Mathematics

We give an example of a finite-dimensional algebra with a 2-cluster tilting module and a simple module which has infinite complexity. This answers a question of Erdmann and Holm.

We define and study an exotic t-structure on the bounded derived category of equivariant coherent sheaves on partial resolutions of the nilpotent cone.

For the discrete series representations of GL(n) over a non-archimedean local field F , we define a notion of functions similar to "zonal spherical functions" for unramified principal series. We prove the existence of such functions in the level 0 case. As for unramified principal series, they give rise to explicit coefficients. We deduce a local proof of Matringe's criterion of distinction of discrete series, in the level 0 case, for the Galois symmetric space GL(n,F)/GL(n, F 0 ) , for any unramified quadratic extension F/ F 0 . We also exhibit explicit test vectors when these representations are distinguished.

We describe all blocks of the category of finite-dimensional q(3) -supermodules by providing their extension quivers. We also obtain two general results about the representation of q(n) : we show that the Ext quiver of the standard block of q(n) is obtained from the principal block of q(n−1) by identifying certain vertices of the quiver and prove a ''virtual'' BGG-reciprocity for q(n) . The latter result is used to compute the radical filtrations of q(3) projective covers.

Let G be a reductive algebraic group over a field k. When k=C, R.K.Brylinski constructed a filtration of weight spaces of a G module, using the action of a principal nilpotent element of the Lie algebra, and proved that this filtration corresponds to Lusztig's q-analogue of the weight multiplicity. Later, Ginzburg discovered that this filtration has an interesting geometric interpretation via the geometric Satake correspondence. The goal of this article is to generalize these results to positive characteristics.

We study the interaction between the block decompositions of reduced universal enveloping algebras in positive characteristic, the PBW filtration, and the nilpotent cone. We provide two natural versions of the PBW filtration on the block subalgebra A λ of the restricted universal enveloping algebra U χ (g) and show these are dual to each other. We also consider a shifted PBW filtration for which we relate the associated graded algebra to the algebra of functions on the Frobenius neighbourhood of 0 in the nilpotent cone and the coinvariants algebra corresponding to λ . In the case of g= sl 2 (k) in characteristic p>2 we determine the associated graded algebras of these filtrations on block subalgebras of U 0 ( sl 2 ) . We also apply this to determine the structure of the adjoint representation of U 0 ( sl 2 ) .

We describe an easy way how to find supercharacter theories for a finite group G , if the character table of G is known. Namely, we show how an arbitrary partition of the conjugacy classes of G or of the irreducible characters of G can be refined to the coarsest partition that belongs to a supercharacter theory. Our constructions emphasize the duality between superclasses and supercharacters. An algorithm is presented to find all supercharacter theories on a given character table. The algorithm is used to compute the number of supercharacter theories for some nonabelian simple groups with up to 26 conjugacy classes.

In this paper, we discuss the generalization of finitary 2 -representation theory of finitary 2 -categories to finitary birepresentation theory of finitary bicategories. In previous papers on the subject, the classification of simple transitive 2 -representations of a given 2 -category was reduced to that for certain subquotients. These reduction results were all formulated as bijections between equivalence classes of 2 -representations. In this paper, we generalize them to biequivalences between certain 2 -categories of birepresentations. Furthermore, we prove an analog of the double centralizer theorem in finitary birepresentation theory.

We prove that every finite dimensional representation of a finite group over a field of characteristic p admits a finite resolution by p-permutation modules. The proof involves a reformulation in terms of derived categories.

It is a well-known result of Auslander and Reiten that contravariant finiteness of the class P fin ∞ (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, in this work we examine the Gorenstein counterpart of Auslander--Reiten condition, namely contravariant finiteness of the class GP fin ∞ (of finitely generated modules of finite Gorenstein projective dimension), and its relation to validity of finitistic dimension conjectures. It is proved that contravariant finiteness of the class GP fin ∞ implies validity of the second finitistic dimension conjecture over left artinian rings. In the more special setting of Artin algebras, however, it is proved that the Auslander--Reiten sufficient condition and its Gorenstein counterpart are virtually equivalent in the sense that contravariant finiteness of the class GP fin ∞ implies contravariant finiteness of the class P fin ∞ over any Artin algebra, and the converse holds for Artin algebras over which the class GP fin 0 (of finitely generated Gorenstein projective modules) is contravariantly finite.