The purpose of the present paper is to discuss the following conjecture of Fel'shtyn and Hill, which is a generalization of the classical Burnside theorem:
Let G be a countable discrete group, f its automorphism, R(f) the number of f-conjugacy classes (Reidemeister number), S(f):=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other.
This conjecture plays a very important role in the theory of twisted conjugacy classes having a long history and has very serious consequences in Dynamics, while its proof needs rather fine results from Functional and Non-commutative Harmonic Analysis. It was proved for finitely generated groups of type I in a previous paper.
In the present paper this conjecture is disproved for non-type I groups. More precisely, an example of a group and its automorphism is constructed such that the number of fixed irreducible representations is greater than the Reidemeister number. But the number of fixed finite-dimensional representations (i.e. the number of invariant finite-dimensional characters) in this example coincides with the Reidemeister number.
The present paper consists of two parts. In the first part, we prove a noncommutative analogue of the Riesz(-Markov-Kakutani) theorem on representation of functionals on an algebra of continuous functions by regular measures on the underlying space. In the second part, using this result, we prove a weak version of Burnside type theorem for twisted conjugacy for arbitrary discrete groups.
This is a (mostly expository) paper on Reidemeister classes, twisted Burnside-Frobenius theory, congruences, R-infinity property and all that. It was written in 2005 and published in 2008. We post it as it was, only the bibliography data is updated. For some of the recent progress see e.g. arXiv:0903.4533, arXiv:0903.3455, arXiv:0802.2937, arXiv:0712.2601, arXiv:0704.3411, arXiv:math/0703744, arXiv:math/0606725, arXiv:math/0606764, arXiv:0805.1371 and references there.
We prove that the restricted wreath product ${\mathbb{Z}_n \mathbin{\mathrm{wr}} \mathbb{Z}^k}$ has the $R_\infty$-property, i. e. every its automorphism $φ$ has infinite Reidemeister number $R(φ)$, in exactly two cases: (1) for any $k$ and even $n$; (2) for odd $k$ and $n$ divisible by 3. In the remaining cases there are automorphisms with finite Reidemeister number, for which we prove the finite-dimensional twisted Burnside--Frobenius theorem (TBFT): $R(φ)$ is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action ${[ρ]\mapsto…
For a wide class of groups including polycyclic and finitely generated polynomial growth groups it is proved that the Reidemeister number of an automorphism f is equal to the number of finite-dimensional fixed points of the induced map f^ on the unitary dual, if one of these numbers is finite. This theorem is a natural generalization of the classical Burnside-Frobenius theorem to infinite groups. This theorem also has important consequences in topological dynamics and in some sense is a reply to a remark of J.-P. Serre. The main technical results proved in the paper yield a tool for a further …
Let $R(ϕ)$ be the number of $ϕ$-conjugacy (or Reidemeister) classes of an endomorphism $ϕ$ of a group $G$. We prove for several classes of groups (including polycyclic) that the number $R(ϕ)$ is equal to the number of fixed points of the induced map of an appropriate subspace of the unitary dual space $\widehat G$, when $R(ϕ)<\infty$. Applying the result to iterations of $ϕ$ we obtain Gauss congruences for Reidemeister numbers. In contrast with the case of automorphisms, studied previously, we have a plenty of examples having the above finiteness condition, even among groups with $R_\infty$ pr…
The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A.Fel'shtyn and R.Hill, which is a generalization of the classical Burnside theorem. Let G be a countable discrete group, f one of its automorphisms, R(f) the number of f-conjugacy classes, and S(f)=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other. This conjecture plays an important role in the theory of twisted conjugacy classes and has very important consequences in Dyn…
We introduce new zeta functions related to an endomorphism $ϕ$ of a discrete group $Γ$. They are of two types: counting numbers of fixed ($ρ\sim ρ\circϕ^n$) irreducible representations for iterations of $ϕ$ from an appropriate dual space of $Γ$ and counting Reidemeister numbers $R(ϕ^n)$ of different compactifications. Many properties of these functions and their coefficients are obtained. In many cases it is proved that these zeta functions coincide. The Gauss congruences are proved. Useful asymptotic formulas for the zeta functions are found. Rationality is proved for some examples, which giv…
It is proved for Abelian groups that the Reidemeister coincidence number of two endomorphisms $ϕ$ and $ψ$ is equal to the number of coincidence points of $\whϕ$ and $\whψ$ on the unitary dual, if the Reidemeister number is finite. An affirmative answer to the bitwisted Dehn conjugacy problem for almost polycyclic groups is obtained. Finally we explain why the Reidemeister numbers are always infinite for injective endomorphisms of Baumslag-Solitar groups.
