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Alpha-determinant cyclic modules and Jacobi polynomials. With an appendix by Kazufumi Kimoto. (English) Zbl 1189.17009

The Lie algebra \(\mathfrak{gl}_n(\mathbb C)\), and hence its universal enveloping algebra \(\mathcal U\), acts on the space of complex polynomials in \(n^2\) variables \(\{x_{ij}\}_{1\leq i,j\leq n}\) by differential operators (this is the differential of the translation action by \(\text{GL}(n,\mathbb C)\)). In an earlier paper, S. Matsumoto and M. Wakayama [J. Lie Theory 16, No. 2, 393–405 (2006; Zbl 1102.17004)] determined the decomposition of the submodule \(V_n^{(\alpha)}\) generated by a certain polynomial, the \(\alpha\)-determinant: \[ \det^{(\alpha)}(X)=\sum_{\sigma\in\mathfrak S_n}\alpha^{\nu(\sigma)}\prod_{i=1}^nx_{i\sigma(i)}. \] Here \(\alpha\) is a complex number, \(\mathfrak S_n\) is the symmetric group on \(n\) letters, and \(\nu(\sigma)\) denotes \(n\) minus the number of cycles in \(\sigma\). The \(\alpha\)-determinant is a common generalization of the determinant and the permanent of \(X=(x_{i,j})\) in the sense that \(\text{per}(X)=\det^{(1)}(X)\) and \(\det(X)=\det^{(-1)}(X)\). For generic values of \(\alpha\), each of these modules is equivalent to the \(n\)-fold tensor product of the standard representation \(\mathbb C^n\).
In this paper, the authors study the cyclic modules \(V_{n,l}^{(\alpha)}\) generated by polynomials of the form \(\det^{(\alpha)}(X)^l\) for positive integers \(l\). They find that, for all but a finite number of exceptional choices of \(\alpha\), the module \(V_{n,l}^{(\alpha)}\) is isomorphic (as a \(\mathcal U\)-module) to the space \(\mathcal S^l(\mathbb C^n)^{\otimes n}\). For an “exceptional” value of \(\alpha\), the multiplicities of some of the irreducible subrepresentations in \(V_{n,l}^{(\alpha)}\) are smaller than those in the full module of symmetric \(l\)-tensors on \(\mathbb C^n\); in each case, the multiplicity is given by the rank of a certain “transition matrix” \(F^{\lambda}_{n,l}(\alpha)\). Here \(\lambda\) is the partition of \(nl\) determining the irreducible representation. This matrix has size \(K_{\lambda(l^2)}\), the appropriate Kostka number, which agrees with the multiplicity of the representation in the full module \(\mathcal S^l(\mathbb C^n)^{\otimes n}\). The authors give a definition of the transition matrix; however, this matrix and its rank are very difficult to compute, and in general, no effective method to evaluate these multiplicities is given. For the case \(n=2\), however, the matrix is a scalar which is given by a classical Jacobi polynomial. These polynomials are unitary, i.e., their zeros all lie on the unit circle.
Finally, in the appendix, the authors show that the entries of the transition matrices are given by a variation of the spherical Fourier transformation of a certain class function on \(\mathfrak S_{nl}\) with respect to the subgroup \((\mathfrak S_l)^n\), and describe the case \(n=2\) by using zonal spherical functions on the Gelfand pair \((\mathfrak S_{2l},\mathfrak S_l^2)\).

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
43A90 Harmonic analysis and spherical functions

Citations:

Zbl 1102.17004
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References:

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