Alphabeta Math

Representation Theory of Groups

64 pages in 2 parts

Representation theory studies groups through their linear actions on vector spaces and modules. This collection develops methods that apply across families of groups: character theory, induction and restriction, fields of definition, projective representations and Clifford theory. The modular branch studies blocks, defect groups, relative projectivity and the Green and Brauer correspondences, with coefficient characteristic and semisimplicity hypotheses kept explicit.

The topological branch begins with Haar measure and convolution, then constructs unitary representations, positive-type functions and cyclic representations. Averaging leads to complete reducibility for compact groups; Peter–Weyl theory supplies the density and decomposition statements needed for their harmonic analysis. Induction for locally compact groups leads toward imprimitivity, group operator algebras and direct integrals. Amenability, property T and the representations of the real special linear group provide concrete applications and boundaries.

Specialized theories have four categories: Symmetric Groups and Symmetric Functions; Quantum Groups and Crystal Bases; Hecke Algebras and Categorification; and Category O and Geometric Representation Theory. Braid Groups and Hopf Algebras & Hecke Algebras have their own categories. Page and item publication statuses distinguish available proofs from future work.

Pathway

The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.

  1. Part 1 · General Methods and Group Representations

    30 pages

    Finite-group methods lead from characters and induced characters through Brauer's modular theory to blocks, defect groups and the Green correspondence, while the modular function and L1 group algebras open the analytic half. There, unitary representations, positive type and GNS, complete reducibility and Peter–Weyl, induced representations and Mackey imprimitivity, group C*-algebras and the Fell dual develop the compact and locally compact theory; amenability, direct integral decomposition into factor and type-I representations, and Kazhdan's property (T) supply structural results. The SL2(R) pages compute the principal, complementary and discrete series and the irreducible unitary dual. Geometrically, equivariant constructible perverse foundations and quiver full-flag incidence bundles with their compactifications support correspondence convolution realizing the signed KLR algebra as an equivariant Ext algebra.

  2. Part 2 · Algebra and representation interfaces

    2 pages · after Part 1

    These draft extensions connect the existing foundations to algebraic structures and representation problems motivated by theoretical physics. Each companion supplies the planned examples and boundary cases; exact hypotheses, local proof obligations, and reuse holds remain in its page contract.