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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A Littlewood-Richardson rule for factorial Schur functions
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by Alexander I. Molev and Bruce E. Sagan PDF
Trans. Amer. Math. Soc. 351 (1999), 4429-4443 Request permission

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
  • 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
  • A. Capelli, Sur les opérations dans la théorie des formes algébriques, Math. Ann. 37 (1890), 1–37.
  • Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
  • Sergey Fomin and Curtis Greene, A Littlewood-Richardson miscellany, European J. Combin. 14 (1993), no. 3, 191–212. MR 1215331, DOI 10.1006/eujc.1993.1024
  • 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
  • Roger Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570. MR 986027, DOI 10.1090/S0002-9947-1989-0986027-X
  • Roger Howe and T\B{o}ru Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), no. 3, 565–619. MR 1116239, DOI 10.1007/BF01459261
  • M. H. Peel and G. D. James, Specht series for skew representations of symmetric groups, J. Algebra 56 (1979), no. 2, 343–364. MR 528580, DOI 10.1016/0021-8693(79)90342-9
  • A. N. Kirillov and N. Yu. Reshetikhin, The Bethe ansatz and the combinatorics of Young tableaux, J. Soviet Math. 41 (1988), 925–955.
  • Alain Lascoux, Puissances extérieures, déterminants et cycles de Schubert, Bull. Soc. Math. France 102 (1974), 161–179 (French). MR 364274
  • A. Lascoux, “Interpolation,” Lectures at Tianjin University, June 1996.
  • A. Lascoux and M.-P. Schützenberger, Interpolation de Newton à plusieurs variables, Séminaire d’algèbre Paul Dubreil et Marie-Paule Malliavin, 36ème année (Paris, 1983–1984) Lecture Notes in Math., vol. 1146, Springer, Berlin, 1985, pp. 161–175 (French). MR 873085, DOI 10.1007/BFb0074538
  • D. E. Littlewood and A. R. Richardson, Group characters and algebra, Philos. Trans. Roy. Soc. London Ser. A 233 (1934), 49–141.
  • 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, 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
  • 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.
  • 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.
  • Andrei Okounkov, Quantum immanants and higher Capelli identities, Transform. Groups 1 (1996), no. 1-2, 99–126. MR 1390752, DOI 10.1007/BF02587738
  • Andrei Okounkov, Young basis, Wick formula, and higher Capelli identities, Internat. Math. Res. Notices 17 (1996), 817–839. MR 1420550, DOI 10.1155/S1073792896000505
  • A. Okounkov and G. Olshanski, Shifted Schur functions, St. Petersburg Math. J. 9 (1997), no. 2; q-alg/9605042.
  • B. E. Sagan, “The symmetric group: representations, combinatorial algorithms, and symmetric functions,” 2nd edition, Springer-Verlag, New York, to appear.
  • Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
  • S. Veigneau, “Calcul symbolique et calcul distribué en combinatoire algébrique, Ph.D. thesis, Université de Marne-la-Vallée, Marne-la-Vallée, 1996.
  • 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
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Additional Information
  • Alexander I. Molev
  • Affiliation: School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia
  • MR Author ID: 207046
  • Email: alexm@maths.usyd.edu.au
  • Bruce E. Sagan
  • Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
  • MR Author ID: 152890
  • Email: sagan@math.msu.edu
  • Received by editor(s): September 2, 1997
  • Received by editor(s) in revised form: January 15, 1998
  • Published electronically: February 8, 1999
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 4429-4443
  • MSC (1991): Primary 05E05; Secondary 05E10, 17B10, 17B35, 20C30
  • DOI: https://doi.org/10.1090/S0002-9947-99-02381-8
  • MathSciNet review: 1621694