# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## Partitions, irreducible characters, and inequalities for generalized matrix functionsHTML articles powered by AMS MathViewer

by Thomas H. Pate
Trans. Amer. Math. Soc. 325 (1991), 875-894 Request permission

## 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.
References
Similar Articles
• Retrieve articles in Transactions of the American Mathematical Society with MSC: 15A15
• Retrieve articles in all journals with MSC: 15A15