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Remarks on the $alpha$--permanent

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Published in Mathematical Research Letters. 2010, vol. 17, no. 4, p. 795-802
Abstract We recall Vere-Jones's definition of the $alpha$--permanent and describe the connection between the (1/2)--permanent and the hafnian. We establish expansion formulae for the $alpha$--permanent in terms of partitions of the index set, and we use these to prove Lieb-type inequalities for the $pmalpha$--permanent of a positive semi-definite Hermitian $n imes n$ matrix and the $alpha/2$--permanent of a positive semi-definite real symmetric $n imes n$ matrix if $alpha$ is a nonnegative integer or $alphage n-1$. We are unable to settle Shirai's nonnegativity conjecture for $alpha$--permanents when $alphage 1$, but we verify it up to the $5 imes 5$ case, in addition to recovering and refining some of Shirai's partial results by purely combinatorial proofs.
Stable URL https://archive-ouverte.unige.ch/unige:11986
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arXiv: 0912.1018
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Deposited on : 2010-10-01

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