Remark 1.2. The definition of an FG-module is more technical than the definition of a representation of G, but, as the exercise shows, the two notions are equivalent. Module can be more convenient to work with, because there is less notation, and we can use results from ring theory without any translation. The language of representations is preferable if we want to have an explicit map ρ : G→ G...

We present a generalization of the classical Schur modules of GL(N) exhibiting the same interplay among algebra, geometry, and combinatorics. A generalized Young diagram D is an arbitrary finite subset of N × N. For each D, we define the Schur module SD of GL(N). We introduce a projective variety FD and a line bundle LD, and describe the Schur module in terms of sections of LD. For diagrams wit...

Let L be a free Lie algebra of finite rank r over a field . For each positive integer n, denote the degree n homogeneous component of L by L. The group of graded algebra automorphisms of L may be identified with GLr; ‘ in such a way that L1 becomes the natural module, and then the L are referred to as the Lie powers of this module. Understanding the GLr; ‘-module structure of the L may be tho...

We give several resolutions of the Steinberg representation Stn for the general linear group over a principal ideal domain, in particular over Z. We compare them, and use these results to prove that the computations in [AGM11] are definitive. In particular, in [AGM11] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4,Z). One complex is based on Voronoi’s ...

Let A be a unital C*-algebra. We describe K-skeleton factorizations of all invertible operators on a Hilbert C*-module HA, in particular on H = l 2, with the Fredholm index as an invariant. We then outline the isomorphisms K0(A) ∼= π2k([p]0) ∼= π2k(GL p r(A)) and K1(A) ∼= π2k+1([p]0) ∼= π2k+1(GL p r(A)) for k ≥ 0, where [p]0 denotes the class of all compact perturbations of a projection p in th...

Let F̄p be an algebraic closure of a finite field of characteristic p. Let ρ be a continuous homomorphism from the absolute Galois group of Q to GL(3, F̄p) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the Serre conductor of ρ is squarefree, we prove that ρ is attached to a Hecke eigenclass in the homology of an a...

We prove a Schur-Weyl duality between the quantum enveloping algebra of $\mathfrak{gl}_m$ and certain quotient algebras Ariki-Koike algebras, which we give explicitly. The involves several algebraically independent parameters is realized through tensor product parabolic universal Verma module power natural representation $\mathfrak{gl}_m$. also new presentation by generators relations generaliz...

For a finite abelian group G ⊂ GL(n, k), we describe the coherent component Yθ of the moduli space Mθ of McKay quiver representations. This is a not-necessarily-normal toric variety that admits a projective birational morphism Yθ → A/G obtained by variation of GIT quotient. We present a simple calculation to determine the quiver representation corresponding to any point of Yθ, and describe the ...