Seminar Representation Theory, WS 2021/22

Time and place: Wednesdays 10–12 in room V4-112

The seminar covers different topics within the field of representation theory.

The spectrum of $\mathbb R$

Tilting subcategories of exact categories

Deformed preprojective algebras

Higher Auslander-Reiten theory I

Schur-Weyl duality

Abstract: There seems to be no text book which treats Schur-Weyl duality in full generality (e.g. over any infinite field). The aim of this talk is to sketch a possible „textbook proof“. The main result is valid for any commutative base ring, and a crucial ingredient is the fact that any Schur algebra is quasi-hereditary.

Shapes of Auslander Reiten triangles in the stable category of modules over repetitive algebras.

Abstract: Let $\Bbbk$ be an algebraically closed field, let $\Lambda$ be a finite dimensional $\Bbbk $-algebra, and let $\widehat{\Lambda}$ be the repetitive algebra of $\Lambda$. For the stable category of finitely generated left $\widehat{\Lambda}$-modules $\widehat{\Lambda} $-$\underline{mod}$, we show that the irreducible morphisms fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. We next use this fact in order to describe the shape of the Auslander-Reiten triangles in $\widehat{\Lambda}$-$\underline{mod}$. We use the fact (and prove) that every Auslander-Reiten triangle in $\widehat{\Lambda}$-$\underline{mod}$ is induced from an Auslander-Reiten sequence of finitely generated left $\widehat{\Lambda}$-modules. Finally, we will talk about applications of this last result.

Perfect complexes with small homology over $\mathbb{F}_2[(\mathbb{Z}/2)^2\rtimes Q]$

Abstract: Let $G$ be an extension of a group of odd order by an elementary abelian $2$-group of rank $2$. It is known that the total homology of an arbitrary perfect complex over $\mathbb{F}_2[G]$ is either zero or at least four-dimensional. I will report on joint work with Henrik Rüping in which we study perfect complexes with four-dimensional total homology. As an application, I will provide restrictions for the existence of free $A_4$-actions on a product of two spheres. This is also joint with Ergün Yalcin.