Littlewood-Richardson rules for Grassmannians. (English) Zbl 1053.05121

One of the equivalent forms of the classical Littlewood-Richardson rule describes the structure constants obtained when the cup product of two Schubert classes in the cohomology ring of a complex Grassmannian is written as a linear combination of Schubert classes. In the paper under review the authors give a short and self-contained argument which shows that this rule is a direct consequence of the Pieri formula [M. Pieri, Lomb. Ist. Rend. (2) XXVI. 534–546 (1893), XXVII. 258–273 (1894; JFM 25.1038.02)] for the product of a Schubert class with a special Schubert class.
The Littlewood-Richardson rule has a generalization [J. R. Stembridge, Adv. Math. 74, No. 1, 87–134 (1989; Zbl 0677.20012)] for the Grassmannians which parametrize maximal isotropic subspaces of \({\mathbb C}^n\) equipped with a symplectic or orthogonal form. The same arguments work equally well in more difficult cases and give a simple derivation of the rule of Stembridge from the analogues of the Pieri formula in [H. Hiller and B. Boe, Adv. Math. 62, 49–67 (1986; Zbl 0611.14036)].


05E10 Combinatorial aspects of representation theory
05E05 Symmetric functions and generalizations
14M15 Grassmannians, Schubert varieties, flag manifolds
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[1] W. Fulton, Young Tableaux, L.M.S. Student Texts, Vol. 35, Cambridge University Press, Cambridge, 1997.
[2] Hiller, H.; Boe, B., Pieri formula for SO2n+1/un and spn/un, Adv. math., 62, 49-67, (1986)
[3] Hoffman, P.N.; Humphreys, J.F., Projective representations of the symmetric groups; Q-functions and shifted tableaux, (1992), Oxford University Press New York · Zbl 0777.20005
[4] A. Knutson, T. Tao, Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J., to appear. · Zbl 1064.14063
[5] A. Knutson, T. Tao, C. Woodward, The honeycomb model of GLn tensor products II: puzzles determine facets of the Littlewood-Richardson cone, J. Amer. Math. Soc., to appear. · Zbl 1043.05111
[6] Littlewood, D.E.; Richardson, A.R., Group characters and algebra, Phil. trans. R. soc. A, 233, 99-141, (1934) · Zbl 0009.20203
[7] Pieri, M., Sul problema degli spazi secanti. nota 1^{a}, Rend. ist. lombardo (2), 26, 534-546, (1893) · JFM 25.1038.02
[8] P. Pragacz, Algebro-geometric applications of Schur S- and Q-polynomials, Topics in Invariant Theory (Paris, 1989/1990), Lecture Notes in Mathematics, Vol. 1478, Springer, Berlin, 1991, pp. 130-191. · Zbl 0783.14031
[9] Remmel, J.B.; Shimozono, M., A simple proof of the Littlewood-Richardson rule and applications, Discrete math., 193, 1-3, 257-266, (1998) · Zbl 1061.05507
[10] Sagan, B., Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley, J. combin. theory ser. A, 45, 62-103, (1987) · Zbl 0661.05010
[11] Schur, I., Über die darstellung der symmetrischen und der alternierenden gruppe durch gebrochene lineare substitutionen, J. reine angew. math., 139, 155-250, (1911) · JFM 42.0154.02
[12] Shimozono, M., Multiplying Schur Q-functions, J. combin. theory ser. A, 87, 198-232, (1999) · Zbl 0978.05073
[13] Stembridge, J.R., Shifted tableaux and the projective representations of symmetric groups, Adv. math., 74, 87-134, (1989) · Zbl 0677.20012
[14] M.A.A. van Leeuwen, The Littlewood-Richardson rule, and related combinatorics, in: Interactions of Combinatorics and Representation Theory, MSJ Memoirs Vol. 11, Math. Soc. Japan, Tokyo, 2001, pp. 95-145. · Zbl 0991.05101
[15] D.R. Worley, A theory of shifted Young tableaux, Ph.D. Thesis, M.I.T., 1984.
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