# zbMATH — the first resource for mathematics

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)].

##### MSC:
 05E10 Combinatorial aspects of representation theory 05E05 Symmetric functions and generalizations 14M15 Grassmannians, Schubert varieties, flag manifolds
Full Text:
##### References:
 [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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.