Algebraic and combinatorial properties of minimal border strip tableaux

Abstract

Motivated by results and conjectures of Stanley concerning minimal border strip tableaux of partitions, we present three results. First we generalize the rank of a partition [lambda] to the rank of a shifted partition S([lambda]).We show that the number of bars required in a minimal bar tableau of S([lambda]) is max(o, e + (ℓ([lambda]) mod 2)), where o and e are the number of odd and even rows of [lambda]. As a consequence we show that the irreducible negative characters of [tilde]S[sub]n vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur's Q[sub][lambda] symmetric functions in terms of the power sum symmetric functions. The second result gives a basis for the space spanned by the lowest degree terms in the expansion of the Schur symmetric functions in terms of the power sum symmetric functions. These lowest degree terms studied by Stanley correspond to minimal border strip tableaux of [lambda]. The Hilbert series of these spaces is the generating function giving the number of partitions of n into parts differing by at least 2. Applying the Rogers-Ramanujan identity, the generating function also counts the number of partitions of n into parts 5k + 1 and 5k - 1. Finally for each [lambda] we give a relation between the power sum symmetric functions and the monomial symmetric functions; the terms are indexed by the types of minimal border strip tableaux of [lambda].