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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., Comment: 20 pages. This paper replaces paper the second half of [1] below, and [2] replaces the first half of [1]. [1] Peter Webb, Consequences of the existence of Auslander-Reiten triangles with applications to perfect complexes for self-injective algebras, arXiv:1301.4701 [2] Peter Webb, Bilinear forms on Grothendieck groups of triangulated categories, arXiv:1709.03889

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., Comment: 80 pages

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., Comment: 28 pages

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., Comment: arXiv admin note: substantial text overlap with arXiv:1301.4701

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.

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.

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.

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., Comment: The first half of this paper is replaced by Bilinear forms on Grothendieck groups of triangulated categories, arXiv:1709.03889 and the second half of this paper is replaced by The Auslander-Reiten Quiver of Perfect Complexes for a Self-Injective Algebra

Research into workforce development (WD) considered the relationship between corporate assessments of workers' development needs and WD strategies; how learning at work takes place; and what learning methods are used and their effectiveness. Focus was on practice in small and medium-sized enterprises (SMEs). Methodology included a literature review to identify the scope and extent of WD; survey of WD being undertaken in a sample of SMEs; follow-up interviews with managers in SMEs; and focus group meetings with employees in SMEs. Findings indicated that, while there are still many barriers to progress, WD seems to be gaining ground as a concept with both individuals and employers; securing greater employer involvement in learning within SMEs requires an emphasis to be placed on business support, rather than learning in itself; a wide range of learning styles and forms of learning must be accommodated, but a central theme is the need for flexible, time-efficient solutions; companies are finding innovative solutions; space as well as time may be a limiting factor in the extent to which certain types of learning can occur in the workplace; and very small firms usually face different problems requiring different solutions from those faced by bigger companies. Appendixes include: a review of related WD initiatives; and 19 tables illustrating characteristics of focus group participants. (Contains 46 endnotes.) (YLB)

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., Comment: The applications and Examples in section 4 have been extended