-theoretic Schubert calculus

Hugh Thomas and Alexander Yong

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Abstract

We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of Schützenberger (1977) 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 and others. This rule naturally generalizes to give a conjectural root-system uniform rule for any minuscule flag variety GP, extending recent work of Thomas and Yong. We also present analogues of results of Fomin, Haiman, Schensted and Schützenberger.

Article information

Source
Algebra Number Theory, Volume 3, Number 2 (2009), 121-148.

Dates
Received: 4 November 2007
Revised: 17 September 2008
Accepted: 29 November 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513797353

Digital Object Identifier
doi:10.2140/ant.2009.3.121

Mathematical Reviews number (MathSciNet)
MR2491941

Subjects
Primary: 05E10: Combinatorial aspects of representation theory [See also 20C30]
Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]

Keywords
Schubert calculus K-theory jeu de taquin

Citation

Thomas, Hugh; Yong, Alexander. A jeu de taquin theory for increasing tableaux, with applications to {\textsl K}\hskip-2pt-theoretic Schubert calculus. Algebra Number Theory 3 (2009), no. 2, 121--148. doi:10.2140/ant.2009.3.121. https://projecteuclid.org/euclid.ant/1513797353


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References

  • G. Benkart, F. Sottile, and J. Stroomer, “Tableau switching: algorithms and applications”, J. Combin. Theory Ser. A 76:1 (1996), 11–43.
  • M. Brion, “Lectures on the geometry of flag varieties”, pp. 33–85 in Topics in cohomological studies of algebraic varieties, edited by P. Pragacz, Birkhäuser, Basel, 2005. http://www.ams.org/mathscinet-getitem?mr=2006f:14058MR 2006f:14058
  • A. S. Buch, “Grothendieck classes of quiver varieties”, Duke Math. J. 115:1 (2002), 75–103.
  • A. S. Buch, “A Littlewood–Richardson rule for the $K$-theory of Grassmannians”, Acta Math. 189:1 (2002), 37–78.
  • A. S. Buch, “Alternating signs of quiver coefficients”, J. Amer. Math. Soc. 18:1 (2005), 217–237.
  • A. S. Buch, “Combinatorial $K$-theory”, pp. 87–103 in Topics in cohomological studies of algebraic varieties, edited by P. Pragacz, Birkhäuser, Basel, 2005.
  • A. S. Buch and W. Fulton, “Chern class formulas for quiver varieties”, Invent. Math. 135:3 (1999), 665–687.
  • A. S. Buch, A. Kresch, and H. Tamvakis, “Littlewood–Richardson rules for Grassmannians”, Adv. Math. 185:1 (2004), 80–90.
  • A. S. Buch, A. Kresch, H. Tamvakis, and A. Yong, “Grothendieck polynomials and quiver formulas”, Amer. J. Math. 127:3 (2005), 551–567.
  • A. S. Buch, A. Kresch, M. Shimozono, H. Tamvakis, and A. Yong, “Stable Grothendieck polynomials and $K$-theoretic factor sequences”, Math. Ann. 340:2 (2008), 359–382.
  • C. Chindris, H. Derksen, and J. Weyman, “Counterexamples to Okounkov's log-concavity conjecture”, Compos. Math. 143:6 (2007), 1545–1557.
  • S. Fomin, W. Fulton, C.-K. Li, and Y.-T. Poon, “Eigenvalues, singular values, and Littlewood–Richardson coefficients”, Amer. J. Math. 127:1 (2005), 101–127.
  • S. Griffeth and A. Ram, “Affine Hecke algebras and the Schubert calculus”, European J. Combin. 25:8 (2004), 1263–1283.
  • M. D. Haiman, “Dual equivalence with applications, including a conjecture of Proctor”, Discrete Math. 99:1-3 (1992), 79–113.
  • A. Knutson and E. Miller, “Gröbner geometry of Schubert polynomials”, Ann. of Math. $(2)$ 161:3 (2005), 1245–1318.
  • A. Knutson and A. Yong, “A formula for $K$-theory truncation Schubert calculus”, Int. Math. Res. Not. 70 (2004), 3741–3756.
