REPRESENTATIONS OF Aut(A(Γ)) ACTING ON HOMOGENEOUS COMPONENTS OF A(Γ) AND A(Γ)

ثبت نشده
چکیده

In this paper we will study the structure of algebras A(Γ) associated to two directed, layered graphs Γ. These are algebras associated with Hasse graphs of n-gons and the algebras Qn related to pseudoroots of noncommutative polynomials. We will find the filtration preserving automorphism group of these algebras and then we will find the multiplicities of the irreducible representations of Aut(A(Γ)) acting on the homogeneous components of A(Γ) and A(Γ). Introduction In this paper we will be considering directed graphs Γ satisfying certain hypotheses. There exists an algebra A(Γ), graded algebra grA(Γ), and dual algebra A(Γ) over a field k associated with each of these graphs. In Section 0 we will give some preliminaries on how these algebras are built from the graphs as well as a basis for A(Γ). The definition of subalgebras that will play an integral role in what follows will be given in Section 1. In Section 2 we will introduce the two algebras A(ΓDn) and Qn that we will be describing throughout this paper. The automorphism group of the graph injects into the automorphism group AutA(Γ) of A(Γ). Furthermore, the nonzero scalars inject into the automorphism group of the algebra. Thus, one is naturally led to ask how these automorphism groups are related. This question will be answered in Section 3. A second question that we are led to is to describe the homogeneous components of A(Γ) and A(Γ). We will decompose the algebra and its dual into irreducible Aut(A(Γ))-modules by calculating the graded trace generating functions. These graded trace generating functions are actually the graded dimensions of certain subalgebras of grA(Γ). Hence, the technique for calculating the graded trace generating functions is to abstract the problem into finding the graded dimension of subalgebras. In Section 4 we will explain why this is true and how these graded dimensions are found. The graded traces of our two particular algebras will be given in Section 5 and their decompositions in Section 6. The decomposition can be found using the graded trace and the characters of the automorphism group of the algebra. The dual algebras of these two examples will be considered in Section 7. 0. Preliminaries A(Γ) Certain algebras, denoted A(Γ), associated to directed graphs were first defined by Gelfand, Retakh, Serconek, and Wilson [GRSW]. We recall the definitions of A(Γ) and grA(Γ) following the development found in [[RSW], §2]. Let k be a field and for any set W let T (W ) be the free associative algebra on W over k. Let Γ = (V,E) be a directed graph where V is the set of vertices, E the set of edges, and there are functions t, h : E → V (tail and head of e). We say Γ is a layered graph if V = n ⋃

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Groups Acting on Products of Trees, Tiling Systems and Analytic K-Theory

Let T1 and T2 be homogeneous trees of even degree ≥ 4. A BM group Γ is a torsion-free discrete subgroup of Aut(T1) × Aut(T2) which acts freely and transitively on the vertex set of T1 × T2. This article studies dynamical systems associated with BM groups. A higher rank Cuntz-Krieger algebra A(Γ) is associated both with a 2-dimensional tiling system and with a boundary action of a BM group Γ. An...

متن کامل

Generic Representations of Finitely Generated Groups on Fraisse Structures

For finitely generated groups Γ and ultrahomogeneous countable relational structures M we study the space Rep(Γ,M) of all representations of Γ by automorphisms on M equipped with the topology it inherits seen as a closed subset of Aut(M)Γ. When Γ is the free group Fn on n generators this space is just Aut(M)n, but is in general significantly more complicated. We prove that when Γ is finitely ge...

متن کامل

A Product Formula for Spherical Representations of a Group of Automorphisms of a Homogeneous Tree, I

Let G = Aut(T ) be the group of automorphisms of a homogeneous tree T , and let Γ be a lattice subgroup of G. Let π be the tensor product of two spherical irreducible unitary representations of G. We give an explicit decomposition of the restriction of π to Γ. We also describe the spherical component of π explicitly, and this decomposition is interpreted as a multiplication formula for associat...

متن کامل

Weak containment in the space of actions of a free group

(A) We consider measure preserving actions of an infinite, countable (discrete) group Γ on non-atomic standard measure spaces (X,μ), i.e., standard Borel spaces equipped with a non-atomic probability Borel measure. (All such measure spaces are isomorphic to ([0, 1], λ), where λ is Lebesgue measure.) We denote by A(Γ, X, μ) the space of such actions. If a ∈ A(Γ, X, μ) and γ ∈ Γ, we denote by γ(x...

متن کامل

Homogeneous factorisations of Johnson graphs

For a graph Γ, subgroups M < G 6 Aut(Γ), and an edge partition E of Γ, the pair (Γ, E) is a (G, M)-homogeneous factorisation if M is vertex-transitive on Γ and fixes setwise each part of E , while G permutes the parts of E transitively. A classification is given of all homogeneous factorisations of finite Johnson graphs. There are three infinite families and nine sporadic examples.

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2007