Equivariant Littlewood-Richardson skew tableaux

Kreiman, Victor

doi:10.1090/S0002-9947-09-04862-4

Equivariant Littlewood-Richardson skew tableaux
HTML articles powered by AMS MathViewer

by Victor Kreiman PDF

Trans. Amer. Math. Soc. 362 (2010), 2589-2617 Request permission

Abstract:

We give a positive equivariant Littlewood-Richardson rule also discovered independently by Molev. Our proof generalizes a proof by Stembridge of the classical Littlewood-Richardson rule. We describe a weight-preserving bijection between our indexing tableaux and trapezoid puzzles which restricts to a bijection between positive indexing tableaux and Knutson-Tao puzzles.

References

Sara C. Billey, Kostant polynomials and the cohomology ring for $G/B$, Duke Math. J. 96 (1999), no. 1, 205–224. MR 1663931, DOI 10.1215/S0012-7094-99-09606-0
Edward A. Bender and Donald E. Knuth, Enumeration of plane partitions, J. Combinatorial Theory Ser. A 13 (1972), 40–54. MR 299574, DOI 10.1016/0097-3165(72)90007-6
L. C. Biedenharn and J. D. Louck, A new class of symmetric polynomials defined in terms of tableaux, Adv. in Appl. Math. 10 (1989), no. 4, 396–438. MR 1023942, DOI 10.1016/0196-8858(89)90023-7
L. C. Biedenharn and J. D. Louck, Inhomogeneous basis set of symmetric polynomials defined by tableaux, Proc. Nat. Acad. Sci. U.S.A. 87 (1990), no. 4, 1441–1445. MR 1037509, DOI 10.1073/pnas.87.4.1441
William Y. C. Chen and James D. Louck, The factorial Schur function, J. Math. Phys. 34 (1993), no. 9, 4144–4160. MR 1233264, DOI 10.1063/1.530032
William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
—, Equivariant cohomology in algebraic geometry, Eilenberg lectures, Columbia University, 2007. Available at http://www.math.lsa.umich.edu/dandersn/eilenberg.
Ian Goulden and Curtis Greene, A new tableau representation for supersymmetric Schur functions, J. Algebra 170 (1994), no. 2, 687–703. MR 1302864, DOI 10.1006/jabr.1994.1361
William Graham, Positivity in equivariant Schubert calculus, Duke Math. J. 109 (2001), no. 3, 599–614. MR 1853356, DOI 10.1215/S0012-7094-01-10935-6
T. Ikeda and H. Naruse, Excited Young diagrams and equivariant Schubert calculus, arXiv:math.AG/0703637.
Allen Knutson and Terence Tao, Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J. 119 (2003), no. 2, 221–260. MR 1997946, DOI 10.1215/S0012-7094-03-11922-5
Allen Knutson, Terence Tao, and Christopher Woodward, The honeycomb model of $\textrm {GL}_n(\Bbb C)$ tensor products. II. Puzzles determine facets of the Littlewood-Richardson cone, J. Amer. Math. Soc. 17 (2004), no. 1, 19–48. MR 2015329, DOI 10.1090/S0894-0347-03-00441-7
V. Kreiman, Schubert classes in the equivariant K-theory and equivariant cohomology of the Grassmannian, arXiv:math.AG/0512204.
—, Products of factorial Schur functions, Electron. J. Combin. 15 (2008), no. 1, Research paper 84, 12 pp. (electronic).
Alain Lascoux, Puissances extérieures, déterminants et cycles de Schubert, Bull. Soc. Math. France 102 (1974), 161–179 (French). MR 364274, DOI 10.24033/bsmf.1776
Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 13, 447–450 (French, with English summary). MR 660739
Alain Lascoux and Marcel-Paul 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 (1982), no. 11, 629–633 (French, with English summary). MR 686357
D. E. Littlewood and A. R. Richardson, Group characters and algebra, Philos. Trans. Roy. Soc. London Ser. A 233 (1934).
I. G. Macdonald, Schur functions: theme and variations, Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992) Publ. Inst. Rech. Math. Av., vol. 498, Univ. Louis Pasteur, Strasbourg, 1992, pp. 5–39. MR 1308728, DOI 10.1108/EUM0000000002757
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Peter J. McNamara, Factorial Grothendieck polynomials, Electron. J. Combin. 13 (2006), no. 1, Research Paper 71, 40. MR 2240776
Leonardo Constantin Mihalcea, Giambelli formulae for the equivariant quantum cohomology of the Grassmannian, Trans. Amer. Math. Soc. 360 (2008), no. 5, 2285–2301. MR 2373314, DOI 10.1090/S0002-9947-07-04245-6
A. I. Molev, Littlewood-Richardson polynomials, arXiv:0704.0065.
Alexander Molev, Factorial supersymmetric Schur functions and super Capelli identities, Kirillov’s seminar on representation theory, Amer. Math. Soc. Transl. Ser. 2, vol. 181, Amer. Math. Soc., Providence, RI, 1998, pp. 109–137. MR 1618747, DOI 10.1090/trans2/181/04
Alexander I. Molev and Bruce E. Sagan, A Littlewood-Richardson rule for factorial Schur functions, Trans. Amer. Math. Soc. 351 (1999), no. 11, 4429–4443. MR 1621694, DOI 10.1090/S0002-9947-99-02381-8
Maxim Nazarov, Yangians and Capelli identities, Kirillov’s seminar on representation theory, Amer. Math. Soc. Transl. Ser. 2, vol. 181, Amer. Math. Soc., Providence, RI, 1998, pp. 139–163. MR 1618751, DOI 10.1090/trans2/181/05
Andrei Okounkov, Quantum immanants and higher Capelli identities, Transform. Groups 1 (1996), no. 1-2, 99–126. MR 1390752, DOI 10.1007/BF02587738
A. Okun′kov and G. Ol′shanskiĭ, Shifted Schur functions, Algebra i Analiz 9 (1997), no. 2, 73–146 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 2, 239–300. MR 1468548
Jeffrey B. Remmel and Mark Shimozono, A simple proof of the Littlewood-Richardson rule and applications, Discrete Math. 193 (1998), no. 1-3, 257–266. Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 1661373, DOI 10.1016/S0012-365X(98)00145-9
Shawn Robinson, A Pieri-type formula for $H^\ast _T(\textrm {SL}_n(\Bbb C)/B)$, J. Algebra 249 (2002), no. 1, 38–58. MR 1887984, DOI 10.1006/jabr.2001.9067
Bruce E. Sagan, The symmetric group, 2nd ed., Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001. Representations, combinatorial algorithms, and symmetric functions. MR 1824028, DOI 10.1007/978-1-4757-6804-6
John R. Stembridge, A concise proof of the Littlewood-Richardson rule, Electron. J. Combin. 9 (2002), no. 1, Note 5, 4. MR 1912814
Ravi Vakil, A geometric Littlewood-Richardson rule, Ann. of Math. (2) 164 (2006), no. 2, 371–421. Appendix A written with A. Knutson. MR 2247964, DOI 10.4007/annals.2006.164.371
A. V. Zelevinsky, A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence, J. Algebra 69 (1981), no. 1, 82–94. MR 613858, DOI 10.1016/0021-8693(81)90128-9