Sibylle Schroll

GROUP ACTIONS AND COVERINGS OF BRAUER GRAPH ALGEBRAS

We develop a theory of group actions and coverings on Brauer graphs that parallels the theory of group actions and coverings of algebras. In particular, we show that any Brauer graph can be covered by a tower of coverings of Brauer graphs such that the topmost covering has multiplicity function identically one, no loops, and no multiple edges. Furthermore, we classify the coverings of Brauer graph algebras that are again Brauer graph algebras.

THE EXT ALGEBRA OF A BRAUER GRAPH ALGEBRA

In this paper we study finite generation of the Ext algebra of a Brauer graph algebra by determining the degrees of the generators. As a consequence we characterize the Brauer graph algebras that are Koszul and those that are K_2.

Special multiserial algebras are quotients of symmetric special multiserial algebras

In this paper we give a new definition of symmetric special multiserial algebras in terms of defining cycles. As a consequence, we show that every special multiserial algebra is a quotient of a symmetric special multiserial algebra.

On decomposition numbers and Alvis-Curtis duality

We show that for general linear groups GLn(q) as well as for q-Schur algebras the knowledge of the modular Alvis�Curtis duality over fields of characteristic l, l q, is equivalent to the knowledge of the decomposition numbers.

Brauer graph algebras

This survey starts with a motivation of the study of Brauer graph algebras by relating them to special biserial algebras. The definition of Brauer graph algebras is given in great detail with many examples to illustrate the concepts. An interpretation of Brauer graphs as decorated ribbon graphs is included. A section on gentle algebras and their associated ribbon graph, trivial extensions of gentle algebras, admissible cuts of Brauer graph algebras and a first connection of Brauer graph algebras with Jacobian algebras associated to triangulations of marked oriented surfaces follows. The interpretation of flips of diagonals in triangulations of marked oriented surfaces as derived equivalences of Brauer graph algebras and the comparison of derived equivalences of Brauer graph algebras with derived equivalences of frozen Jacobian algebras is the topic of the next section. In the last section, after defining Green’s walk around the Brauer graph, a complete description of the Auslander Reiten quiver of a Brauer graph algebra is given.

A circular order on edge-coloured trees and RNA m-diagrams

We study a circular order on labelled, m-edge-coloured trees with k vertices, and show that the set of such trees with a fixed circular order is in bijection with the set of RNA m-diagrams of degree k, combinatorial objects which can be regarded as RNA secondary structures of a certain kind. We enumerate these sets and show that the set of trees with a fixed circular order can be characterized as an equivalence class for the transitive closure of an operation which, in the case m=3, arises as an induction in the context of interval exchange transformations.

ON THE COXETER COMPLEX AND ALVIS–CURTIS DUALITY FOR PRINCIPAL ℓ-BLOCKS OF GLn(q)

M. Cabanes and J. Rickard showed in [3] that the Alvis–Curtis character duality of a finite group of Lie type is induced in non defining characteristic l by a derived equivalence given by tensoring with a bounded complex X, and they further conjecture that this derived equivalence should actually be a homotopy equivalence. Following a suggestion of R. Kessar, we show here for the special case of principal blocks of general linear groups with abelian Sylow-l-subgroups that this is true, by an explicit verification relating the complex X to the Coxeter complex of the corresponding Weyl group.