Space Of Homomorphisms Research Articles

Modular Representation Theory and Commutative Banach Algebras

In a recent paper of Benson and Symonds, a new invariant was introduced for modular representations of a finite group. An interpretation was given as a spectral radius with respect to a Banach algebra completion of the representation ring. Our purpose here is to take these notions further, and investigate the structure of the resulting Banach algebras. Some of the material in that paper is repeated here in greater generality, and for clarity of exposition. We give an axiomatic definition of an abstract representation ring, and representation ideal. The completion is then a commutative Banach algebra, and the techniques of Gelfand from the 1940s are applied in order to study the space of algebra homomorphisms to C \mathbb {C} . One surprising consequence of this investigation is that the Jacobson radical and the nil radical of a (complexified) representation ring always coincide. These notes are intended for representation theorists. So background material on commutative Banach algebras is given in detail, whereas representation theoretic background is more condensed.

Algebra environments II. Algebra homomorphisms and derivations

Algebra environments provide requisites for studying objects of interest in operator algebra theory, group representation theory, spin geometry, Clifford analysis, and several variable operator theory. The concept is analyzed by developing an algebraic geometry approach. Specific algebraic sets, called structure manifolds of algebra environments, and their Zariski tangent spaces are introduced and described by using as critical tools derivations on algebras. Structure manifolds of tensor environments in particular yield spaces of algebra homomorphisms. Consequently, such spaces could be investigated as algebraic manifolds. Related issues include characterizations of their Zariski tangent spaces and of derivations that preserve algebra homomorphisms.

The space of commuting elements in a Lie group and maps between classifying spaces

Let $\pi$ be a discrete group, and let $G$ be a compact-connected Lie group. Then, there is a map $\Theta \colon \mathrm {Hom}(\pi,G)_0\to \mathrm {map}_*(B\pi,BG)_0$ between the null components of the spaces of homomorphisms and based maps, which sends a homomorphism to the induced map between classifying spaces. Atiyah and Bott studied this map for $\pi$ a surface group, and showed that it is surjective in rational cohomology. In this paper, we prove that the map $\Theta$ is surjective in rational cohomology for $\pi =\mathbb {Z}^m$ and the classical group $G$ except for $SO(2n)$ , and that it is not surjective for $\pi =\mathbb {Z}^m$ with $m\ge 3$ and $G=SO(2n)$ with $n\ge 4$ . As an application, we consider the surjectivity of the map $\Theta$ in rational cohomology for $\pi$ a finitely generated nilpotent group. We also consider the dimension of the cokernel of the map $\Theta$ in rational homotopy groups for $\pi =\mathbb {Z}^m$ and the classical groups $G$ except for $SO(2n)$ .

On homomorphisms into Weyl modules corresponding to partitions with two parts

AbstractLet K be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ be partitions, where $\mu$ has two parts. We find sufficient arithmetic conditions on $p, \lambda, \mu$ for the existence of a nonzero homomorphism $\Delta(\lambda) \to \Delta (\mu)$ of Weyl modules for the general linear group $GL_n(K)$ . Also, for each p we find sufficient conditions so that the corresponding homomorphism spaces have dimension at least 2.

Isolated points of spaces of homomorphisms from ordered AL-algebras

We study isolated points of spaces comprised of homomorphisms between a given ordered AL-algebra and a given unital normed algebra. An ordered AL-algebra is a complex AL-space and simultaneously a Banach algebra such that the positive cone associated with

Hyperreflexivity of the space of module homomorphisms between non-commutative Lp-spaces

Let M be a von Neumann algebra, and let 0<p,q≤∞. Then the space HomM(Lp(M),Lq(M)) of all right M-module homomorphisms from Lp(M) to Lq(M) is a reflexive subspace of the space of all continuous linear maps from Lp(M) to Lq(M). Further, the space HomM(Lp(M),Lq(M)) is hyperreflexive in each of the following cases: (i) 1≤q<p≤∞; (ii) 1≤p,q≤∞ and M is injective, in which case the hyperreflexivity constant is at most 8.

Fine structure of the homomorphisms of the lattice of uniformly continuous functions on the line

We provide a representation of the homomorphisms \(U\longrightarrow {\mathbb {R}}\), where U is the lattice of all uniformly continuous functions on the line The resulting picture is sharp enough to describe the fine topological structure of the space of such homomorphisms.

