Using holonomy decomposition, the absence of certain types of cycles in automata has been characterized. In the direction of studying the structure of automata with cycles, this paper focuses on a special class of semi-flower automata and establish the holonomy decomposition of certain circular semiflower automata. In particular, we show that the transformation monoid of a circular semi-flower ...

Suppose we are given a homogeneous tree Tq of degree q ≥ 3, where at each vertex sits a lamp, which can be switched on or off. This structure can be described by the wreath product (Z/2) ≀ Γ, where Γ = ∗ i=1 Z/2 is the free product group of q factors Z/2. We consider a transient random walk on a Cayley graph of (Z/2) ≀ Γ, for which we want to compute lower and upper bounds for the rate of escap...

Generalising Segal’s approach to 1-fold loop spaces, the homotopy theory of n-fold loop spaces is shown to be equivalent to the homotopy theory of reduced Θn-spaces, where Θn is an iterated wreath-product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternative description of the Segal-spectrum associated to a Γ-space. In particular, each Eilenberg-MacLane space ...

We define an excedance number for the multi-colored permutation group i.e. the wreath product (Zr1 × · · · ×Zrk) o 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 ...

We determine all positive harmonic functions for a large class of “semiisotropic” random walks on the lamplighter group, i.e., the wreath product Zq ≀Z, where q ≥ 2. This is possible via the geometric realization of a Cayley graph of that group as the Diestel-Leader graph DL(q, q). More generally, DL(q, r) (q, r ≥ 2) is the horocyclic product of two homogeneous trees with respective degrees q+1...

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...

Let CAn=C[S2?S2???S2] be the group algebra of an n-step iterated wreath product. We prove some structural properties An such as their centers, centralizers, and right double cosets. apply these results to explicitly write down Mackey theorem for groups give a partial description natural transformations between induction restriction functors on representations product tower by computing certain ...

In recent work, Bacher and de la Harpe define and study conjugacy growth series for finitary permutation groups. In two subsequent papers, Cotron, Dicks, and Fleming study the congruence properties of some of these series. We define a new family of conjugacy growth series for the finitary alternating wreath product that are related to sums of modular forms of integer and half-integral weights, ...

Harmonic analysis is at the heart of much of signal and image processing. This is mainly the harmonic analysis of abelian groups, and ultimately, after sampling, and quantizing, finite abelian groups. Nevertheless, this general group theoretic viewpoint has proved fruitful, yielding a natural group-based multiresolution framework as well as some new and potentially useful nonabelian examples. T...

We provide elementary proofs of the Nielsen-Schreier Theorem and the Kurosh Subgroup Theorem via wreath products. Our proofs are diagrammatic in nature and work simultaneously in the abstract and profinite categories. A new proof that open subgroups of quasifree profinite groups are quasifree is also given.