On a question of Külshammer for representations of finite groups in reductive groups

Let G be a simple algebraic group of type G₂ over an algebraically closed field of characteristic 2. We give an example of a finite group Γ with Sylow 2-subgroup Γ₂ and an infinite family of pairwise non-conjugate homomorphisms ρ: Γ → G whose restrictions to Γ₂ are all conjugate. This answers a question of Burkhard Külshammer from 1995. We also give an action of Γ on a connected unipotent group V such that the map of 1-cohomologies H¹(Γ, V) → H¹(Γ_p, V) induced by restriction of 1-cocycles has an infinite fibre.     

On the normalizer of a Gaschütz system of a finite soluble group

The notion of a Gaschütz system of a finite soluble group was introduced by S. F. Kamornikov in 2008 (this is a set of complements of crowns of pairwise nonisomorphic non-Frattini factors of a chief series of the group). In the present paper, properties of Gaschütz systems are investigated. In particular, we calculate the number of Gaschütz systems in a finite soluble group and prove their conjugacy, obtain a connection between mathfrakN mathfrak{N} -prefrattini subgroups and normalizers of Gaschütz systems, and investigate factorizations of the normalizer of a Gaschütz system.     

On {phi}-contractibility of the Lebesgue–Fourier algebra of a locally compact group

For a locally compact group G, we present some characterizations for f{phi}-contractibility of the Lebesgue–Fourier algebra LA(G){mathcal{L}A(G)} endowed with convolution or pointwise product.     

What does a group algebra of a free group “know” about the group?

We describe solutions to the problem of elementary classification in the class of group algebras of free groups. We will show that unlike free groups, two group algebras of free groups over infinite fields are elementarily equivalent if and only if the groups are isomorphic and the fields are equivalent in the weak second order logic. We will show that the set of all free bases of a free group F is 0-definable in the group algebra $K(F)$ when K is an infinite field, the set of geodesics is definable, and many geometric properties of F are definable in $K(F)$. Therefore $K(F)$ “knows” some very important information about F. We will show that similar results hold for group algebras of limit groups.     

On the non-normal Padé table in a non-commutative algebra

In the non-commutative algebra the blocks in the table of orthogonal polynomials and therefore in the Padé table are not square, and generally it is impossible to say anything on the structure of these blocks except for infinite blocks. This last case is extensively studied here for the non-normal Padé table, the non-normal table P of orthogonal polynomials, and the non-normal ϵ-table. Some examples of illustration of different situations are given.     

Combinatorial properties of the Temperley–Lieb algebra of a Coxeter group

We study two families of polynomials that play the same role in the Temperley–Lieb algebra of a Coxeter group as the Kazhdan–Lusztig and R-polynomials play in the Hecke algebra of the group. Our results include recursions, non-recursive formulas, symmetry properties and expressions for the constant term. We focus mainly on non-branching Coxeter graphs.     

Hochschild Homology Invariants of Külshammer Type of Derived Categories

For a perfect field k of characteristic p > 0 and for a finite dimensional symmetric k-algebra A Külshammer studied a sequence of ideals of the centre of A using the p-power map on degree 0 Hochschild homology. In joint work with Bessenrodt and Holm we removed the condition to be symmetric by passing through the trivial extension algebra. If A is symmetric, then the dual to the Külshammer ideal structure was generalised to higher Hochschild homology in earlier work [23 Zimmermann , A. ( 2007 ). Fine Hochschild invariants of derived categories for symmetric algebras . Journal of Algebra 308 : 350 – 367 . [Google Scholar]]. In the present article we follow this program and propose an analogue of the dual to the Külshammer ideal structure on the degree m Hochschild homology theory also to not necessarily symmetric algebras.     

On the T∘-slices of a finite group

A slice $(G,S)$ of finite groups is a pair consisting of a finite group G and a subgroup S of G. In this paper, we show that some properties of finite groups extend to slices of finite groups. In particular, by analogy with B-groups, we introduce the notion of $T^{°}$-slice, and show that any slice of finite groups admits a largest quotient $T^{°}$-slice.     

On the unit group and the class number¶of certain composita of two real quadratic fields

The purpose of this paper is to exhibit a new family of real bicyclic biquadratic fields K for which we can write the Hasse unit index of the group generated by the units of the three quadratic subfields in the unit group E _K of K. As a byproduct, one can explicitly relate the class number of K with the product of the class numbers of the three quadratic subfields. Received: 25 July 2000 / Revised version: 12 December 2000     

Külshammer Ideals of Graded Categories and Hochschild Cohomology

We generalize the notion of Külshammer ideals to the setting of a graded category. This allows us to define and study some properties of Külshammer type ideals in the graded center of a triangulated category and in the Hochschild cohomology of an algebra, providing new derived invariants. Further properties of Külshammer ideals are studied in the case where the category is d-Calabi-Yau.     

On the Chermak–Delgado lattice of a finite group

Abstract

By imposing conditions upon the index of a self-centralizing subgroup of a group, and upon the index of the center of the group, we are able to classify the Chermak-Delgado lattice of the group. This is our main result. We use this result to classify the Chermak-Delgado lattices of dicyclic groups and of metabelian p-groups of maximal class.     

On the universal cover of certain exotic Kähler surfaces of negative curvature

We study a class of examples of negatively curved compact Kähler surfaces that are not diffeomorphic to any locally symmetric space. From the analysis of certain totally geodesic curves on these surfaces we deduce that, for infinitely many examples, the natural representation of the fundamental group into PU(2,1) is non-faithful. We also give a new construction of bounded holomorphic functions on the universal cover of our surfaces, based on lifting maps to compact Riemann surfaces.     

On τ-tilting finiteness of the Schur algebra

We determined the τ-tilting finiteness of Schur algebras over an algebraically closed field of arbitrary characteristic, except for a few small cases.