Rational Cherednik Algebras and Schubert Cells

The representation theory of rational Cherednik algebras of type A at t = 0 gives rise, by considering supports, to a natural family of smooth Lagrangian subvarieties of the Calogero-Moser space. The goal of this article is to make precise the relationship between these Lagrangian families and Schubert cells in the adelic Grassmannian. In order to do this we show that the isomorphism, as constructed by Etingof and Ginzburg, from the spectrum of the centre of the rational Cherednik algebra to the Calogero-Moser space is compatible with the factorization property of both of these spaces. As a consequence, the space of homomorphisms between certain representations of the rational Cherednik algebra can be identified with functions on the intersection of some Schubert cells.

Invariants of the special orthogonal group and an enhanced Brauer category

We first give a short intrinsic, diagrammatic proof of the First Fundamental Theorem of invariant theory (FFT) for the special orthogonal group SO _m(\mathbb C) , given the FFT for O _m(\mathbb C) . We then define, by means of a presentation with generators and relations, an enhanced Brauer category \widetilde{\mathcal B}(m) by adding a single generator to the usual Brauer category \mathcal B(m) , together with four relations. We prove that our category \widetilde{\mathcal B}(m) is actually (and remarkably) equivalent to the category of representations of SO _m generated by the natural representation. The FFT for SO _m amounts to the surjectivity of a certain functor \mathcal F on Hom spaces, while the Second Fundamental Theorem for SO _m says simply that \mathcal F is injective on Hom spaces. This theorem provides a diagrammatic means of computing the dimensions of spaces of homomorphisms between tensor modules for SO _m (for any m ).

Warmth and edge spaces of graphs

In this paper we study a pair of numerical parameters associated to a graph G. On the one hand, we consider the (topological) connectivity of the polyhedral complex Hom(K2,G), a space of homomorphisms from a edge K2 into G. This approach dates back to the neighborhood complexes introduced by Lovász in his proof of the Kneser conjecture. In another direction we study the warmth ζ(G) of a graph G, a parameter introduced by Brightwell and Winkler based on the asymptotic behavior of d-branching walks in G and inspired by constructions in statistical physics. Both the warmth of G and the connectivity of Hom(K2,G) provide lower bounds on the chromatic number of G.Here we seek to relate these two constructions, and in particular we provide evidence for the conjecture that the warmth of a graph G is always less than three plus the connectivity of Hom(K2,G). We succeed in establishing a first nontrivial case of the conjecture, by showing that ζ(G)≤3 if Hom(K2,G) has an infinite first homology group. We also calculate warmth for a family of ‘twisted toroidal’ graphs that are important extremal examples in the context of Hom complexes. Finally we show that ζ(G)≤n−1 if a graph G does not have the complete bipartite graph Ka,b for a+b=n. This provides an analogue for a similar result in the context of Hom complexes.

Classifying spaces for 1–truncated compact Lie groups

A 1-truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of $Map_*(BG,BH)$, $Map(BG,BH)$, and $Map(EG, B_GH)^G$ for compact Lie groups $G$ and $H$ with $H$ 1-truncated, showing that they are computed entirely in terms of spaces of homomorphisms from $G$ to $H$. These results generalize the well-known case when $H$ is finite, and the case of $H$ compact abelian due to Lashof, May, and Segal.

Mixing properties for hom-shifts and the distance between walks on associated graphs

Let $\mathcal H$ be a finite connected undirected graph and $\mathcal H_{walk}$ be the graph of bi-infinite walks on $\mathcal H$; two such walks $\{x_i\}_{i\in \mathbb Z}$ and $\{y_i\}_{i \in \mathbb Z}$ are said to be adjacent if $x_i$ is adjacent to $y_i$ for all $i \in \mathbb Z$. We consider the question: Given a graph $\mathcal H$ when is the diameter (with respect to the graph metric) of $\mathcal H_{walk}$ finite? Such questions arise while studying mixing properties of hom-shifts (shift spaces which arise as the space of graph homomorphisms from the Cayley graph of $\mathbb Z^d$ with respect to the standard generators to $\mathcal H$) and are the subject of this paper.

Affine representations of $${\ell}$$ ℓ -groups and MV-algebras

Affine representations for archimedean \({\ell}\)-groups and semisimple MV-algebras via embedding theorems are presented; they are simple to work with but powerful enough to express significant properties of our studied objects. Indeed, we focus on the space of particular homomorphisms between an archimedean \({\ell}\)-group (a semisimple MV-algebra, respectively) and a vector lattice (a Riesz MV-algebra, respectively), i.e., the set of the generalized states, providing a general framework.

