×

Partitions, irreducible characters, and inequalities for generalized matrix functions. (English) Zbl 0729.15008

The main results of the paper are theorems 3 and 6. Theorem 3 provides a stepping-up inequality: it shows how, given an irreducible character \(\psi\) of \(S_ n\), to find a second character \({\mathcal E}\), induced from a Young subgroup of \(S_ n\), such that \(\psi\leq {\mathcal E}\). In the same sense theorem 6 provides a stepping inequality. The author combines theorems 3 and 6 to obtain theorem 7. The result is as follows. Given a partition \(\alpha =\{\alpha_ 1,\alpha_ 2,...,\alpha_ s\}\), \(\alpha_ 1\geq \alpha_ 2\geq...\geq \alpha_ s\), of n the author lets \(X_{\alpha}\) denote the derived irreducible character of \(S_ n\), and he associates with \(\alpha\) a derived partition \[ \alpha '=\{\alpha_ 1-1,\alpha_ 2-1,...,\alpha_ t-1,\alpha_{t+1},...,\alpha_ s,1^ t\} \] where t denotes the smallest positive integer such that \(\alpha_ t>\alpha_{t+1}\) \((\alpha_{s+1}=0)\). He shows that if Y is a decomposable \({\mathbb{C}}\)-valued n-linear function on \({\mathbb{C}}^ m\times {\mathbb{C}}^ m\times...\times {\mathbb{C}}^ m\) (n-copies) then \(<X_{\alpha}Y,Y>\geq <X_{\alpha '}Y,Y>\). The author obtains an inequality involving the generalized matrix functions \(d_{X_{\alpha}}\) and \(d_{X_{\alpha '}}\), namely that \[ (X_{\alpha}(e))^{-1}d_{X_{\alpha}}(B)\geq (X_{\alpha '}(e))^{- 1}d_{X_{\alpha '}}(B) \] 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.

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
15A69 Multilinear algebra, tensor calculus
15A45 Miscellaneous inequalities involving matrices
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. W. Neuberger, Norm of symmetric product compared with norm of tensor product, Linear and Multilinear Algebra 2 (1974), 115 – 121. · Zbl 0288.15034 · doi:10.1080/03081087408817047
[2] Thomas H. Pate, A continuous analogue of the Lieb-Neuberger inequality, Houston J. Math. 12 (1986), no. 2, 225 – 234. · Zbl 0636.46030
[3] I. Schur, Über endliche Gruppen und Hermitesche Formen, Math. Z. 1 (1918), no. 2-3, 184 – 207 (German). · JFM 46.0174.03 · doi:10.1007/BF01203611
[4] Marvin Marcus, On two classical results of I. Schur, Bull. Amer. Math. Soc. 70 (1964), 685 – 688. · Zbl 0263.15004
[5] Marvin Marcus, The Hadamard theorem for permanents, Proc. Amer. Math. Soc. 15 (1964), 967 – 973. · Zbl 0166.29903
[6] Elliott H. Lieb, Proofs of some conjectures on permanents, J. Math. Mech. 16 (1966), 127 – 134. · Zbl 0144.26802
[7] Marvin Marcus, Finite dimensional multilinear algebra. Part II, Marcel Dekker, Inc., New York, 1975. Pure and Applied Mathematics, Vol. 23. · Zbl 0339.15003
[8] Gordon D. James and Martin W. Liebeck, Permanents and immanants of Hermitian matrices, Proc. London Math. Soc. (3) 55 (1987), no. 2, 243 – 265. · Zbl 0657.15006 · doi:10.1093/plms/s3-55_2.243
[9] Thomas H. Pate, Permanental dominance and the Soules conjecture for certain right ideals in the group algebra, Linear and Multilinear Algebra 24 (1989), no. 2, 135 – 149. · Zbl 0738.15005 · doi:10.1080/03081088908817906
[10] Peter Heyfron, Immanant dominance orderings for hook partitions, Linear and Multilinear Algebra 24 (1988), no. 1, 65 – 78. · Zbl 0678.15009 · doi:10.1080/03081088808817899
[11] Russell Merris and William Watkins, Inequalities and identities for generalized matrix functions, Linear Algebra Appl. 64 (1985), 223 – 242. · Zbl 0563.15006 · doi:10.1016/0024-3795(85)90279-4
[12] M. A. Naĭmark and A. I. Štern, Theory of group representations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 246, Springer-Verlag, New York, 1982. Translated from the Russian by Elizabeth Hewitt; Translation edited by Edwin Hewitt.
[13] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. · Zbl 0355.20006
[14] Thomas H. Pate, Generalizing the Fischer inequality, Linear Algebra Appl. 92 (1987), 1 – 15. · Zbl 0618.15011 · doi:10.1016/0024-3795(87)90247-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.