On the Kronecker Product s_(n-p;p) * s_l
by
Cristina Ballantine
College of the Holy Cross
Coauthors: Rosa Orellana (Dartmouth College)

The Kronecker product of two Schur functions s_l¸ and s_m, denoted s_l*s_m, is defined as the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group indexed by partitions of n, l and m, respectively. The coefficient, g_{l, m, n} of s_n in s_l*s_m is equal to the multiplicity of the irreducible representation indexed by n in the tensor product. Let l = (l₁, l₂, ¼, l_l). We show that the coefficients in the expansion of s_(n-p;p) * s_l do not depend on n if l₁-l₂ ³ 2p. In this case, we give an algorithm for expanding the Kronecker product s_(n-p;p) * s_l, where p is a positive integer and l₁-l₂ ³ 2p. We also give a simple combinatorial interpretation for g_{l, (n-p, p), n} if l ³ 2p-1 or if l₁ ³ 2p - 1; i.e., if l is not a partition inside the 2(p - 1) ×2(p - 1) square.

Date received: April 2, 2005