We describe the generalized Matsuda's theorem, and some results of a Burnside ring extend a partial Burnside ring. In particular, we give isomorphism between partial Burnside rings of different groups. Moreover, we consider the relationship between image of Frobenius-Wielandt homomorphism, a partial Burnside ring, and structure of a group.
Let A be an associative algebra over a field, and let M be a finite family of right A-modules. Study of the noncommutative deformation functor of the family M leads to the construction of the algebra of observables and the Generalized Burnside Theorem, due to Laudal. In this paper, we give an overview of aspects of noncommutative deformations closely connected to the Generalized Burnside Theorem.
A well-known theorem of Burnside says that if $ρ$ is a faithful representation of a finite group $G$ over a field of characteristic $0$, then every irreducible representation of $G$ appears as a constituent of a tensor power of $ρ$. In 1962, R. Steinberg gave a module theoretic proof that simultaneously removed the constraint on the characteristic, and allowed the group to be replaced by a monoid. Brauer subsequently simplified Burnside's proof and, moreover, showed that if the character of $ρ$ takes on $r$ distinct values, then the first $r$ tensor powers of $ρ$ already contain amongst them a…
In this paper, we first show that the irreducible characters of a quotient table algebra modulo a normal closed subset can be viewed as the irreducible characters of the table algebra itself. Furthermore, we define the character products for table algebras and give a condition in which the products of two characters are characters. Thereafter, as a main result we state and prove the Burnside-Brauer Theorem on finite groups for table algebras.
Balmer and Dell'Ambrogio introduced the pseudo-functor $P$ from the bicategory of $k$-linear Mackey $2$-motives to the bicategory of $k$-linear cohomological Mackey $2$-motives over a commutative ring $k$. They showed that $P$ maps the general Mackey $2$-motives to the cohomological Mackey $2$-motives by using the ring homomorphism from the crossed Burnside ring of a finite group $G$ over $k$ to the center $ ZkG$ of group algebra $kG$ ([BD21, Theorem 5.3]). We study the behavior of motivic decomposition of cohomological Mackey $2$-motives as images by $P$ of motivic decomposition of Mackey $2$…
We study 2-representations, i.e., actions of 2-groups on 2-vector spaces. Our main focus is character theory for 2-representations. To this end we employ the technique of extended Burnside rings. Our main theorem is that the Ganter-Kapranov 2-character is a particular mark homomorphism of the Burnside ring. As an application we give a new proof of Osorno formula for the Ganter-Kapranov 2-character of a finite group.
For a finite group $G$, we introduce a multiplication on the $\QQ$-vector space with basis $\scrS_{G\times G}$, the set of subgroups of $G\times G$. The resulting $\QQ$-algebra $\Atilde$ can be considered as a ghost algebra for the double Burnside ring $B(G,G)$ in the sense that the mark homomorphism from $B(G,G)$ to $\Atilde$ is a ring homomorphism. Our approach interprets $\QQ B(G,G)$ as an algebra $eAe$, where $A$ is a twisted monoid algebra and $e$ is an idempotent in $A$. The monoid underlying the algebra $A$ is again equal to $\scrS_{G\times G}$ with multiplication given by composition o…
The purpose of this expository paper is to present new directions in the classical Nielsen-Reidemeister fixed point theory. We describe twisted Burnside-Frobenius theorem, groups with $R_\infty$ \emph{property} and a connection between Nielsen fixed point theory and symplectic Floer homology.
