Partitions, Irreducible Characters, and Inequalities for Generalized Matrix Functions

Abstract

Given a partition $\alpha = \{ {\alpha _1},{\alpha _2}, \ldots ,{\alpha _s}\}$, ${\alpha _1} \geq {\alpha _2} \geq \cdots \geq {\alpha _s}$, of $n$ we let ${X_\alpha }$ denote the derived irreducible character of ${S_n}$, and we associate with $\alpha$ a derived partition \[ \alpha \prime = \{ {\alpha _1} - 1,{\alpha _2} - 1, \ldots ,{\alpha _t} - 1,{\alpha _{t + 1}}, \ldots ,{\alpha _s},{1^t}\} \] where $t$ denotes the smallest positive integer such that ${\alpha _t} > {\alpha _{t + 1}}\;({\alpha _{s + 1}} = 0)$. We show that if $Y$ is a decomposable $\mathbb {C}$-valued $n$-linear function on ${\mathbb {C}^m} \times {\mathbb {C}^m} \times \cdots \times {\mathbb {C}^m}$ ($n$-copies) then $\left \langle {{X_\alpha }Y,Y} \right \rangle \geq \left \langle {{X_\alpha },Y,Y} \right \rangle$. Translating into the notation of matrix theory we obtain an inequality involving the generalized matrix functions ${d_{{X_\alpha }}}$ and ${d_{{X_{\alpha \prime }}}}$, namely that \[ {({X_\alpha }(e))^{ - 1}}{d_{{X_\alpha }}}(B) \geq {({X_{\alpha \prime }}(e))^{ - 1}}{d_{{X_{\alpha \prime }}}}(B)\] for each $n \times n$ positive semidefinite Hermitian matrix $B$. This result generalizes a classical result of I. Schur and includes many other known inequalities as special cases.