  • A. Knutson, T. Tao, and C. Woodward, “A positive proof of the Littlewood–Richardson rule using the octahedron recurrence”, Electron. J. Combin. 11:1 (2004), Research Paper 61, 18 pp.
  • A. Knutson, E. Miller, and M. Shimozono, “Four positive formulae for type $A$ quiver polynomials”, Invent. Math. 166:2 (2006), 229–325.
  • A. Knutson, E. Miller, and A. Yong, “Tableau complexes”, Israel J. Math. 163 (2008), 317–343.
  • A. Knutson, E. Miller, and A. Yong, “Gröbner geometry of vertex decompositions and of flagged tableaux”, J. Reine Angew. Math. (2009).
  • T. Lam and P. Pylyavskyy, “Combinatorial Hopf algebras and $K$-homology of Grassmannians”, Int. Math. Res. Not. IMRN 24 (2007), Art. ID rnm125, 48 pp.
  • T. Lam, A. Postnikov, and P. Pylyavskyy, “Schur positivity and Schur log-concavity”, Amer. J. Math. 129:6 (2007), 1611–1622.
  • A. Lascoux, “Transition on Grothendieck polynomials”, pp. 164–179 in Physics and combinatorics. 2000 (Nagoya, 2000), edited by A. N. Kirillov and N. Liskova, World Sci. Publ., River Edge, NJ, 2001.
  • A. Lascoux and M.-P. Schützenberger, “Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variété de drapeaux”, C. R. Acad. Sci. Paris Sér. I Math. 295:11 (1982), 629–633.
  • M. A. A. van Leeuwen, “Flag varieties and interpretations of Young tableau algorithms”, J. Algebra 224:2 (2000), 397–426.
  • C. Lenart, “Combinatorial aspects of the $K$-theory of Grassmannians”, Ann. Comb. 4:1 (2000), 67–82.
  • C. Lenart and A. Postnikov, “Affine Weyl groups in $K$-theory and representation theory”, Int. Math. Res. Not. IMRN 12 (2007), Art. ID rnm038, 65.
  • L. Manivel, Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence, Cours Spécialisés 3, Société Mathématique de France, Paris, 1998. Translated in English by John R. Swallow.
  • E. Miller, “Alternating formulas for $K$-theoretic quiver polynomials”, Duke Math. J. 128:1 (2005), 1–17.
  • A. Okounkov, “Why would multiplicities be log-concave?”, pp. 329–347 in The orbit method in geometry and physics (Marseille, 2000), edited by C. Duval et al., Progr. Math. 213, Birkhäuser, Boston, 2003.
  • N. Perrin, “Small resolutions of minuscule Schubert varieties”, Compos. Math. 143:5 (2007), 1255–1312.
  • R. Proctor, “d-Complete posets generalize Young diagrams for the jeu de taquin property”, preprint, 2004, http://www.math.unc.edu/Faculty/rap/.
  • K. Purbhoo and F. Sottile, “The recursive nature of cominuscule Schubert calculus”, Adv. Math. 217:5 (2008), 1962–2004.
  • V. Reiner and M. Shimozono, “Plactification”, J. Algebraic Combin. 4:4 (1995), 331–351.
  • M.-P. Schützenberger, “La correspondance de Robinson”, pp. 59–113 in Combinatoire et représentation du groupe symétrique (Strasbourg, 1976), edited by D. Foata, Lecture Notes in Math. 579, Springer, Berlin, 1977.
  • D. Speyer, “A matroid invariant via the K-theory of the Grassmannian”, preprint, 2006.
  • R. P. Stanley, Enumerative combinatorics, vol. 2, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, Cambridge, 1999.
  • H. Thomas and A. Yong, “A combinatorial rule for (co)minuscule Schubert calculus”, preprint, 2006.
  • H. Thomas and A. Yong, “Cominuscule tableau combinatorics”, preprint, 2007.
  • H. Thomas and A. Yong, “An $S_3$-symmetric Littlewood–Richardson rule”, Math. Res. Lett. 15:5 (2008), 1027–1037.
  • H. Thomas and A. Yong, “Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm”, preprint, 2008.
  • R. Vakil, “A geometric Littlewood–Richardson rule”, Ann. of Math. $(2)$ 164:2 (2006), 371–421.
  • M. Willems, “$K$-théorie équivariante des tours de Bott. Application à la structure multiplicative de la $K$-théorie équivariante des variétés de drapeaux”, Duke Math. J. 132:2 (2006), 271–309.