# Seventy-eighth BLOC meeting

**BLOC**

**Bristol Leicester Oxford Colloquium**

The seventy-eighth meeting will be held on 23 - 24 August 2017 in the Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford. This celebrates the 20th anniversary of BLOC.

**Programme*** (Abstracts below)***: **

**Wednesday 23 August**

**12.00 **Lunch

**2.00-3.00 Dave Benson** (Aberdeen) *Representations and cohomology of finite group schemes and finite supergroup schemes*

**3.00-4.00 Nadia Mazza** (Lancaster) *On endotrivial modules for finite reductive groups*

Tea

**4.45-5.45 Eleonore Faber** (Michigan/ Leeds) *A McKay correspondence for reflection groups*

Dinner

**Thursday 24 August**

**10.00-11.00 Sibylle Schroll** (Leicester) *New varieties for algebras*

Coffee

**11.30-12.30 Alastair King** (Bath) *Quivers and Conformal Field Theory: preprojective algebras and beyond*

Lunch

**2.00-3.00 Lleonard Rubio y Degrassi** (City University, London) *On Hochschild cohomology and global/local structures*

**3.00-4.00 Jeremy Rickard** (Bristol) *Unbounded derived categories and the finitistic dimension conjecture*

Tea

**Abstracts**

**Dave Benson:** I shall describe recent work with Srikanth Iyengar, Henning Krause and Julia Pevtsova on the representation theory and cohomology of finite group schemes and finite supergroup schemes. Particular emphasis will be placed on the role of generic points, detection of projectivity for modules, and detection modulo nilpotents for cohomology.

**Nadia Mazza:** Joint work with Carlson, Grodal, Nakano. In this talk, we will present some recent results on an "important" class of modular representations for an "important" class of finite groups. For convenience of the audience, we'll briefly review the notion of an endotrivial module and present the main results pertaining to endotrivial modules and finite reductive groups which we use in our ongoing work.

**Eleonore Faber:** This is joint work with Ragnar-Olaf Buchweitz and Colin Ingalls. The classical McKay correspondence relates the geometry of so-called Kleinian surface singularities with the representation theory of finite subgroups of SL(2,C). M.Auslander observed an algebraic version of this correspondence: let *G* be a finite subgroup of SL(2,*K*) for a field *K* whose characteristic does not divide the order of *G*. The group acts linearly on the polynomial ring *S*=*K*[*x*,*y*] and then the so-called skew group algebra *A*=*G*S* can be seen as an incarnation of the correspondence. In particular, *A* is isomorphic to the endomorphism ring of *S* over the corresponding Kleinian surface singularity.

Our goal is to establish an analogous result when *G* in GL(*n*,*K*) is a finite group generated by reflections, assuming that the characteristic of *K* does not divide the order of the group. Therefore we will consider a quotient of the skew group ring *A*=*S*G*, where *S* is the polynomial ring in *n* variables. We show that our construction yields a generalization of Auslander's result, and moreover, a noncommutative resolution of the discriminant of the reflection group *G*.

**Sibylle Schroll:** In this talk, we will introduce new affine algebraic varieties for algebras given by quiver and relations. Each variety contains a distinguished element in the form of a monomial algebra. The properties and characteristics of this monomial algebra govern those of all other algebras in the variety. We will show how amongst other things this gives rise to a new way to determine whether an algebra is quasi hereditary. This is a report on joint work both with Ed Green and with Ed Green and Lutz Hille.

**Alastair King:** I will describe how the ADE preprojective algebras appear in certain Conformal Field Theories, namely SU(2) WZW models, and explain the generalisation to the SU(3) case, where 'almost CY3' algebras appear.

**Lleonard Rubio y Degrassi:** In this talk I will discuss the interplay between the local and the global invariants in modular representation theory with a focus on the first Hochschild cohomology HH^{1}(*B*) of a block algebra *B*. In particular, I will show the compatibility between *r*-integrable derivations and stable equivalences of Morita type. I will also show that if HH^{1}(*B*) is a simple Lie algebra such that *B* has a unique isomorphism class of simple modules, then *B* is nilpotent with an elementary abelian defect group *P* of order at least 3. The second part is joint work with M. Linckelmann.

**Jeremy Rickard:** If A is a finite dimensional algebra, and D(A) the unbounded derived category of the full module category Mod-A, then it is straightforward to see that D(A) is generated (as a "localizing subcategory") by the indecomposable projectives, and by the simple modules. It is not so obvious whether it is generated by the indecomposable injectives. In 2001, Keller gave a talk in which he remarked that "injectives generate" would imply several of the well-known homological conjectures, such as the Nunke condition and hence the generalized Nakayama conjecture, and asked if there was any relation to the finitistic dimension conjecture. I'll show that an algebra that satisfies "injectives generate" also satisfies the finitistic dimension conjecture and discuss some examples. I'll present things in a fairly concrete way, so most of the talk won't assume much knowledge of derived categories.

Please let Karin Erdmann know, preferably by 31st July, but no later than 5th August, if you would like accommodation. (Some rooms have been reserved at Somerville College, which are subsidised - please ask Karin for details.) Please also let Karin know if you will be there for lunch and/or dinner.

BLOC is supported by an LMS Scheme 3 grant, which is used to help offset travel costs for participants, with priority given to covering travel and accommodation costs for speakers and postgraduates. Funding for BLOC is very limited this year, so please email both Karin Erdmann and Nicole Snashall if you require funding (travel/accommodation) so that we can coordinate this!

Travel information is HERE.

There will be time for informal discussions.

For further information about BLOC please contact one of the BLOC organisers: