Affine hyperplane arrangements and Jordan classes

The aim of this article is to contribute to a question of R. Brauer that “when do non-isomorphic groups have isomorphic complex group algebras?” Let H and G be finite groups where and let denote the first column of the complex character table of H. In this article, we show that if then provided that q + 1 divides neither n nor n – 1. Consequently, it is shown that G is uniquely determined by the structure of its complex group algebra. This in particular extends a recent result of Bessenrodt et al. [Algebra Number Theory 9 (2015), 601–628] to the almost simple groups of arbitrary rank.

On a central algorithm for calculation of the inverse of the harmonic oscillator in the spaces of orbits

Suppose a locally compact group G acts on a separable metrizable space M, preserving a finite measure #. Then there is an associated unitary representation of G on LZ(M,/~)o, the square integrable functions orthogonal to the constants. This representation will in general be far from irreducible, and in fact, when G is not compact need not have any irreducible subrepresentations. Thus, in the decomposition of LZ(G)o into irreducible constituents, one not only requires a direct integral rather than a direct sum, but the associated spectral measure on the unitary dual G (say for type I groups) will, except in special circumstances, have no atoms. The point of this paper is to give natural and general conditions, mostly involving just rq(M), under which there will be an infinite atomic spectrum. (I.e., the spectral measure has infinitely many atoms.) The basic example of actions in which the spectrum is purely atomic, i.e., when the representation is a direct sum of irreducible representations, is the case of M = G/F, where F c G is a cocompact discrete subgroup. In this case, assuming 9 G to be simply connected, we have 7tl(M ) = F. The first results of this paper show that when G is a simple Lie group with R-rank(G)=> 2, we can deduce that any action of G on a compact manifold M with fundamental group isomorphic to such a F, and satisfying a natural hypothesis of either a geometric or ergodic theoretic nature, must also have an infinite atomic spectrum. In fact, under certain natural hypotheses the same is true as long as there is a surjection of hi(M) onto F. To formulate this more precisely, let F c Ad(G) be a cocompact lattice, and let Pr be the unitary representation of Ad(G) on L2(Ad(G)/F)o . We recall that an action of a group on a manifold (or more generally on a standard space in the sense of [4]) is called engaging if there is no loss of ergodicity in passing to finite covers. (See I'4] for discussion.)

Simple classical groups of Lie type are determined by their character degrees

Let \(G\) be a finite group. The character degree graph of \(G\), which is denoted by \(\Gamma (G)\), is the graph whose vertices are the prime divisors of the character degrees of the group \(G\) and two vertices \(p_1\) and \(p_2\) are joined by an edge if \(p_1p_2\) divides some character degree of \(G\). In this paper we prove that the simple group \(\mathrm{PSL}(2,p^2) \) is uniquely determined by its character degree graph and its order. Let \(X_1(G)\) be the set of all irreducible complex character degrees of \(G\) counting multiplicities. As a consequence of our results we prove that if \(G\) is a finite group such that \(X_1(G)=X_1(\mathrm{PSL}(2,p^2) )\), then \(G\cong \mathrm{PSL}(2,p^2) \). This implies that \(\mathrm{PSL}(2,p^2) \) is uniquely determined by the structure of its complex group algebra.

Almost simple groups with the socle $PSL(2,p^n)$ are determined by their complex group algebras

A ‐tuple of core‐free subgroups of a finite group is said to be regular if has a regular orbit on the Cartesian product . The regularity number of , denoted by , is the smallest positive integer with the property that every such ‐tuple is regular. In this paper, we develop some general methods for studying the regularity of subgroup tuples in arbitrary finite groups, and we determine the precise regularity number of all almost simple groups with an alternating or sporadic socle. For example, we prove that and . We also formulate and investigate natural generalisations of several well‐studied problems on base sizes for finite permutation groups, including conjectures due to Cameron, Pyber and Vdovin. For instance, we extend earlier work of Burness, O'Brien and Wilson by proving that for every almost simple sporadic group, with equality if and only if is the Mathieu group . We also show that every triple of soluble subgroups in an almost simple sporadic group is regular, which generalises recent work of Burness on base sizes for transitive actions of sporadic groups with soluble point stabilisers.

Abstract

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic p ? 0 with natural module W. Let H be a closed subgroup of G and let V be a nontrivial irreducible tensor indecomposable -restricted 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 H is a closed disconnected almost simple positive-dimensional subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classifcation of the irreducible triples (G,H,V ) where G is a simple algebraic group over K, and H is a maximal closed subgroup of positive dimension.

