In this note an ‘extended Burnside ring’ is defined, generated by classes of semisimple module categories over Rep(G) with quasifibre functors. Here G is a finite group and representations are taken over an algebraically closed field of characteristic 0. It is shown that this is equivalent to a ring generated by centrally extended G-sets and hence the name. Ring homomorphisms into the multiplic...

We start by briefly explaining key results on the representation theory of semisimple Lie algebras in large enough positive characteristic. After that, we start a new topic: the complex representation theory of finite groups of Lie type and Hecke algebras. Today we consider the most basic group: G := GLn(Fq). We introduce the Hecke algebra as the endomorphism algebra of the G-module C[B \ G]. W...

In this article we describe the G×G-equivariant K-ring of X, where X is a regular compactification of a connected complex reductive algebraic group G. Furthermore, in the case when G is a semisimple group of adjoint type, and X its wonderful compactification, we describe its ordinary K-ring K(X). More precisely, we prove that K(X) is a free module over K(G/B) of rank the cardinality of the Weyl...

M be a module. In this article we study δ–essential submodules as a dual of δ-small submodules of Zhou where δ = {N ∈ σ[M ] : Rej(N,M) = 0} and M = {N ∈ σ[M ] : N ¿ b N}, and also define μ-M–singular modules as modules N ∈ σ[M ] such that N ∼= K/L for some K ∈ σ[M ] and L is μ–essential in K. By M–M–singular modules and δ–M– singular modules a characterization of GCO–modules, and by FC–M–singul...

Grothendieck [7] showed that finite dimensional vector bundles over the projective line split into direct sums of line bundles (see also Andersen [1]). We begin this paper, after two preliminary sections, by proving this result (Theorem (3.5)) for induced bundles using representation theoretic techniques. The proof yields partial results in the infinite dimensional case and leads to the definit...

This article studies the Albanese kernel TF (E × E), for an elliptic curve E over a p-adic field F . The main result furnishes information, for any odd prime p, about the kernel and image of the Galois symbol map from TF (E ×E)/p to the Galois cohomology group H(F, E[p] ⊗ E[p]), for E/F ordinary, without requiring that the ptorsion points are F -rational, or even that the Galois module E[p] is ...

Kazhdan-Lusztig polynomials Px,w(q) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. In particular, the value Px,w(1) is the dimension of the intersection cohomology sheaf of the Schubert variety Xw at the T -fixed point indexed by x [KL2]. It is also the multiplicity of a certain irreducible module in a Verma module [BeB...

We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra $${\mathcal {M}}(p)$$ , $$p\ge 2$$ . The first category consists all finite-length -modules with atypical composition factors, while the second is subcategory that induce to local triplet {W}}(p)$$ show every irreducible module has a projective cover in these categories, although...