A group is irreducibly represented if it has a faithful irreducible unitary representation. For countable groups, a criterion for irreducible representability is given, which generalises a result obtained for finite groups by W. Gaschütz in 1954. In particular, torsionfree groups and infinite conjugacy class groups are irreducibly represented. We indicate some consequences of this for operator algebras. In particular, we charaterise up to isomorphism the countable subgroups $Δ$ of the unitary group of a separable infinite dimensional Hilbert space $\Cal H$ of which the bicommutants $Δ''$ (in the sense of the theory of von Neumann algebras) coincide with the algebra of all bounded linear operators on $\Cal H$.
Let $G$ be a group. A subset $F \subset G$ is called irreducibly faithful if there exists an irreducible unitary representation $π$ of $G$ such that $π(x) \neq \mathrm{id}$ for all $x \in F \smallsetminus \{e\}$. Otherwise $F$ is called irreducibly unfaithful. Given a positive integer $n$, we say that $G$ has Property $P(n)$ if every subset of size $n$ is irreducibly faithful. Every group has $P(1)$, by a classical result of Gelfand and Raikov. Walter proved that every group has $P(2)$. It is easy to see that some groups do not have $P(3)$. We provide a complete description of the irreducibly …
Let $W$ be a vector space over an algebraically closed field $k$. Let $H$ be a quasisimple group of Lie type of characteristic $p\ne {\rm char}(k)$ acting irreducibly on $W$. Suppose also that $G$ is a classical group with natural module $W$, chosen minimally with respect to containing the image of $H$ under the associated representation. We consider the question of when $H$ can act irreducibly on a $G$-constituent of $W^{\otimes e}$ and study its relationship to the maximal subgroup problem for finite classical groups.
Suppose that a finite solvable group $G$ acts faithfully, irreducibly and quasi-primitively on a finite vector space $V$, and $G$ is not metacyclic. Then $G$ always has a regular orbit on $V$ except for a few "small" cases. We completely classify these cases in this paper.
We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic.
Let $G$ be a connected closed subgroup of $\mathrm{GL}_n(\mathbb{C})$ which is simple as a Lie group and which acts irreducibly on $\mathbb{C}^n$. Regarding both $G$ and its Lie algebra $\mathfrak{g}$ as subsets of $M_n(\mathbb{C})$, we have $G\cap \mathfrak{g}\neq\emptyset$ if and only if $G$ is a classical group and $\mathbb{C}^n$ is a minuscule representation.
We call an element of a finite general linear group $ \textrm{GL}(d,q) $ \emph{fat} if it leaves invariant, and acts irreducibly on, a subspace of dimension greater than $d/2$. Fatness of an element can be decided efficiently in practice by testing whether its characteristic polynomial has an irreducible factor of degree greater than $d/2$. We show that for groups $G$ with $ \textrm{SL}(d,q) \leq G \leq \textrm{GL}(d,q) $ most pairs of fat elements from $G$ generate irreducible subgroups, namely we prove that the proportion of pairs of fat elements generating a reducible subgroup, in the set o…
This article began as a study of the structure of infinite permutation groups G in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-N, the minimal condition on normal subgroups. The groups G are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal non-trivial normal subgroups, or they have a regular normal subgroup M which is a divisible abelian p-group of finite rank. In the latt…
Let G be a simple classical algebraic group over an algebraically closed field K of characteristic $p \ge 0$ with natural module W. Let H be a closed subgroup of G and let V be a nontrivial p-restricted irreducible tensor indecomposable rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V) of this form, where $V \ne W, W^{*}$ and H is a disconnected almost simple positive-dimensional closed subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classification of the irreducible …
In this paper we prove the following result. Let $G$ be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field $F$ of characteristic $p\geq 0$, and let $u\in G$ be a nonidentity unipotent element. Let $ϕ$ be a non-trivial irreducible representation of $G$. Then the Jordan normal form of $ϕ(u)$ contains at most one non-trivial block if and only if $G$ is of type $G_2$, $u$ is a regular unipotent element and $\dim ϕ\leq 7$. Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent eleme…
Let $G$ be a group, $F$ a field, and $A$ a finite-dimensional central simple algebra over $F$ on which $G$ acts by $F$-algebra automorphisms. We study the ideals and subalgebras of $A$ which are preserved by the group action. Let $V$ be the unique simple module of $A$. We show that $V$ is a projective representation of $G$ and $A\cong\text{End}_D(V)$ makes $V$ into a projective representation. We then prove that there is a natural one-to-one correspondence between $G$-invariant $D$-submodules of $V$ and invariant left (and right) ideals of $A$. Under the assumption that $V$ is irreducible, we …
Every finitely generated self-similar group naturally produces an infinite sequence of finite $d$-regular graphs $Γ_n$. We construct self-similar groups, whose graphs $Γ_n$ can be represented as an iterated zig-zag product and graph powering: $Γ_{n+1}=Γ_n^k\mathop{\mbox{\textcircled{$z$}}}Γ$ ($k\geq 1$). Also we construct self-similar groups, whose graphs $Γ_n$ can be represented as an iterated replacement product and graph powering: $Γ_{n+1}=Γ_n^k\mathop{\mbox{\textcircled{$r$}}}Γ$ ($k\geq 1$). This gives simple explicit examples of self-similar groups, whose graphs $Γ_n$ form an expanding fa…
We present new efficient data structures for elements of Coxeter groups of type $A_m$ and their associated Iwahori--Hecke algebras $H(A_m)$. Usually, elements of $H(A_m)$ are represented as simple coefficient list of length $M = (m+1)!$ with respect to the standard basis, indexed by the elements of the Coxeter group. In the new data structure, elements of $H(A_m)$ are represented as nested coefficient lists. While the cost of addition is the same in both data structures, the new data structure leads to a huge improvement in the cost of multiplication in~$H(A_m)$.
A group is combable if it can be represented by a language of words satisfying a fellow traveller property; an automatic group has a synchronous combing which is a regular language. This article surveys results for combable groups, in particular in the case where the combing is a formal language.
In a recent paper by L. A. Bokut, V. V. Chaynikov and K. P. Shum in 2007, Braid group $B_n$ is represented by Artin-Burau's relations. For such a representation, it is told that all other compositions can be checked in the same way. In this note, we support this claim and check all compositions.
It is known that every finite group can be represented as the full group of automorphisms of a suitable compact dessin d'enfant. In this paper, we give a constructive and easy proof that the same holds for any countable group by considering non-compact dessins. Moreover, we show that any tame action of a countable group is so realizable.
We show that each integral Borel cohomology class of a connected Lie group G can be represented by a Borel bounded cocycle if and only if the radical of G is linear. This leads to a generalization of Gromov's boundedness theorem on characteristic classes of flat bundles.
It is known that every finite group can be represented as the full group of automorphisms of a suitable compact origami. In this paper, we provide a short argument to note that the same holds for any countable group by considering origamis on the Loch Ness monster.