Suppose, $G$ is a residually finite group of finite upper rank admitting an automorphism $φ$ with finite Reidemeister number $R(φ)$ (the number of $φ$-twisted conjugacy classes). We prove that such $G$ is soluble-by-finite (in other words, any residually finite group of finite upper rank, which is not soluble-by-finite, has the $R_\infty$ property). This reduction is the first step in the proof of the second main theorem of the paper: suppose, $G$ is a residually finite group of finite Prüfer rank and $φ$ is its automorphism with $R(φ)<\infty$; then $R(φ)$ is equal to the number of equivalence…
Let $B^\times$ be the biset functor over $\mathbb{F}_2$ sending a finite group~$G$ to the group $B^\times(G)$ of units of its Burnside ring $B(G)$, and let $\widehat{B^\times}$ be its dual functor. The main theorem of this paper gives a characterization of the cokernel of the natural injection from $B^\times$ in the dual Burnside functor $\widehat{\mathbb{F}_2B}$, or equivalently, an explicit set of generators $\mathcal{G}_S$ of the kernel $L$ of the natural surjection $\mathbb{F}_2B\to \widehat{B^\times}$. This yields a two terms projective resolution of $\widehat{B^\times}$, leading to some …
We consider converses to the density theorem for irreducible, projective, unitary group representations restricted to lattices using the dimension theory of Hilbert modules over twisted group von Neumann algebras. We show that under the right assumptions, the restriction of a $σ$-projective unitary representation $π$ of a group $G$ to a lattice $Γ$ extends to a Hilbert module over the twisted group von Neumann algebra $\text{L}(Γ,σ)$. We then compute the center-valued von Neumann dimension of this Hilbert module. For abelian groups with 2-cocycle satisfying Kleppner's condition, we show that t…
The analog of the Chern-Gauss-Bonnet theorem is studied for a $C^*$-dynamical system consisting of a $C^*$-algebra $A$ equipped with an ergodic action of a compact Lie group $G$. The structure of the Lie algebra $\mathfrak{g}$ of $G$ is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra $\mathcal{A} \subset A$ as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique $G$-invariant state on $A$, by means of a Weyl conformal factor given by a positive invertible element o…
We prove that for any automorphism $ϕ$ of the restricted wreath product $\mathbb{Z}_2 \mathrm{wr} \mathbb{Z}^k$ and $\mathbb{Z}_3 \mathrm{wr} \mathbb{Z}^{2d}$ the Reidemeister number $R(ϕ)$ is infinite, i.e. these groups have the property $R_\infty$. For $\mathbb{Z}_3 \mathrm{wr} \mathbb{Z}^{2d+1}$ and $\mathbb{Z}_p \mathrm{wr} \mathbb{Z}^k$, where $p>3$ is prime, we give examples of automorphisms with finite Reidemeister numbers. So these groups do not have the property $R_\infty$. For these groups and $\mathbb{Z}_m \mathrm{wr} \mathbb{Z}$, where $m$ is relatively prime to $6$, we prove the t…
In the present paper we extend the Riemann-Roch formalism to structure algebras of moment graphs. We introduce and study the Chern character and pushforwards for twisted fibrations of moment graphs. We prove an analogue of the Riemann-Roch theorem for moment graphs. As an application, we obtain the Riemann-Roch type theorem for equivariant $K$-theory of some Kac-Moody flag varieties.
We analyze families of Markov chains that arise from decomposing tensor products of irreducible representations. This illuminates the Burnside-Brauer Theorem for building irreducible representations, the McKay Correspondence, and Pitman's 2M-X Theorem. The chains are explicitly diagonalizable, and we use the eigenvalues/eigenvectors to give sharp rates of convergence for the associated random walks. For modular representations, the chains are not reversible, and the analytical details are surprisingly intricate. In the quantum group case, the chains fail to be diagonalizable, but a novel analy…
Let us suppose that $\mathbb{Q}_p$ is the field of $p$-adic numbers and $\mathbb{G}$ is a split connected reductive group scheme over $\mathbb{Z}_p$. In this work we will introduce a sheaf of twisted arithmetic differential operators on the formal flag variety of $\mathbb{G}$, associated to a general character. We will generalize the arguments of C. Huyghe and T. Schmidt, concerning the $\mathcal{D}^†$-affinity of the formal flag variety of $\mathbb{G}$, of certain sheaves of $p$-adically complete twisted arithmetic differential operators associated to an algebraic character, and the calculati…
Some general connections between martingales and character ratios of finite groups are developed. As an application we sharpen the convergence rate in a central limit theorem for the character ratio of a random representation of the symmetric group on transpositions. A generalization of these results is given for Jack measure on partitions. We also give a probabilistic proof of a result of Burnside and Brauer on the decomposition of tensor products.
