On Automorphisms and Focal Subgroups of Blocks

Given a p-block B of a finite group with defect group P and fusion system on P, we show that the rank of the group is invariant under stable equivalences of Morita type. The main ingredients are the construction, due to Broué and Puig, a theorem of Weiss on linear source modules, arguments of Hertweck and Kimmerle applying Weiss’ theorem to blocks, and connections with integrable derivations in the Hochschild cohomology of block algebras

Quantum affine Cartan matrices, Poincare series of binary polyhedral groups, and reflection representations

We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a,b,h;p,q,r) that one usually associates with such a group and hence with a simply-laced Coxeter-Dynkin diagram have a meaningful definition for the non-simply-laced diagrams, too, and as a byproduct we extend Saito's formula for the determinant of the Cartan matrix to all cases. Returning to invariant theory we show that for each irreducible representation i of a binary tetrahedral, octahedral, or icosahedral group one can find a homomorphism into a finite complex reflection group whose defining reflection representation restricts to i.Comment: 19 page

Descent of Equivalences and Character Bijections

Categorical equivalences between block algebras of finite groups—such as Morita and derived equivalences—are well known to induce character bijections which commute with the Galois groups of field extensions. This is the motivation for attempting to realise known Morita and derived equivalences over non-splitting fields. This article presents various results on the theme of descent to appropriate subfields and subrings. We start with the observation that perfect isometries induced by a virtual Morita equivalence induce isomorphisms of centres in non-split situations and explain connections with Navarro’s generalisation of the Alperin–McKay conjecture. We show that Rouquier’s splendid Rickard complex for blocks with cyclic defect groups descends to the non-split case. We also prove a descent theorem for Morita equivalences with endopermutation source

Higher S-dualities and Shephard-Todd groups

Abstract: Seiberg and Witten have shown that in N=2$$\mathcal{N}=2$$ SQCD with Nf = 2Nc = 4 the S-duality group PSL2\u2124$$\mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right)$$ acts on the flavor charges, which are weights of Spin(8), by triality. There are other N=2$$\mathcal{N}=2$$ SCFTs in which SU(2) SYM is coupled to strongly-interacting non-Lagrangian matter: their matter charges are weights of E6, E7 and E8 instead of Spin(8). The S-duality group PSL2\u2124$$\mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right)$$ acts on these weights: what replaces Spin(8) triality for the E6, E7, E8root lattices? In this paper we answer the question. The action on the matter charges of (a finite central extension of) PSL2\u2124$$\mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right)$$ factorizes trough the action of the exceptional Shephard-Todd groups G4 and G8 which should be seen as complex analogs of the usual triality group S3 43WeylA2$${\mathfrak{S}}_3\simeq \mathrm{Weyl}\left({A}_2\right)$$. Our analysis is based on the identification of S-duality for SU(2) gauge SCFTs with the group of automorphisms of the cluster category of weighted projective lines of tubular type. \ua9 2015, The Author(s)

The Symbolic and cancellation-free formulae for Schur elements

In this paper we give a symbolical formula and a cancellation-free formula for the Schur elements associated to the simple modules of the degenerate cyclotomic Hecke algebras. As some direct applications, we show that the Schur elements are symmetric with respect to the natural symmetric group action and are integral coefficients polynomials and we give a different proof of Ariki-Mathas-Rui's criterion on the semi-simplicity of degenerate cyclotomic Hecke algebras.Comment: To appear in Monatshefte fur Mathemati

Stochastic flowering phenology in Dactylis Glomerata populations described by Markov chain modelling

Understanding the relationship between flowering patterns and pollen dispersal is important in climate change modelling, pollen forecasting, forestry and agriculture. Enhanced understanding of this connection can be gained through detailed spatial and temporal flowering observations on a population level, combined with modelling simulating the dynamics. Species with large distribution ranges, long flowering seasons, high pollen production and naturally large populations can be used to illustrate these dynamics. Revealing and simulating species-specific demographic and stochastic elements in the flowering process will likely be important in determining when pollen release is likely to happen in flowering plants. Spatial and temporal dynamics of eight populations of Dactylis glomerata were collected over the course of two years to determine high-resolution demographic elements. Stochastic elements were accounted for using Markov Chain approaches in order to evaluate tiller-specific contribution to overall population dynamics. Tiller-specific developmental dynamics were evaluated using three different RV matrix correlation coefficients. We found that the demographic patterns in population development were the same for all populations with key phenological events differing only by a few days over the course of the seasons. Many tillers transitioned very quickly from non-flowering to full flowering, a process that can be replicated with Markov Chain modelling. Our novel approach demonstrates the identification and quantification of stochastic elements in the flowering process of D. glomerata, an element likely to be found in many flowering plants. The stochastic modelling approach can be used to develop detailed pollen release models for Dactylis, other grass species and probably other flowering plants