A Combinatorial Interpretation for the Coefficients in the Kronecker Product s(n−p,p) ∗ sλ

In this paper we give a combinatorial interpretation for the coefficient of sν in the Kronecker product s(n−p,p) ∗ sλ, where λ = (λ1, . . . , λ`(λ)) ` n, if `(λ) ≥ 2p − 1 or λ1 ≥ 2p − 1; that is, if λ is not a partition inside the 2(p − 1) × 2(p − 1) square. For λ inside the square our combinatorial interpretation provides an upper bound for the coefficients. In general, we are able to combinatorially compute these coefficients for all λ when n > (2p−2)2. We use this combinatorial interpretation to give characterizations for multiplicity free Kronecker products. We have also obtained some formulas for special cases. introduction Let χ and χ be the irreducible characters of Sn (the symmetric group on n letters) indexed by the partitions λ and μ of n. The Kronecker product χχ is defined by (χχ)(w) = χ(w)χ(w) for w ∈ Sn. Then χχ is the character that corresponds to the diagonal action of Sn on the tensor product of the irreducible representations indexed by λ and μ. We have χχ = ∑