Alphabeta MathHecke Algebras and Categorification
Hecke representation theory and categorification connect deformed group algebras with graded modules, quantum groups and highest-weight structures. This collection develops ordinary Hecke Specht polytabloids, their comparison with Murphy constructions, and type-A cellular bases. The KLR branch studies induction and restriction, quantum Serre relations on Grothendieck groups, and cyclotomic highest-weight categorification. Cyclotomic Hecke and quiver Hecke algebras provide the bridge that transports grading and categorical information to Hecke modules.
Ariki categorification relates these constructions to decomposition numbers. The degenerate Ariki theorem retains its separate parameter regime and draws on category O and higher-level Schur–Weyl duality. Category O and Geometric Representation Theory supplies that route and the geometric convolution foundations; Quantum Groups and Crystal Bases supplies global bases, highest weights and Fock spaces. Symmetric Groups and Symmetric Functions contains the classical and modular symmetric-group applications. Foundational presentations, bases and structural constructions remain in Hopf Algebras & Hecke Algebras. Every topic keeps its examples companion, and publication statuses distinguish available material from draft work.
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Part 1 · Hecke Algebras and Categorification
7 pagesSpecht polytabloids and cellular bases develop Hecke module constructions. KLR induction and restriction categorify quantum Serre relations; cyclotomic quotients give highest-weight categorification. The Hecke–KLR bridge leads to graded decomposition numbers. The degenerate Ariki theorem follows its distinct category-O route, using the functors and Fock-space dictionary in Category O and Geometric Representation Theory.
This page builds the generic theory of quiver Hecke (KLR) algebras and their categorification of the negative half of a symmetrizable quantum group.
5 definitions, 3 lemmas, 7 theoremsExamples & counterexamples →This page builds the integral comparison between the ordinary Dipper-James polytabloid module P(λ)=z λ H and the sign-twisted Murphy cell C M(λ')^θ over the generic type-A…
1 definition, 7 lemmas, 2 theorems, 1 corollaryExamples & counterexamples →This page builds the cyclotomic quotient of a quiver Hecke (KLR) algebra at a dominant integral weight and proves the highest-weight categorification statements attached to it.
4 definitions, 21 lemmas, 9 theoremsExamples & counterexamples →This page develops the cellular-algebra foundations of the modular representation theory of the type-A Hecke algebra and its q-Schur algebra.
12 definitions, 14 lemmas, 1 proposition, 8 theorems, 2 corollaries, 1 remarkExamples & counterexamples →This page develops the type-A cyclotomic Hecke algebras in the nondegenerate and degenerate branches, their charged finite-quantum-characteristic specializations, the central…
6 definitions, 26 lemmas, 1 proposition, 7 theorems, 1 corollaryExamples & counterexamples →This page records the nondegenerate level-one form of Ariki's categorification theorem transported across the blockwise Hecke--KLR isomorphism, together with the graded…
7 definitions, 12 lemmas, 2 propositions, 7 theorems, 1 corollary, 1 remarkExamples & counterexamples →This page records the degenerate analogue of Ariki's categorification theorem: the finite-dimensional representation theory of the degenerate cyclotomic Hecke tower H^Λ at…
1 definition, 1 proposition, 4 theorems, 1 remarkExamples & counterexamples →