Refinements of the Littlewood-Richardson rule

Haglund, J.; Luoto, K.; Mason, S.; van Willigenburg, S.

doi:10.1090/S0002-9947-2010-05244-4

Refinements of the Littlewood-Richardson rule
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by J. Haglund, K. Luoto, S. Mason and S. van Willigenburg PDF

Trans. Amer. Math. Soc. 363 (2011), 1665-1686 Request permission

Abstract:

In the prequel to this paper, we showed how results of Mason involving a new combinatorial formula for polynomials that are now known as Demazure atoms (characters of quotients of Demazure modules, called standard bases by Lascoux and Schützenberger) could be used to define a new basis for the ring of quasisymmetric functions we call “Quasisymmetric Schur functions” (QS functions for short). In this paper we develop the combinatorics of these polynomials further, by showing that the product of a Schur function and a Demazure atom has a positive expansion in terms of Demazure atoms. We use these techniques, together with the fact that both a QS function and a Demazure character have explicit expressions as a positive sum of atoms, to obtain the expansion of a product of a Schur function with a QS function (Demazure character) as a positive sum of QS functions (Demazure characters). Our formula for the coefficients in the expansion of a product of a Demazure character and a Schur function into Demazure characters is similar to known results and includes in particular the famous Littlewood-Richardson rule for the expansion of a product of Schur functions in terms of the Schur basis.

References

Kaan Akin, David A. Buchsbaum, and Jerzy Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), no. 3, 207–278. MR 658729, DOI 10.1016/0001-8708(82)90039-1
Demazure, M., Désingularisation des variétés de Schubert, Ann. E. N. S., 6 (1974), 163–172.
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
Ira M. Gessel, Multipartite $P$-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 289–317. MR 777705, DOI 10.1090/conm/034/777705
J. Haglund, M. Haiman, and N. Loehr, A combinatorial formula for nonsymmetric Macdonald polynomials, Amer. J. Math. 130 (2008), no. 2, 359–383. MR 2405160, DOI 10.1353/ajm.2008.0015
Haglund, J., Luoto, K., Mason, S., van Willigenburg, S., Quasisymmetric Schur functions, J. Combin. Theory Ser. A, in press. Also available at arXiv:0810.2489.
van der Kallen, W., Lectures on Frobenius splittings and B-modules, Springer, New York, USA, 1993.
Mikhail Kogan, RC-graphs and a generalized Littlewood-Richardson rule, Internat. Math. Res. Notices 15 (2001), 765–782. MR 1849481, DOI 10.1155/S1073792801000393
Axel Kohnert, Multiplication of a Schubert polynomial by a Schur polynomial, Ann. Comb. 1 (1997), no. 4, 367–375. MR 1630751, DOI 10.1007/BF02558487
Venkatramani Lakshmibai and Peter Magyar, Standard monomial theory for Bott-Samelson varieties, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 11, 1211–1215 (English, with English and French summaries). MR 1456289, DOI 10.1016/S0764-4442(99)80401-7
Alain Lascoux and Marcel-Paul Schützenberger, Keys & standard bases, Invariant theory and tableaux (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 19, Springer, New York, 1990, pp. 125–144. MR 1035493
Macdonald, I., A new class of symmetric polynomials, Sém. Lothar. Combin., 372 (1988).
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
I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Astérisque 237 (1996), Exp. No. 797, 4, 189–207. Séminaire Bourbaki, Vol. 1994/95. MR 1423624
Mason, S., A decomposition of Schur functions and an analogue of the Robinson-Schensted-Knuth algorithm, Sém. Lothar. Combin., 57 (2008), B57e.
S. Mason, An explicit construction of type A Demazure atoms, J. Algebraic Combin. 29 (2009), no. 3, 295–313. MR 2496309, DOI 10.1007/s10801-008-0133-4
Patrick Polo, Variétés de Schubert et excellentes filtrations, Astérisque 173-174 (1989), 10–11, 281–311 (French, with English summary). Orbites unipotentes et représentations, III. MR 1021515
Victor Reiner and Mark Shimozono, Key polynomials and a flagged Littlewood-Richardson rule, J. Combin. Theory Ser. A 70 (1995), no. 1, 107–143. MR 1324004, DOI 10.1016/0097-3165(95)90083-7
V. Reiner and M. Shimozono, Flagged Weyl modules for two column shapes, J. Pure Appl. Algebra 141 (1999), no. 1, 59–100. MR 1705973, DOI 10.1016/S0022-4049(99)00066-3
V. Reiner and M. Shimozono, Straightening for standard monomials on Schubert varieties, J. Algebra 195 (1997), no. 1, 130–140. MR 1468886, DOI 10.1006/jabr.1997.7092
Victor Reiner and Mark Shimozono, Percentage-avoiding, northwest shapes and peelable tableaux, J. Combin. Theory Ser. A 82 (1998), no. 1, 1–73. MR 1616579, DOI 10.1006/jcta.1997.2841
Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
Brian D. Taylor, A straightening algorithm for row-convex tableaux, J. Algebra 236 (2001), no. 1, 155–191. MR 1808350, DOI 10.1006/jabr.2000.8495