Unimodality of differences of specialized Schur functions

Reiner, V. ; Stanton, D.

Journal of Algebraic Combinatorics, Tome 7 (1998) no. 1, pp. 91-107.

Résumé

Summary: The centered difference of principally specialized Schur functions $s _{[$( l)$\tilde] } (1, q$, frac14 , $q ^{ n} ) - q ^{ n} s _{[$( l)$\tilde] } (1, q$, frac14 , $q ^{ n} )$ s_tildelambda (1,q,$ \ldots $,q^n ) - q^n s_tildelambda (1,q,$ \ldots $,q^n ) is shown to be a symmetric, unimodal polynomial in q with non-negative coefficients for certain choices of   $lambda, lambda$, and n, in which   $lambda$ is always obtained from $lambda$ by adding two cells, and n is chosen to be odd or even depending on   $lambda, lambda$. The basic technique is to find an injection of representations for the symplectic or orthogonal Lie algebras, and interpret the above difference as the principal specialization of the formal character of the quotient. As a special case, a difference of q-binomial coefficients is shown to be unimodal.

Détail

Keywords: unimodality, peck, principal specialization