Categorical Methods in Representation Theory — Mathematics Knowledge System

Overview

Categorical representation theory applies category theory to organize and generalize classical representation theory. Key structures include: abelian categories of modules over algebras, derived categories of sheaves, triangulated categories, t-structures, and perverse sheaves. The derived equivalences (Fourier-Mukai transforms, Koszul duality) relate seemingly different categories. Categorification -- lifting numerical equalities to equivalences of categories -- produces higher-dimensional algebra: for instance, the Kazhdan-Lusztig polynomials categorify to complexes of Soergel bimodules.

Intuition

Classical representation theory studies how groups/algebras act on vector spaces (numbers and linear maps). Categorical representation theory asks: what if the 'numbers' are replaced by vector spaces, and 'linear maps' by functors? This is categorification. For example, the equation dim L(lambda) = sum_w (-1)^{l(w)} dim M(w*lambda) (from the BGG resolution) becomes an exact sequence of categories. The numerical KL polynomials become actual complexes (Soergel bimodules) whose Euler characteristic gives the polynomial.

Formal Definition

Definition

An abelian category A is a category where every morphism has a kernel and cokernel, and images factor nicely. Rep(G) = category of representations of G over k is the canonical example. The derived category D(A) is obtained from the chain complex category by inverting quasi-isomorphisms (maps inducing isomorphisms on cohomology). Soergel bimodules: for a Coxeter group W with polynomial ring R = k[x_1,...,x_n] and Schubert calculus, the category of Soergel bimodules B is a full subcategory of (R,R)-bimodules categorifying the Hecke algebra H(W).

\mathrm{Rep}(G) = \{k\text{-representations of }G\}

Category of representations

D^b(\mathrm{Rep}(G)) = \text{bounded derived category}

Derived category of representations

[V] - [W] = [X] \text{ in } K_0 \iff 0 \to V \to X \to W \to 0 \text{ (categorification)}

Grothendieck group and Euler characteristic

\bigoplus_w B_w\text{ (Soergel bimodules)} \longrightarrow H(W)\text{ (Hecke algebra)}

Soergel categorification

Notation

Notation	Meaning
$D^b(\mathcal{A})$	Bounded derived category of abelian category A
$K_0(\mathcal{A})$	Grothendieck group: generators = iso classes, relations = short exact sequences
$B_w$	Soergel bimodule indexed by Weyl group element w
$H(W,S)$	Hecke algebra of Coxeter group (W,S)

Theorems

Theorem 1: BGG Koszul Duality

For the projective space P^n, the derived category of coherent sheaves D^b(Coh(P^n)) is equivalent (as triangulated categories) to the derived category of finitely generated modules over the exterior algebra Lambda = k[xi_0,...,xi_n]/(xi_i^2, xi_i*xi_j + xi_j*xi_i) (graded): D^b(Coh(P^n)) is equivalent to D^b(grmod-Lambda). This 'BGG correspondence' is the prototype for Koszul duality.

Theorem 2: Soergel Categorification Theorem

For a Coxeter group W, there is a monoidal category BS of Bott-Samelson bimodules whose Grothendieck group (K_0) is isomorphic to the Hecke algebra H(W) over Z[v,v^{-1}]. The indecomposable Soergel bimodules B_w correspond to the Kazhdan-Lusztig basis elements. Their characters give the KL polynomials.

Theorem 3: Elias-Williamson Positivity Theorem

Using diagrammatic calculus for Soergel bimodules, Elias and Williamson proved that the Kazhdan-Lusztig polynomials have non-negative integer coefficients for all Coxeter groups. This had been conjectured by Kazhdan-Lusztig and proved earlier for Weyl groups via perverse sheaves (using positivity of IC sheaves), but the diagrammatic proof works for arbitrary Coxeter groups.

Worked Examples

1
K_0(Rep(G)) is the free abelian group on isomorphism classes [V] of representations, modulo [V] = [U]+[W] for each short exact sequence 0->U->V->W->0.
2
For G finite and k = C: every SES splits (complete reducibility). So K_0(Rep(G)) = Z^{#irreducibles}, generated by [V_1],...,[V_r] (irreducible reps).
3
K_0(Rep(G)) has a ring structure from tensor product: [V]*[W] = [V tensor W]. This is the representation ring R(G).
$K_0(\mathrm{Rep}(G)) \cong R(G) = \mathbb{Z}[\chi_{V_1},\ldots,\chi_{V_r}]$
4
The character map chi: R(G) -> class functions on G is a ring isomorphism for G finite over C.

✓ Answer

K_0(Rep(G)) = representation ring R(G). For G finite over C, it is a free Z-module on irreducibles, with ring structure from tensor products.

Practice Problems

Hardfree response

What does it mean to 'categorify' the equation [V_j] tensor [V_{j'}] = sum_k [V_k] in the representation ring of SU(2)?

Common Mistakes

Common Mistake

Thinking derived categories are just the same as ordinary module categories.

The derived category D(A) is strictly larger than A: it contains complexes that are not quasi-isomorphic to any single module (i.e., complexes with non-zero cohomology in multiple degrees). D(A) remembers the higher Ext groups that are invisible in A. Derived equivalences (like BGG duality) can relate totally different-looking abelian categories A and B.

Quiz

The Soergel categorification theorem says:

The Hecke algebra is isomorphic to a matrix algebraSoergel bimodules form a monoidal category whose K_0 is the Hecke algebraRepresentations of Weyl groups are all semisimpleThe derived category of the flag variety categorifies the Hecke algebra

Historical Background

Grothendieck's development of abelian and derived categories in the 1950s-60s gave the framework. Verdier duality and triangulated categories formalized homological methods. Beilinson's description of derived categories of coherent sheaves on P^n (1978) showed derived equivalences are fundamental. Bernstein-Gelfand-Gelfand's 1978 derived equivalence between P^n and the exterior algebra exemplified 'Koszul duality'. Soergel's bimodules (1990) categorify Hecke algebra representations and led to modern developments by Elias-Williamson.

1957
Grothendieck introduces abelian categories and the derived functor formalism
Alexander Grothendieck
1977
Verdier introduces triangulated categories and duality
Jean-Louis Verdier
1978
BGG derived equivalence: D^b(Coh(P^n)) = D^b(Lambda-mod)
Joseph Bernstein, Israel Gelfand, Sergei Gelfand
1990
Soergel introduces bimodules categorifying Hecke algebra
Wolfgang Soergel
2012
Elias-Williamson diagrammatic approach categorifies Hecke algebra positivity
Ben Elias, Geordie Williamson

Summary

Categorical rep theory uses abelian/derived/triangulated categories to organize representations.
The Grothendieck group K_0 extracts numerical information (representation ring) from categorical data.
Koszul duality (BGG): D^b(Coh(P^n)) = D^b(grmod-Ext^*(k,k)^op) -- a derived equivalence.
Soergel bimodules categorify the Hecke algebra; Elias-Williamson proved KL positivity diagrammatically.

References

BookMazorchuk, V. Lectures on Algebraic Categorification. European Mathematical Society, 2012.
BookRiche, S. and Williamson, G. Tilting Modules and the p-Canonical Basis. Asterisque, 2018.
WebsiteWikipedia -- Categorification