A Gelfand model for Weyl groups of type \(D_{2n}\).
Summary: A Gelfand model for a finite group \(G\) is a complex representation of \(G\), which is isomorphic to the direct sum of all irreducible representations of \(G\). When \(G\) is isomorphic to a subgroup of \(\text{GL}_n(\mathbb C)\), where \(\mathbb C\) is the field of complex numbers, it has been proved that each \(G\)-module over \(\mathbb C\) is isomorphic to a \(G\)-submodule in the polynomial ring \(\mathbb C[x_1,\dots,x_n]\), and taking the space of zeros of certain \(G\)-invariant operators in the Weyl algebra, a finite-dimensional \(G\)-space \(\mathcal N_G\) in \(\mathbb C[x_1,\dots,x_n]\) can be obtained, which contains all the simple \(G\)-modules over \(\mathbb C\). This type of representation has been named polynomial model. It has been proved that when \(G\) is a Coxeter group, the polynomial model is a Gelfand model for \(G\) if, and only if, \(G\) has not an irreducible factor of type \(D_{2n}\), \(E_7\), or \(E_8\). This paper presents a model of Gelfand for a Weyl group of type \(D_{2n}\) whose construction is based on the same principles as the polynomial model.
scientific article; zbMATH DE number 6078649
scientific article