Remarks on the $alpha$--permanent

Frenkel, Peter Erno

Scientific article

English

Remarks on the $alpha$--permanent

Contributors

Frenkel, Peter Erno

Published in

Mathematical research letters, vol. 17, no. 4, p. 795-802

Publication date

2010

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.

Classification

arxiv : math.AC

UNIGE affiliation entities

Faculté des sciences / Section de mathématiques

Citation (ISO format)

FRENKEL, Peter Erno. Remarks on the $alpha$--permanent. In: Mathematical research letters, 2010, vol. 17, n° 4, p. 795–802.

Article (Published version)

Identifiers

PID : unige:11986
arXiv : 0912.1018

Journal ISSN

1073-2780

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23/09/2010 13:24:00

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14/03/2023 16:07:04

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29/10/2024 17:12:05

Remarks on the $alpha$--permanent

Contributors

Published in

Publication date

Abstract

Classification

UNIGE affiliation entities

Citation (ISO format)

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Journal ISSN

Technical informations

Creation

First validation

Update

Status update

Last indexation