Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A Littlewood-Richardson rule for
factorial Schur functions

Authors: Alexander I. Molev and Bruce E. Sagan
Journal: Trans. Amer. Math. Soc. 351 (1999), 4429-4443
MSC (1991): Primary 05E05; Secondary 05E10, 17B10, 17B35, 20C30
Published electronically: February 8, 1999
MathSciNet review: 1621694
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Applications to Capelli operators and quantum immanants are also given.

References [Enhancements On Off] (What's this?)

  • [BL] L. Biedenharn and J. Louck, A new class of symmetric polynomials defined in terms of tableaux, Advances in Appl. Math. 10 (1989), 396-438. MR 91c:05189
  • [C] A. Capelli, Sur les opérations dans la théorie des formes algébriques, Math. Ann. 37 (1890), 1-37.
  • [D] J. Dixmier, ``Algèbres enveloppantes,'' Gauthier-Villars, Paris, 1974. MR 58:16803a
  • [FG] S. Fomin and C. Greene, A Littlewood-Richardson miscellany, European J. Combin. 14 (1993), 191-212. MR 94b:05213
  • [GG] I. Goulden and C. Greene, A new tableau representation for supersymmetric Schur functions, J. Algebra 170 (1994), 687-703. MR 96f:05187
  • [H] R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), 539-570. MR 90h:22015a
  • [HU] R. Howe and T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), 569-619. MR 92j:17004
  • [JP] G. D. James and M. H. Peel, Specht series for skew representations of symmetric groups, J. Algebra 56 (1979), 343-364. MR 80h:20021
  • [KR] A. N. Kirillov and N. Yu. Reshetikhin, The Bethe ansatz and the combinatorics of Young tableaux, J. Soviet Math. 41 (1988), 925-955.
  • [L1] A. Lascoux, Puissances extérieures, déterminants et cycles de Schubert, Bull Soc. Math. France 102 (1974), 161-179. MR 51:529
  • [L2] A. Lascoux, ``Interpolation,'' Lectures at Tianjin University, June 1996.
  • [LS] A. Lascoux and M.-P. Schützenberger, Interpolation de Newton à plusieurs variables, in ``Séminaire d'Algèbre P. Dubreil et M.-P. Malliavin, 36ème annie (Paris, 1983-1984),'' Lecture Notes in Math., Vol. 1146, Springer-Verlag, New York, NY, 1985, 161-175. MR 88h:05020
  • [LR] D. E. Littlewood and A. R. Richardson, Group characters and algebra, Philos. Trans. Roy. Soc. London Ser. A 233 (1934), 49-141.
  • [M1] I. G. Macdonald, ``Symmetric functions and Hall polynomials,'' 2nd edition, Oxford University Press, Oxford, 1995. MR 96h:05207
  • [M2] I. G. Macdonald, Schur functions: theme and variations, in ``Actes 28-e Séminaire Lotharingien,'' I.R.M.A., Strasbourg, 1992, 5-39. MR 95m:05245
  • [M3] A. Molev, Factorial supersymmetric Schur functions and super Capelli identities, in ``A. A. Kirillov Seminar on Representation Theory,'' S. Gindikin, ed., Amer. Math. Soc.Transl., Amer. Math. Soc., Providence, 1998, 109-137. CMP 98:12
  • [N] M. Nazarov, Yangians and Capelli identities, in ``A. A. Kirillov Seminar on Representation Theory,'' S. Gindikin, ed., Amer. Math. Soc. Transl., Amer. Math.Soc., Providence, 1998, 139-163. CMP 98:2
  • [O1] A. Okounkov, Quantum immanants and higher Capelli identities, Transformation Groups 1 (1996), 99-126. MR 97j:17010
  • [O2] A. Okounkov, Young basis, Wick formula, and higher Capelli identities, Int. Math. Research Notes (1996), 817-839. MR 98b:17009
  • [OO] A. Okounkov and G. Olshanski, Shifted Schur functions, St. Petersburg Math. J. 9 (1997), no. 2; q-alg/9605042.
  • [S1] B. E. Sagan, ``The symmetric group: representations, combinatorial algorithms, and symmetric functions,'' 2nd edition, Springer-Verlag, New York, to appear.
  • [S2] S. Sahi, The spectrum of certain invariant differential operators associated to a Hermitian symmetric space, in ``Lie Theory and Geometry,'' J.-L. Brylinski, R. Brylinski, V. Guillemin, V. Kac eds., Progress in Math., Vol. 123, Birkhäuser, Boston, 1994, 569-576. MR 94d:43013
  • [V] S. Veigneau, ``Calcul symbolique et calcul distribué en combinatoire algébrique, Ph.D. thesis, Université de Marne-la-Vallée, Marne-la-Vallée, 1996.
  • [Z] A. V. Zelevinsky, A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence, J. Algebra 69 (1981), 82-94. MR 82j:20028

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 05E05, 05E10, 17B10, 17B35, 20C30

Retrieve articles in all journals with MSC (1991): 05E05, 05E10, 17B10, 17B35, 20C30

Additional Information

Alexander I. Molev
Affiliation: School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia

Bruce E. Sagan
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027

Keywords: Capelli operator, factorial Schur function, Littlewood-Richardson rule, quantum immanant, Young tableau
Received by editor(s): September 2, 1997
Received by editor(s) in revised form: January 15, 1998
Published electronically: February 8, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society