Alphabeta MathSpecial Topics in Representation Theory
This collection develops representation theories whose constructions depend on a particular algebraic family or combinatorial structure. Its symmetric-group branch starts with Young diagrams, tableaux, permutation modules and Specht modules, then develops branching, hook lengths and the Robinson–Schensted correspondence. Symmetric functions and the Frobenius characteristic connect characters to combinatorial bases, while Jucys–Murphy elements and integral Specht modules prepare the modular theory.
The subsequent programme includes skew modules, Littlewood–Richardson and Murnaghan–Nakayama rules, blocks and modular branching, Hecke and quiver Hecke representation theory, Fock spaces and categorification. Representation stability, Kronecker coefficients and asymptotic Young diagrams form a separate branch. Quantum groups and crystal bases, together with Kazhdan–Lusztig and Soergel theory, remain identifiable tracks within this category and retain their source and proof obligations.
General methods for finite, compact and locally compact groups belong to Representation Theory of Groups. Braid Groups remains a separate category. Hopf Algebras & Hecke Algebras develops the foundational algebraic structures, while the specialized applications here keep their established page identities. Draft and planned entries describe future work and do not certify completed proofs.
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.
Part 1 · Symmetric Groups and Their Representation Theory
9 pagesBegin with partitions, tableaux and ordinary Specht modules. Branching and the hook formula prepare the character dictionary; symmetric functions, seminormal forms and integral Specht modules develop the specialized theory. The asymptotic page remains a draft.
This page sets up the combinatorial and permutation-module apparatus of the symmetric group.
7 definitions, 5 lemmasExamples & counterexamples →This page constructs the Specht module S^λ inside the tabloid permutation module by applying the signed column antisymmetrizer to tabloids.
3 definitions, 8 lemmas, 5 theorems, 1 corollaryExamples & counterexamples →This page develops the stable graded ring of symmetric functions from compatible finite-rank specializations.
6 definitions, 1 lemma, 2 propositions, 6 theorems, 1 corollaryExamples & counterexamples →This page develops the integral and modular theory of Specht modules for the symmetric group S n over a fixed splitting p-modular system (K,O,k).
4 definitions, 4 lemmas, 1 proposition, 4 theorems, 1 remarkExamples & counterexamples →- The Branching Rule and the Young Graph20 results
This page develops the branching rule for the symmetric group, its bookkeeping by the Young graph, Young's rule for permutation modules, and the Schur–Weyl decomposition of a tensor power.
4 definitions, 9 lemmas, 5 theorems, 2 corollariesExamples & counterexamples → This page proves the hook length formula and builds the Robinson-Schensted correspondence on the combinatorial base of young-diagrams-tableaux-and-permutation-modules.
4 definitions, 9 lemmas, 4 theorems, 3 corollariesExamples & counterexamples →This page builds the classical dictionary between the ordinary character theory of the symmetric groups and the ring of symmetric functions.
3 definitions, 4 lemmas, 2 propositions, 2 theorems, 1 corollaryExamples & counterexamples →This page develops the Jucys–Murphy elements X k=∑ j<k(j k) of the symmetric group algebra and the representation theory they generate.
3 definitions, 6 lemmas, 9 theorems, 1 corollaryExamples & counterexamples →This page studies the Plancherel measure on Young diagrams, the asymptotic shape of a typical diagram, and Kerov's central limit theorem for the normalized cycle characters…
7 definitions, 7 lemmas, 4 propositions, 5 theorems, 1 remarkExamples & counterexamples →