The twisted q-Yangians are coideal subalgebras of the quantum affine algebra associated with gl(N). We prove a classification theorem for finite-dimensional irreducible representations of the twisted q-Yangians associated with the symplectic Lie algebras sp(2n). The representations are parameterized by their highest weights or by their Drinfeld polynomials. In the simplest case of sp(2) we give an explicit description of all the representations as tensor products of evaluation modules. We prove analogues of the Poincare-Birkhoff-Witt theorem for the quantum affine algebra and for the twisted q…
We determine a basis of the (twisted) cocenter of graded affine Hecke algebras with arbitrary parameters. In this setting, we prove that the kernel of the (twisted) trace map is the commutator subspace (Density theorem) and that the image is the space of good forms (trace Paley-Wiener theorem).
The strong Macdonald theorems state that, for $L$ reductive and $s$ an odd variable, the cohomology algebras $H^*(L[z]/z^N)$ and $H^*(L[z,s])$ are freely generated, and describe the cohomological, $s$-, and $z$-degrees of the generators. The resulting identity for the $z$-weighted Euler characteristic is equivalent to Macdonald's constant term identity for a finite root system. We calculate $H^*(\mathfrak{p} / z^N \mathfrak{p})$ and $H^*(\mathfrak{p}[s])$ for $\mathfrak{p}$ a standard parahoric in a twisted loop algebra, giving strong Macdonald theorems that take into account both a parabolic …
In joint work with M. Hopkins and C. Teleman we find a new description of the Verlinde algebra associated to a compact Lie group. In this expository account we describe twisted K-theory, prove the theorem for the group SU(2), and motivate the general theorem using ideas in topological field theory. The full proof will appear elsewhere.
Analogs of the classical Sylvester theorem have been known for matrices with entries in noncommutative algebras including the quantized algebra of functions on GL(N) and the Yangian for gl(N). We prove a version of this theorem for the twisted Yangians Y(g(N)) associated with the orthogonal and symplectic Lie algebras g(N)=o(N) or sp(N). This gives rise to representations of the twisted Yangian Y(g(N-M)) on the space of g(M)-homomorphisms Hom(W,V), where W and V are finite-dimensional irreducible modules over g(M) and g(N), respectively. In the symplectic case these representations turn out to…
We classify the primitive idempotents of the $p$-local complex representation ring of a finite group $G$ in terms of the cyclic subgroups of order prime to $p$ and show that they all come from idempotents of the Burnside ring. Our results hold without adjoining roots of unity or inverting the order of $G$, thus extending classical structure theorems. We then derive explicit group-theoretic obstructions for tensor induction to be compatible with the resulting idempotent splitting of the representation ring Mackey functor. Our main motivation is an application in homotopy theory: we conclude tha…
In this paper, we prove certain multiplicity one theorems and define twisted gamma factors for irreducible generic cuspidal representations of split $G_2$ over finite fields $k$ of odd characteristic. Then we prove the first converse theorem for exceptional groups, namely, $GL_1$ and $GL_2$-twisted gamma factors will uniquely determine an irreducible generic cuspidal representation of $G_2(k)$.
Twisted K-theory has received much attention recently in both mathematics and physics. We describe some models of twisted K-theory, both topological and geometric. Then we state a theorem which relates representations of loop groups to twisted equivariant K-theory. This is joint work with Michael Hopkins and Constantin Teleman.
