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Categorification and Geometric Representation Theory

122 pages in 9 parts

Categorification and geometric representation theory studies algebraic structures through categories, functors, and geometry. These pages develop Hopf and Hecke algebras, quantum enveloping algebras, integral forms, highest-weight modules, crystal lattices, tensor-product crystals, and global bases. Quantum affine representations and Fock spaces connect these constructions to KLR algebras, highest-weight categorification, and Ariki decomposition-number theorems. Kazhdan–Lusztig theory, Soergel methods, projective functors, category O, and localization provide categorical and geometric approaches to characters and multiplicities. Hall algebras organize extension data in finitary, derived, motivic, and cohomological settings; further topics include Yangians, root-of-unity quantum groups, tilting quotients, braided tensor structures, stability, wall crossing, sewing, and defects.

Abstract Algebra, Category Theory, and Homological Algebra supply ring, module, tensor, and derived foundations. Coxeter Groups, Lie Theory, and Representation Theory of Groups provide reflection and highest-weight methods; Algebraic Geometry supports descent and convolution. Symmetric Groups and Symmetric Functions supplies branching and partition combinatorics, while Braid Groups contributes categorical actions and diagram constructions. The reading pathway follows these prerequisite chains through quantum groups and crystals to categorification, and through Hall integration and tensor categories to geometric and topological applications. Related Clifford, diagram-algebra, and vertex-algebra topics remain in Quantum Algebra and Representation Theory. Example companions accompany the theory pages, with each construction retaining its own hypotheses and conventions.

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 · Algebraic and geometric foundations

    6 pages

    Begin with tensor coherence, quantum enveloping algebras, ordinary Hecke Specht constructions, Kazhdan–Lusztig bases, proper descent, and equivariant convolution. These are independent entry points within the collection; their external prerequisites remain on the linked subject pages.

  2. Part 2 · Hecke bases and quantum integral forms

    5 pages · after Part 1

    Develop coalgebras and generic Hecke bases alongside divided powers, integral forms, and quantum highest-weight modules. Soergel Hodge theory and the quantum-symmetry interface prepare the categorical and structural branches.

  3. Part 3 · Affine actions and crystal lattices

    8 pages · after Parts 1 and 2

    Continue to comodules, bialgebras, Hecke induction and traces, affine Hecke algebras, and Yangians. Kashiwara operators and crystal lattices begin the crystal branch. Projective functors and noninvertible symmetries develop their categorical applications.

  4. Part 4 · Hopf modules, characters, and tensor crystals

    10 pages · after Parts 1, 2 and 3

    Study cyclotomic quotients, specialization, Hopf modules, and duality. Tensor-product crystals and highest-weight uniqueness build on crystal lattices, while character formulas, unequal-parameter cells, and modular categories continue their respective branches.

  5. Part 5 · Global bases and finite structures

    5 pages · after Part 4

    Balanced triples construct global crystal bases with their lattice and normalization conventions. Finite Hopf integrals, quasitriangular structures, singular and parabolic multiplicities, and higher defects extend the structural and geometric branches.

  6. Part 6 · Categorification and quantum affine representations

    7 pages · after Parts 1 and 5

    Use global bases in KLR induction and quantum-Serre categorification before cyclotomic highest-weight constructions. Quantum affine type A develops its Fock-space action; root-of-unity quantum groups lead to tilting quotients. Parabolic category O, Specht modules, Drinfeld doubles, and Frobenius sewing follow their prerequisites.

  7. Part 7 · Cyclotomic structures and Fock-space comparisons

    6 pages · after Parts 1, 2, 4 and 6

    Develop cyclotomic KLR categorification, cellular Murphy bases, Hecke tensor actions, and affine Hecke algebras beyond type A. Residue crystals and higher-level Schur–Weyl duality supply the Fock-space comparisons used in the decomposition-number theorems.

  8. Part 8 · Hecke–KLR and Hall constructions

    4 pages · after Parts 1, 6 and 7

    The cyclotomic Hecke–KLR bridge follows its Specht, quantum affine, and categorification inputs. Finitary Hall algebras begin the counting branch after their declared prerequisites. Double-affine and rational Cherednik constructions continue the affine representation-theory routes.

  9. Part 9 · Decomposition numbers, integration, and localization

    10 pages · after Parts 1, 3, 5, 7 and 8

    Read Ariki categorification and its separate degenerate theorem before the category-O/KLR/Hecke route and localization. Green coproducts, derived and cohomological Hall algebras precede Hall modules and motivic integration; stability and wall crossing follow the integration constructions.