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For positive integers n and l, we study the cyclic GL(n)-module generated by the l-th power of the α-determinant. This cyclic module is isomorphic to the n-th tensor space of the symmetric l-th tensor... space of the vector representation of GL(n) for all but finite exceptional values of α. If αa is exceptional, then the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in the aforementioed tensor space. 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 αa with rational coefficients. Especially, we determine explicitly the matrix when n equals 2. In that case, the matrix becomes a scalar and is essentially given by the classical Jacobi polynomial. Moreover, we prove that these polynomials are unitary.続きを見る
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Mendeley出力
