The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes
Mercedes H. Rosas
DOI: 10.1023/A:1011942029902
Abstract
The Kronecker product of two Schur functions s 
and s 
, denoted by s * s , is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions and . The coefficient of s 
in this product is denoted by 
, and corresponds to the multiplicity of the irreducible character 
in .
and s , denoted by s * s , is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions and . The coefficient of s in this product is denoted by , and corresponds to the multiplicity of the irreducible character in . We use Sergeev”s Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for s [ XY] to find closed formulas for the Kronecker coefficients when is an arbitrary shape and and are hook shapes or two-row shapes.
[ XY] to find closed formulas for the Kronecker coefficients when is an arbitrary shape and and are hook shapes or two-row shapes.Pages: 153–173
Keywords: Kronecker product internal product; sergeev”s formula
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References
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2. A.M. Garsia and J.B. Remmel, “Shuffles of permutations and Kronecker products,” Graphs Combin. 1(3) (1985), 217-263.
3. I.M. Gessel, “Multipartite P-partitions and inner products of Schur functions,” Contemp. Math. (1984), 289- 302.
4. A. Lascoux, “Produit de Kronecker des representations du group symmetrique,” Lecture Notes in Mathematics Springer Verlag, 795 (1980), 319-329.
5. D.E. Littlewood, “The Kronecker product of symmetric group representations,” J. London Math. Soc. 31 (1956), 89-93.
6. D.E. Littlewood, “Plethysm and inner product of S-functions,” J. London Math. Soc. 32 (1957), 18-22.
7. I.G. Macdonald, Symmetric Functions and Hall Polynomials, second edition, Oxford: Oxford University Press, 1995.
8. J.H. Redfield, “The theory of group reduced distribution,” Amer. J. Math. 49 (1927), 433-455.
9. J.B. Remmel, “A formula for the Kronecker product of Schur functions of hook shapes,” J. Algebra 120 (1989), 100-118.
10. J.B. Remmel, “Formulas for the expansion of the Kronecker products S(m,n) \otimes S(1p - r ,r) and S(1k2l) \otimes S(1p - r ,r),” Discrete Math. 99 (1992), 265-287.
11. J.B. Remmel and T. Whitehead, “On the Kronecker product of Schur functions of two row shapes,” Bull. Belg. Math. Soc. Simon Stevin 1 (1994), 649-683.
12. M.H. Rosas, “A combinatorial overview of the theory of MacMahon symmetric functions and a study of the Kronecker product of Schur functions,” Ph.D. Thesis, Brandeis University (1999).
13. B.E. Sagan, The Symmetric Group, Wadsworth & Brooks/Cole, Pacific Grove, California, 1991.
14. A.N. Sergeev, “The tensor algebra of the identity representation as a module over the Lie superalgebras gl(n, m) and Q(n),” Math. USSR Sbornik, 51, pp. 419-427.
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