Plane Partitions and Characters of the Symmetric Group

In this paper we show that the existence of plane partitions, which are minimal in a sense to be defined, yields minimal irreducible summands in the Kronecker product χλ ⊗ χμ of two irreducible characters of the symmetric group S(n). The minimality of the summands refers to the dominance order of partitions of n. The multiplicity of a minimal summand χν equals the number of pairs of Littlewood-Richardson multitableaux of shape (λ, μ), conjugate content and type ν. We also give lower and upper bounds for these numbers.

[1]  P. Fishburn,et al.  Sets Uniquely Determined by Projections on Axes I , 1990 .

[2]  H. Ryser.  Combinatorial Properties of Matrices of Zeros and Ones , 1957, Canadian Journal of Mathematics.

[3]  Ernst Snapper,et al.  Group characters and nonnegative integral matrices , 1971 .

[4]  Thomas Brylawski,et al.  The lattice of integer partitions , 1973, Discret. Math..

[5]  Ernesto Vallejo.  Reductions of additive sets, sets of uniqueness and pyramids , 1997, Discret. Math..

[6]  Adriano M. Garsia,et al.  Shuffles of permutations and the Kronecker product , 1985, Graphs Comb..

[7]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[8]  J. Saxl.  The complex characters of the symmetric groups that remain irreducible in subgroups , 1987 .

[9]  Tsit Yuen Lam,et al.  Young diagrams, Schur functions, the Gale-Ryser theorem and a conjecture of snapper , 1977 .

[10]  Jeffrey B. Remmel,et al.  A formula for the Kronecker products of Schur functions of hook shapes , 1989 .

[11]  J. Remmel,et al.  A combinatorial interpretation of the inverse kostka matrix , 1990 .

[12]  D. Gale.  A theorem on flows in networks , 1957 .

[13]  R. H. Makar.  On the Analysis of the Kronecker Product of Irreducible Representations of the Symmetric Group , 1949 .

[14]  Ilan Zisser.  The character covering numbers of the alternating groups , 1992 .

[15]  Jeffrey B. Remmel.  Formulas for the expansion of the Kronecker products S(m, n) * S(1p-r, r) and S(1k2l) * S(1p-r, r) , 1992, Discret. Math..

[16]  Yoav Dvir.  A Family of Z Bases for the Ring of Sn Characters and Applications to the Decomposition of Kronecker Products , 1994, Eur. J. Comb..

[17]  A. J. Coleman,et al.  Induced representations with applications to Sn and GL(n) , 1966 .

[18]  D. E. Littlewood,et al.  The Kronecker Product of Symmetric Group Representations , 1956 .

[19]  Jeffrey B. Remmel,et al.  On the Kronecker product of Schur functions of two row shapes , 1994 .

[20]  F. Murnaghan.  The Analysis of the Kronecker Product of Irreducible Representations of the Symmetric Group , 1938 .

[21]  Y. Dvir,et al.  On the Kronecker Product of Sn Characters , 1993 .

[22]  A. Torres-Chazaro,et al.  Sets of Uniqueness and Minimal Matrices , 1998 .

[23]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[24]  B. Sagan.  The Symmetric Group , 2001 .