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. One of the major theorems in the area is Borovik's trichotomy theorem. The ‘trichotomy’ here is a case division of the generic minimal counterexamples within odd type, that is, groups with a large and divisible Sylow° 2-subgroup. The so-called ‘uniqueness case’ in the trichotomy theorem is the existence of a proper 2-generated core. It is our aim to drive the presence of a proper 2-generated core to a contradiction, and hence bind the complexity of the Sylow° 2-subgroup of a minimal counterexample to the Cherlin–Zilber conjecture. This paper shows that the group in question is a minimal connected simple group and has a strongly embedded subgroup, a far stronger uniqueness case. As a corollary, a tame counterexample to the Cherlin–Zilber conjecture has Prüfer rank at most two.

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $$S^3.$$ S 3 . Such involutions are called hyperelliptic as the manifolds admitting such an action. We prove that the sectional 2-rank of a finite group acting on a 3-manifold and containing a hyperelliptic involution whose fixed-point set has two components has sectional 2-rank at most four; this upper bound is sharp. The cases where the hyperelliptic involution has a fixed-point set with a number of components different from 2 have been already considered in the literature. Our result completes the analysis and we obtain general results where the number of the components of the fixed-point set is not fixed. In particular, we obtain that a finite group acting on a 3-manifold and containing a hyperelliptic involution has 2-rank at most four, and four is the best possible upper bound. Finally, we restrict to the basic case of simple groups acting on hyperelliptic 3-manifolds: we use our result about the sectional 2-rank to prove that a simple group containing a hyperelliptic involution is isomorphic to PSL(2, q) for some odd prime power q, or to one of four other small simple groups.

Let $\Sigma$ be a finite type surface, and $G$ a complex algebraic simple Lie group with Lie algebra $\mathfrak{g}$. The quantum moduli algebra of $(\Sigma,G)$ is a quantization of the ring of functions of $X_G(\Sigma)$, the variety of $G$-characters of $\pi_1(\Sigma)$, introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid '90s. It can be realized as the invariant subalgebra of so-called graph algebras, which are $U_q(\mathfrak{g})$-module-algebras associated to graphs on $\Sigma$, where $U_q(\mathfrak{g})$ is the quantum group corresponding to $G$. We study the structure of the quantum moduli algebra in the case where $\Sigma$ is a sphere with $n+1$ open disks removed, $n\geq 1$, using the graph algebra of the "daisy" graph on $\Sigma$ to make computations easier. We provide new results that hold for arbitrary $G$ and generic $q$, and develop the theory in the case where $q=\epsilon$, a primitive root of unity of odd order, and $G={\rm SL}(2,{\mathbb C})$. In such a situation we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring $\mathcal{O}(G^n)$. We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on $\mathcal{O}(G^n)$. We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ${\mathbb C}[X_G(\Sigma)]$ endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra $K_{\zeta}(\Sigma)$ at $\zeta:={\rm i}\epsilon^{1/2}$ with this quantum moduli algebra specialized at $q=\epsilon$.

The structure of the automorphism group of a simple TAI algebra is studied. In particular, we show that \({\frac{\overline{ \rm{Inn}}(A) }{ \overline{\rm{Inn}}_{0} ( A ) }}\) is isomorphic (as a topological group) to an inverse limit of discrete abelian groups for a unital, simple, AH algebra with bounded dimension growth. Consequently, \({\frac{\overline{\rm{Inn}}(A)}{\overline{\rm{Inn}}_{0}(A) }}\) is totally disconnected.Another consequence of our results is the following: Suppose A is the transformation group C*-algebra of a minimal Furstenberg transformation with a unique invariant probability measure. Then the automorphism group of A is an extension of a simple topological group by the discrete group \({\rm{Aut}(\underline{K}(A))_{+,1}}\).

Short antichains in root systems, semi-Catalan arrangements, and B-stable subspaces

Let $G$ be a finite simple group. By a theorem of Guralnick and Kantor, $G$ contains a conjugacy class $C$ such that for each non-identity element $x \in G$, there exists $y \in C$ with $G = \langle x,y\rangle$. Building on this deep result, we introduce a new invariant $\gamma_u(G)$, which we call the uniform domination number of $G$. This is the minimal size of a subset $S$ of conjugate elements such that for each $1 \ne x \in G$, there exists $s \in S$ with $G = \langle x, s \rangle$. (This invariant is closely related to the total domination number of the generating graph of $G$, which explains our choice of terminology.) By the result of Guralnick and Kantor, we have $\gamma_u(G) \leqslant |C|$ for some conjugacy class $C$ of $G$, and the aim of this paper is to determine close to best possible bounds on $\gamma_u(G)$ for each family of simple groups. For example, we will prove that there are infinitely many non-abelian simple groups $G$ with $\gamma_u(G) = 2$. To do this, we develop a probabilistic approach, based on fixed point ratio estimates. We also establish a connection to the theory of bases for permutation groups, which allows us to apply recent results on base sizes for primitive actions of simple groups.