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A Gel’fand model for the symmetric group. (English) Zbl 1015.20009

Summary: A Gel’fand model for a finite group \(G\) is a complex representation of \(G\) which is isomorphic to the direct sum of all the irreducible representations of \(G\) [see J. Soto-Andrade, Proc. Symp. Pure Math. 47, 305-316 (1987; Zbl 0652.20047)]. Gel’fand models for the linear groups over a finite field can be found in [A. A. Klyachko, Mat. Sb., N. Ser. 120(162), No. 3, 371-376 (1983; Zbl 0526.20033)]. In this work we describe a Gel’fand model for the symmetric group \({\mathfrak S}_n\). When \(K\) is a field of characteristic zero and \(G={\mathfrak S}_n\), we give a finite dimensional \(K\)-subspace \(\mathcal N\) of the polynomial ring \(K[x_1,\dots,x_n]\). If \(K\) is the field of complex numbers, then \(\mathcal N\) provides a Gel’fand model for \(G\). The space \(\mathcal N\) can be obtained as the zeros of certain differential operators (symmetrical operators) in the Weyl algebra.

MSC:

20C30 Representations of finite symmetric groups
20C15 Ordinary representations and characters
16S36 Ordinary and skew polynomial rings and semigroup rings
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References:

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