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Given a partition a = {al, a2 X... Xas} X al > a2 > .> a, of n we let XQ denote the derived irreducible character of Sn, and we associate with a a derived partition a = {al a,2X**X a tat+l as.. , 1 } where t denotes the smallest positive integer such that at > at+I (as+1 = 0). We show that if Y is a decomposable C-valued n-linear function on Cxm x(Cm X... xCm (n-copies) then (X Y, Y) > (X_ Y, Y) . Translating into the notation of matrix theory we obtain an inequality involving the generalized matrix functions dx and dx , namely that (X. (e)) dxa (B) > (X , (e)) 1dx (B) for each n x n positive semidefinite Hermitian matrix B. This result generalizes a classical result of I. Schur and includes many other known inequalities as special cases.

Partitions, irreducible characters, and inequalities for generalized matrix functions