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Zbl 1234.20012
Adin, Ron M.; Postnikov, Alexander; Roichman, Yuval
A Gelfand model for wreath products. (English)
[J] Isr. J. Math. 179, 381-402 (2010). ISSN 0021-2172; ISSN 1565-8511/e

From the introduction: A Gelfand model for wreath products $\Bbb Z_r\wr S_n$ is constructed. The proof relies on a combinatorial interpretation of the characters of the model, extending a classical result of Frobenius and Schur.\par 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 $\Bbb Z_r\wr S_n$, 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 {\it V. Kodiyalam} and {\it D.-N. Verma} [A natural representation model for symmetric groups, preprint (2004)] and {\it R. M. Adin, A. Postnikov} and {\it 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.\par 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 $G(r,n)=\Bbb Z_r\wr S_n$, the wreath product of a cyclic group $\Bbb Z_r$ 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.\par 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.

MSC 2000:: *20C30 Representations of finite symmetric groups
20C15 Ordinary representations and characters of groups
05E10 Tableaux, etc.

Keywords: symmetric groups; Gelfand models; irreducible characters; wreath products; irreducible representations; Murnaghan-Nakayama rule; generalized Robinson-Schensted algorithm; Frobenius-Schur character formula

Citations: Zbl 1172.20009

Cited in: Zbl pre06127521

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