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Immanants of Totally Positive Matrices are Nonnegative

Immanants of Totally Positive Matrices are Nonnegative IMMANANTS OF TOTALLY POSITIVE MATRICES ARE NONNEGATIVE JOHN R. STEMBRIDGE Introduction Let M {k) denote the algebra of n x n matrices over some field k of characteristic zero. For each A>valued function / on the symmetric group S , we may define a corresponding matrix function on M (k) in which w a (fljji— • > X\ )&i (i)'" ( )' (U weS If / is an irreducible character of S , these functions are known as immanants; if/ is an irreducible character of some subgroup G of S (extended trivially to all of S by n n defining /(vv) = 0 for w$G), these are known as generalized matrix functions. Note that the determinant and permanent are obtained by choosing / to be the sign character and trivial character of S , respectively. We should point out that it is more traditional to use /(vv) in (1) where we have used /(W ) . This change can be undone by transposing the matrix. If/ happens to be -1 a character, then /(w ) = x(w), so the generalized matrix function we have indexed by / is the complex conjugate of the traditional one. Since the characters of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Immanants of Totally Positive Matrices are Nonnegative

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References (11)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/23.5.422
Publisher site
See Article on Publisher Site

Abstract

IMMANANTS OF TOTALLY POSITIVE MATRICES ARE NONNEGATIVE JOHN R. STEMBRIDGE Introduction Let M {k) denote the algebra of n x n matrices over some field k of characteristic zero. For each A>valued function / on the symmetric group S , we may define a corresponding matrix function on M (k) in which w a (fljji— • > X\ )&i (i)'" ( )' (U weS If / is an irreducible character of S , these functions are known as immanants; if/ is an irreducible character of some subgroup G of S (extended trivially to all of S by n n defining /(vv) = 0 for w$G), these are known as generalized matrix functions. Note that the determinant and permanent are obtained by choosing / to be the sign character and trivial character of S , respectively. We should point out that it is more traditional to use /(vv) in (1) where we have used /(W ) . This change can be undone by transposing the matrix. If/ happens to be -1 a character, then /(w ) = x(w), so the generalized matrix function we have indexed by / is the complex conjugate of the traditional one. Since the characters of

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Sep 1, 1991

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