Lower and upper bounds of Schur characters

Abstract

Characterizations of lower and upper bounds for dual characters of flagged Weyl modules have attracted considerable interest. In this paper, we establish explicit pattern avoidance characterizations for lower and upper bounds of Schur characters, namely, the dual characters of Weyl modules associated with arbitrary diagrams. This setting extends the corresponding extremal problems for dual characters of flagged Weyl modules and includes skew Schur polynomials and, through Rothe diagrams, Stanley symmetric functions. Specifically, for a permutation $w$, we show that the Stanley symmetric function $F_w$ attains the lower bound if and only if $w$ avoids $321,2143,2413,3142$, and $3412$, and attains the upper bound if and only if $w$ avoids $312$ and $321$.