Applications of the Frobenius Formulas for the Characters of the Symmetric Group and the Hecke Algebras of Type A
Arun Ram
and Jeffrey B. Remmel
dagger
DOI: 10.1023/A:1008696218125
Abstract
We give a simple combinatorial proof of Ram's rule for computing the characters of the Hecke Algebra. We also establish a relationship between the characters of the Hecke algebra and the Kronecker product of two irreducible representations of the Symmetric Group which allows us to give new combinatorial interpretations to the Kronecker product of two Schur functions evaluated at a Schur function of hook shape or a two row shape. We also give a formula for the regular representation of the Hecke algebra.
Pages: 59–87
Keywords: character; symmetric group; Hecke algebra; Kronecker product
Full Text: PDF
References
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15. A. Ram, J.B. Remmel, and S.T. Whitehead, Combinatorics of the q-Basis of Symmetric Functions. Preprint.
16. J.B. Remmel, “A formula for the Kronecker products of Schur functions of hook shape,” J. Alg. 120 (1989), 100-118.
17. J.B. Remmel, “Formulas for the expansions of the Kronecker products S(m,n) \otimes S(1p - r ,r) and S(1k,2 ) \otimes S(1p - r ,r),” Discrete Math 99 (1992), 265-287.
18. J.B. Remmel and S.T. Whitehead, “On the Kronecker product of Schur functions of two row shape,” Bult. Belgian Math. Soc. 1 (1994), 649-683.
19. J.B. Remmel and R. Whitney, “Multiplying Schur functions,” J. Algorithms 5 (1984), 471-487.
20. J. van der Jeugt, “An algorithm for characters of Hecke algebras Hn(q) of type An - 1,” J. Phys. A: Math. Gen. 24 (1991), 3719-3725.
2. N. Bourbaki, Groupes et Alg`ebres de Lie, Chaps. 4-6, Hermann, Paris, 1960.
3. C.W. Curtis and I. Reiner, Methods of Representation Theory-With Applications to Finite Groups and Orders, Vol. II, Wiley-Interscience, New York, 1987.
4. O. Egecioglu and J.B. Remmel, “The combinatorics of the inverse Kostka matrix,” Linear and Multilinear Alg. 26 (1990), 59-84.
5. F.G. Frobenius, “ \ddot Uber die Charactere der symmetrischen Gruppe,” Sitzungberichte der K\ddot oniglich Preussischen Akademie der Wissenschaften zu Berlin (1900), 516-534; reprinted in Gessamelte Abhandlungen 3, 148-166.
6. A.M. Garsia and J.B. Remmel, “Shuffles of permutations and the Kronecker product,” Graphs and Combinatorics 1 (1985), 217-263.
7. A. Gyojia, “Topological invariants of links and representations of Hecke algebras,” Proceedings of a Conference on Three Dimensional Manifolds and Algebraic Invariants of Links, 1986.
8. P.N. Hoefsmit, “Representations of Hecke algebras of finite groups with B N pairs of classical types,” Dissertation, University of British Columbia, 1974.
9. R.C. King and B.G. Wybourne, “Representations and traces of the Hecke algebras Hn(q) of type An - 1,” J. Math. Physics 3 (1992), 4-14.
10. D. M. Knutson, “λ-rings and the representation theory of the symmetric group,” Lecture Notes in Mathematics 308, Springer-Verlag, Berlin, New York, 1973.
11. A. Lascoux and M.P. Sch\ddot utzenberger, “Formulaire raisonné de fonctions symétriques,” Publ. Math. Univ. Paris 7(1985).
12. D.E. Littlewood, Theory of Group Characters, Oxford University Press, 1940.
13. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979.
14. A. Ram, “A Frobenius formula for the characters of the Hecke algebras,” Inventiones Mathematicae 106 (1991), 461-488.
15. A. Ram, J.B. Remmel, and S.T. Whitehead, Combinatorics of the q-Basis of Symmetric Functions. Preprint.
16. J.B. Remmel, “A formula for the Kronecker products of Schur functions of hook shape,” J. Alg. 120 (1989), 100-118.
17. J.B. Remmel, “Formulas for the expansions of the Kronecker products S(m,n) \otimes S(1p - r ,r) and S(1k,2 ) \otimes S(1p - r ,r),” Discrete Math 99 (1992), 265-287.
18. J.B. Remmel and S.T. Whitehead, “On the Kronecker product of Schur functions of two row shape,” Bult. Belgian Math. Soc. 1 (1994), 649-683.
19. J.B. Remmel and R. Whitney, “Multiplying Schur functions,” J. Algorithms 5 (1984), 471-487.
20. J. van der Jeugt, “An algorithm for characters of Hecke algebras Hn(q) of type An - 1,” J. Phys. A: Math. Gen. 24 (1991), 3719-3725.