Remarks on the alpha--permanent
- Autores: Péter E. Frenkel
- Localización: Mathematical research letters, ISSN 1073-2780, Vol. 17, Nº 4, 2010, págs. 795-802
- Idioma: inglés
- DOI: 10.4310/mrl.2010.v17.n4.a17
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Resumen
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 $\pm\alpha$--permanent of a positive semi-definite Hermitian $n\times n$ matrix and the $\alpha/2$--permanent of a positive semi-definite real symmetric $n\times n$ matrix if $\alpha$ is a nonnegative integer or $\alpha\ge n-1$. We are unable to settle Shirai's nonnegativity conjecture for $\alpha$--permanents when $\alpha\ge 1$, but we verify it up to the $5\times 5$ case, in addition to recovering and refining some of Shirai's partial results by purely combinatorial proofs.
