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.


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