A Combinatorial Interpretation for the Coefficients in the Kronecker Product s(n-p,p)*s\lambda

Cristina M. Ballantine ; Rosa C. Orellana

Séminaire lotharingien de combinatoire, 54A (2005-2007)

Résumé

In this paper we give a combinatorial interpretation for the coefficient of s_\nu in the Kronecker product s_(n-p,p)*s_\lambda, where \lambda=(\lambda₁, ..., \lambda_l(\lambda)) is a partition of n, if l(\lambda)>=2p-1 or \lambda₁>=2p-1; that is, if \lambda is not a partition inside the 2(p-1) x 2(p-1) square. For \lambda 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 \lambda 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.

Corrigendum

On page 25, line -6, in Corollary 4.13, $ \displaystyle m_4=\min\left\{ s,p-s, \left\lfloor\frac{p+s-t}{2}\right\rfloor\right\}$ should be replaced by $ \displaystyle m_4=\min\left\{ s,p-s-1, \left\lfloor\frac{p+s-t}{2}\right\rfloor\right\}$.