Length two extensions of modules for the Witt algebra

ABSTRACTIn this paper, we establish an explicit classification of length two extensions of tensor modules for the Witt algebra using the cohomology of the Witt algebra with coefficients in the module of the space of homomorphisms between the two modules of interest. To do this we extended our module to a module that has a compatible action of the commutative algebra of Laurent polynomials in one variable. In this setting, we are be able to directly compute all possible 1-cocycles.

Cosimplicial groups and spaces of homomorphisms

Let $G$ be a real linear algebraic group and $L$ a finitely generated cosimplicial group. We prove that the space of homomorphisms $Hom(L_n,G)$ has a homotopy stable decomposition for each $n\geq 1$. When $G$ is a compact Lie group, we show that the decomposition is $G$-equivariant with respect to the induced action of conjugation by elements of $G$. The spaces $Hom(L_n,G)$ assemble into a simplicial space $Hom(L,G)$. When $G=U$ we show that its geometric realization $B(L,U)$, has a non-unital $E_\infty$-ring space structure whenever $Hom(L_0,U(m))$ is path connected for all $m\geq1$.

Decomposition of Tensor Products Involving a Steinberg Module

We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group G and when restricted to either a Frobenius kernel G r or a finite Chevalley group $G(\mathbb {F}_q)$ . In all three cases, we give formulas reducing this to standard character data for G. Along the way, we use a bilinear form on the characters of finite dimensional G-modules to give formulas for the dimension of homomorphism spaces between certain G-modules when restricted to either G r or $G(\mathbb {F}_q)$ . Further, this form allows us to give a new proof of the reciprocity between tilting modules and simple modules for G which has slightly weaker assumptions than earlier such proofs. Finally, we prove that in a suitable formulation, this reciprocity is equivalent to Donkin’s tilting conjecture.

Strong Spatial Mixing in Homomorphism Spaces

Given a countable graph $\mathcal{G}$ and a finite graph $H$, we consider $Hom(\mathcal{G}, H)$ the set of graph homomorphisms from $\mathcal{G}$ to $H$ and we study Gibbs measures supported on $Hom(\mathcal{G}, H)$. We develop some sufficient and other necessary conditions for the existence of Gibbs specifications on $Hom(\mathcal{G}, H)$ satisfying strong spatial mixing (with exponential decay rate). We relate this with previous work of Brightwell and Winkler, who showed that a graph $H$ has a combinatorial property called dismantlability iff for every $\mathcal{G}$ of bounded degree, there exists a Gibbs specification with unique Gibbs measure. We strengthen their result by showing that such Gibbs specification can be chosen to have weak spatial mixing. In addition, we exhibit a subfamily of graphs $H$ for which there exists Gibbs specifications satisfying strong spatial mixing, but we also show that there exist dismantlable graphs for which no Gibbs specification has strong spatial mixing.

Cross products, invariants, and centralizers

An algebra V with a cross product × has dimension 3 or 7. In this work, we use 3-tangles to describe, and provide a basis for, the space of homomorphisms from V⊗n to V⊗m that are invariant under the action of the automorphism group Aut(V,×) of V, which is a special orthogonal group when dimV=3, and a simple algebraic group of type G2 when dimV=7. When m=n, this gives a graphical description of the centralizer algebra EndAut(V,×)(V⊗n), and therefore, also a graphical realization of the Aut(V,×)-invariants in V⊗2n equivalent to the First Fundamental Theorem of Invariant Theory. We show how the 3-dimensional simple Kaplansky Jordan superalgebra can be interpreted as a cross product (super)algebra and use 3-tangles to obtain a graphical description of the centralizers and invariants of the Kaplansky superalgebra relative to the action of the special orthosymplectic group.

Representation spaces for central extensions and almost commuting unitary matrices

Let $\Gamma$ denote a central extension of the form $1\to \mathbb{Z}^r\to\Gamma\to \mathbb{Z}^n\to 1$. In this paper we describe the topology of the spaces of homomorphisms $\text{Hom}(\Gamma, U(m))$ and the associated moduli spaces $\text{Rep}(\Gamma, U(m))$, where $U(m)$ is the group of $m\times m$ unitary matrices.

On spaces of commuting elements in Lie groups

AbstractThe main purpose of this paper is to introduce a method to “stabilise” certain spaces of homomorphisms from finitely generated free abelian groups to a Lie groupG, namely Hom(ℤn,G). We show that this stabilised space of homomorphisms decomposes after suspending once with “summands” which can be reassembled, in a sense to be made precise below, into the individual spaces Hom(ℤn,G) after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilised space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilised space also allows the description of the additive reduced homology of the individual spaces Hom(ℤn,G), with the order of the Weyl group inverted.