Alpha-determinant cyclic modules and Jacobi polynomials

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dc.creatorKimoto, Kazufumi
dc.creatorMatsumoto, Sho
dc.creatorWakayama, Masato
dc.date2007-10-19
dc.date2008-09-01
dc.date.accessioned2026-07-07T09:59:19Z
dc.date.available2026-07-07T09:59:19Z
dc.descriptionWe study the cyclic $U(\mathfrak{gl}_n)$-module generated by the $l$-th power of the $α$-determinant. When $l$ is a non-negative integer, for all but finite exceptional values of $alpha$, one shows that this cyclic module is isomorphic to the $n$-th tensor space $(S^l(\mathbb{C}^n))^{\otimes n}$ of the symmetric $l$-th tensor space of $\mathbb{C}^n$. If $alpha$ is exceptional, then the structure of the module changes drastically, i.e. some irreducible representations which are the irreducible components of the decomposition of $(S^l(\mathbb{C}^n))^{\otimes n}$ disappear in the decomposition of the cyclic module. The degeneration of each isotypic component of the cyclic module is described by a matrix whose size is given by a Kostka number and entries are polynomials in $alpha$ with rational coefficients. As a special case, we determine the matrix in a full of the detail for the case where $n=2$; the matrix becomes a scalar and is essentially given by the classical Jacobi polynomial. Moreover, we prove that these polynomials are unitary.
dc.description24 pages, to appear in Trans. Amer. Math. Soc
dc.identifierhttps://arxiv.org/abs/0710.3669
dc.identifierhttp://arxiv.org/abs/0710.3669
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/167979
dc.subjectRepresentation Theory
dc.subject22E47; 33C45
dc.titleAlpha-determinant cyclic modules and Jacobi polynomials
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