Scientific article

Remarks on the $alpha$--permanent

Published inMathematical research letters, vol. 17, no. 4, p. 795-802
Publication date2010

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.

  • arxiv : math.AC
Citation (ISO format)
FRENKEL, Peter Erno. Remarks on the $alpha$--permanent. In: Mathematical research letters, 2010, vol. 17, n° 4, p. 795–802.
Main files (1)
Article (Published version)
ISSN of the journal1073-2780

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