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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Partitions, irreducible characters, and inequalities for generalized matrix functions
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by Thomas H. Pate PDF
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.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 325 (1991), 875-894
  • MSC: Primary 15A15
  • DOI: https://doi.org/10.1090/S0002-9947-1991-0998356-3
  • MathSciNet review: 998356