arXiv subjects

Peter Webb

Publications and source records attributed to Peter Webb.

10 recordsLinked to original sources

arXiv2026

Synthetic Buildings for Finite Groups

We introduce synthetic buildings for each finite group $G$ and prime $p$. These are $G$-simplicial complexes, among which are the $p$-subgroups complex of K.S. Brown, and also the buildings of finite groups of Lie type in characteristic $p$. Synthetic buildings are useful because of special properties, including providing a formula for the cohomology of $G$. Our goal is to describe the kinds of synthetic buildings that can arise, and ideally to classify them. The \textit{minimal} synthetic buildings play a special role and for most groups there are not very many of these. However, we show that it is possible to find sequences of finite groups with increasingly many minimal synthetic buildings as we move along the sequence, and also groups with minimal synthetic buildings of different dimensions. It is also possible to find groups with infinitely many non-minimal synthetic buildings of a given dimension. In other situations, we show that there are no minimal synthetic buildings of equivariant homotopy type distinct from those of Brown's complex and the complex consisting of a single point. We make a number of conjectures.

math.GR↗

arXiv2023

The Auslander-Reiten quiver of perfect complexes for a self-injective algebra

We consider the homotopy category of perfect complexes for a finite dimensional self-injective algebra over a field, identifying many aspects of perfect complexes according to their position in the Auslander-Reiten quiver. Short complexes lie close to the rim. We characterize the position in the quiver of complexes of lengths 1, 2 and 3, as well as rigid complexes and truncated projective resolutions. We describe completely the quiver components that contain projective modules (complexes of length 1). We obtain relationships between the homology of complexes at different places in the quiver, deducing that every self-injective algebra of radical length at least 3 has indecomposable perfect complexes with arbitrarily large homology in any given degree. We show that homology stabilizes, in a certain sense, away from the rim of the quiver.

math.RT↗

arXiv2023

Biset functors for categories

We introduce the theory of biset functors defined on finite categories. Previously, biset functors have been defined on groups, and in that context they are closely related to Mackey functors. Standard examples on groups include representation rings, the Burnside ring and group cohomology. The new theory allows these same examples, but defined for arbitrary finite categories, thus including, for instance, the representation rings of finite EI categories, among which are posets, and the free categories associated to quivers without oriented cycles. Group homology with trivial coefficients is replaced by the homology of the category with constant coefficients and this is a biset functor when bisets that are representable on one side are used. We give a definition of the Burnside ring of an arbitrary finite category. It is a biset functor that plays a key role throughout the theory. We discuss properties of the simple biset functors on categories, including their parametrization and calculation. We describe the symmetric monoidal structure on biset functors, Green biset functors and an approach to fibered biset functors for categories, together with the technicalities these entail. Various examples are given, the most elaborate showing a connection between the correspondence functors of Bouc and Th\'evenaz and biset functors on Boolean lattices.

math.RT↗

arXiv2020

Young Modules and Schur algebras

We compute explicitly the submodule structure of the Young modules for symmetric groups $S_n$ over fields of characteristic 2, when $n \le 7$. We use this information to compute the submodule structure of indecomposable projectives for the corresponding Schur algebras when $n \le 5$, and we give give partial information when n=6,7, including the Gabriel quiver and the structure of the Weyl modules. We resolve all Morita equivalences between blocks of these algebras.

math.RT↗

arXiv2017

The bounded derived category of a poset

We introduce a new combinatorial condition on a subinterval of a poset P (a clamped subinterval) that allows us to relate the Auslander-Reiten quiver of the bounded derived category of P to that of the subinterval. Applications include the determination of when a poset is fractionally Calabi-Yau and the computation of the Auslander-Reiten quivers of both the bounded derived category and of the module category.

math.RT↗

arXiv2017

Bilinear forms on Grothendieck groups of triangulated categories

We extend the theory of bilinear forms on the Green ring of a finite group developed by Benson and Parker to the context of the Grothendieck group of a triangulated category with Auslander-Reiten triangles, taking only relations given by direct sum decompositions. We examine the non-degeneracy of the bilinear form given by dimensions of homomorphisms, and show that the form may be modified to give a Hermitian form for which the standard basis given by indecomposable objects has a dual basis given by Auslander-Reiten triangles. An application is given to the homotopy category of perfect complexes over a symmetric algebra, with a consequence analogous to a result of Erdmann and Kerner.

math.RT↗

arXiv2015

The Graded Center of a Triangulated Category

With applications in mind to the representations and cohomology of block algebras, we examine elements of the graded center of a triangulated category when the category has a Serre functor. These are natural transformations from the identity functor to powers of the shift functor that commute with the shift functor We show that such natural transformations which have support in a single shift orbit of indecomposable objects are necessarily of a kind previously constructed by Linckelmann. Under further conditions, when the support is contained in only finitely many shift orbits, sums of transformations of this special kind account for all possibilities. Allowing infinitely many shift orbits in the support, we construct elements of the graded center of the stable module category of a tame group algebra of a kind that cannot occur with wild block algebras. We use functorial methods extensively in the proof, developing some of this theory in the context of triangulated categories.

math.RT↗

arXiv2015

Combinatorial Restrictions on the Tree Class of the Auslander-Reiten Quiver of a Triangulated Category

We show that if a connected, Hom-finite, Krull-Schmidt triangulated category has an Auslander-Reiten quiver component with Dynkin tree class then the category has Auslander-Reiten triangles and that component is the entire quiver. This is an analogue for triangulated categories of a theorem of Auslander, and extends a previous result of Scherotzke. We also show that if there is a quiver component with extended Dynkin tree class, then other components must also have extended Dynkin class or one of a small set of infinite trees, provided there is a non-zero homomorphism between the components. The proofs use the theory of additive functions.

math.RT↗

arXiv2013

Consequences of the existence of Auslander-Reiten triangles with applications to perfect complexes for self-injective algebras

In a k-linear triangulated category (where k is a field) we show that the existence of Auslander-Reiten triangles implies that objects are determined, up to shift, by knowing dimensions of homomorphisms between them. In most cases the objects themselves are distinguished by this information, a conclusion which was also reached under slightly different hypotheses in a theorem of Jensen, Su and Zimmermann. The approach is to consider bilinear forms on Grothendieck groups which are analogous to the Green ring of a finite group. We specialize to the category of perfect complexes for a self-injective algebra, for which the Auslander-Reiten quiver has a known shape. We characterize the position in the quiver of many kinds of perfect complexes, including those of lengths 1, 2 and 3, rigid complexes and truncated projective resolutions. We describe completely the quiver components which contain projective modules. We obtain relationships between the homology of complexes at different places in the quiver, deducing that every self-injective algebra of radical length at least 3 has indecomposable perfect complexes with arbitrarily large homology in any given degree. We find also that homology stabilizes away from the rim of the quiver. We show that when the algebra is symmetric, one of the forms considered earlier is Hermitian, and this allows us to compute its values knowing them only on objects on the rim of the quiver.

math.RT↗

arXiv2008

Extending the Coinvariant Theorems of Chevalley, Shephard--Todd, Mitchell and Springer

We extend in several directions invariant theory results of Chevalley, Shephard and Todd, Mitchell and Springer. Their results compare the group algebra for a finite reflection group with its coinvariant algebra, and compare a group representation with its module of relative coinvariants. Our extensions apply to arbitrary finite groups in any characteristic.

math.AC↗