The recently developed theory of Schur rings over a finite cyclic group is generalized to Schur rings over a ring R being a product of Galois rings of coprime characteristics. It is proved that if the characteristic of R is odd, then as in the cyclic group case any pure Schur ring over R is the tensor product of a pure cyclotomic ring and Schur rings of rank 2 over non-fields. Moreover, it is s...

Let W be the wreath product of a symmetric group with a cyclic group of order l. The corresponding restricted rational Cherednik algebra is a finite dimensional algebra whose block structure has a combinatorial description in terms of J-hearts. We show that this description is equivalent to one given in terms of residues of multipartitions. This establishes links with Rouquier families for the ...

From the irreducible decompositions’ point of view, the structure of the cyclic GLn(C)-module generated by the α-determinant degenerates when α = ± 1 k (1 ≤ k ≤ n − 1) (see [6]). In this paper, we show that − 1 k -determinant shares similar properties which the ordinary determinant possesses. From this fact, one can define a new (relative) invariant called a wreath determinant. Using (GLm, GLn)...

In this note we give a description of the representation of the wreath product Sn[G] of the symmetric group Sn and a finite group G on the homology of the product of n copies of a partially ordered set (poset for short) P on which G acts as a group of automorphisms. In the sequel all posets will be finite. We will consider three types of product constructions. The direct product P ×Q of two pos...

Abstract. We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products Γ ≀ Sn and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf ...

Herein we generalize the holonomy theorem for finite semigroups (see [7]) to arbitrary semigroups, S, by embedding s^ into an infinite Zeiger wreath product, which is then expanded to an infinite iterative matrix semigroup. If S is not finite-J-above (where finite-J-above means every element has only a finite number of divisors), then S is replaced by g3, the triple Schtitzenberger product, whi...

Given a finite group G and a G-space X, we show that a direct sum FG(X) = ⊕ n≥0KGn(X n) ⊗ C admits a natural graded Hopf algebra and λ-ring structure, where Gn denotes the wreath product G ∼ Sn. FG(X) is shown to be isomorphic to a certain supersymmetric product in terms of KG(X) ⊗ C as a graded algebra. We further prove that FG(X) is isomorphic to the Fock space of an infinite dimensional Heis...

We define an excedance number for the multi-colored permutation group i.e. the wreath product Zr1 × · · ·×Zrk ≀Sn and calculate its multi-distribution with some natural parameters. We also compute the multi-distribution of the parameters exc(π) and fix(π) over the sets of involutions in the multi-colored permutation group. Using this, we count the number of involutions in this group having a fi...

The symmetric group S2n and the hyperoctaheadral group Hn is a Gelfand triple for an arbitrary linear representation φ of Hn. Their φ-spherical functions can be caught as transition matrix between suitable symmetric functions and the power sums. We generalize this triplet in the term of wreath product. It is shown that our triplet are always to be a Gelfand triple. Furthermore we study the rela...

A generating set for the wreath product Zr ≀ Sn which leads to a nicely behaved weak order is presented. It is shown that the resulting poset has properties analogous to those of the weak order on the symmetric group: it is a self-dual lattice, ranked by the Foata–Han flag inversion number; any two maximal chains are connected via Tits-type pseudo-Coxeter moves; and its intervals have the desir...