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Partitions, irreducible characters, and inequalities for generalized matrix functions


Author: Thomas H. Pate
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
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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

$\displaystyle \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

$\displaystyle {({X_\alpha }(e))^{ - 1}}{d_{{X_\alpha }}}(B) \geq {({X_{\alpha\prime}}(e))^{ - 1}}{d_{{X_{\alpha\prime}}}}(B)$

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

DOI: https://doi.org/10.1090/S0002-9947-1991-0998356-3
Keywords: Generalized matrix function, tensor product, induced character, partition
Article copyright: © Copyright 1991 American Mathematical Society

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