From the introduction: A Gelfand model for wreath products \(\mathbb Z_r\wr S_n\) is constructed. The proof relies on a combinatorial interpretation of the characters of the model, extending a classical result of Frobenius and Schur.
A complex representation of a group \(G\) is called a Gelfand model for \(G\), or simply a model, if it is equivalent to the multiplicity-free direct sum of all the irreducible representations of \(G\). In this paper, we determine an explicit and simple combinatorial action which gives a model for wreath products \(\mathbb Z_r\wr S_n\), and, in particular, for the Weyl groups of type \(B\). For \(r=1\) (i.e., for the symmetric group) the construction is identical with the one given by V. Kodiyalam and D.-N. Verma [A natural representation model for symmetric groups, preprint (2004)] and R. M. Adin, A. Postnikov and Y. Roichman [J. Algebra 320, No. 3, 1311-1325 (2008; Zbl 1172.20009)]. The proof relies on a combinatorial interpretation of the characters, extending a classical result of Frobenius and Schur.
If all the (irreducible) representations of a finite group are real, then, by a result of Frobenius and Schur, the character-value of a model at a group element is the number of square roots of this element in the group. We are concerned in this paper with \(G(r,n)=\mathbb Z_r\wr S_n\), the wreath product of a cyclic group \(\mathbb Z_r\) with a symmetric group \(S_n\). For \(r>2\) this group is not real, and Frobenius’ theorem does not apply. It will be shown that the character-value of a model at an element of \(G(r,n)\) is the number of “absolute square roots” of this element in the group; see Theorem 3.4 below.
The rest of the paper is organized as follows. The construction of the model is described in Subsection 1.1. Necessary preliminaries and notation are given in Section 2. The combinatorial interpretation of the characters of the model is described in Section 3, Theorem 3.4. Two proofs for this interpretation are given. A direct combinatorial proof, using the Murnaghan-Nakayama rule, is given in Section 4. The second proof combines the properties of the generalized Robinson-Schensted algorithm for wreath products, due to Stanton and White, with a generalized Frobenius-Schur formula due to Bump and Ginzburg; see Section 5. The main theorem, Theorem 1.2, is proved in Section 6. The proof applies generalized Frobenius-Schur character formula, Theorem 3.4, together with Corollary 4.3. Section 7 ends the paper with final remarks and open problems.