Alphabeta MathSymmetric Groups and Symmetric Functions
Representations of symmetric groups connect finite-group symmetry with partitions, tableaux and symmetric functions. This collection begins with Young diagrams, permutation modules and Specht modules, then develops the branching rule, the Young graph, hook lengths and the Robinson–Schensted correspondence. Symmetric functions and the Frobenius characteristic translate characters into Schur bases. Outer products and skew Specht modules lead to Littlewood–Richardson coefficients, while rim hooks and the Murnaghan–Nakayama rule compute character values.
Jucys–Murphy elements and integral Specht lattices prepare the modular branch. Cores, quotients and blocks organize characteristic-dependent behavior; the good-node crystal describes modular branching. Internal tensor products lead to Kronecker coefficients, FI-modules and Murnaghan stability. Plancherel measure supplies the asymptotic branch through random Young diagrams. Quantum Groups and Crystal Bases provides the crystal and Fock-space machinery, and Hecke Algebras and Categorification develops the deformed and categorical applications. General finite-group methods remain in Representation Theory of Groups. Each topic has an examples companion, and page and item statuses identify the available content.
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Part 1 · Symmetric Groups and Symmetric Functions
16 pagesYoung diagrams and permutation modules lead to Specht modules, branching and hook lengths. Symmetric functions translate characters into Schur bases; outer products and rim hooks give character rules. Integral lattices, cores and blocks prepare modular branching. Kronecker coefficients, FI-modules and Plancherel measure develop the stable and asymptotic branches. Crystal prerequisites are linked from Quantum Groups and Crystal Bases.
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 develops the modular branching theory of the symmetric groups in the residue convention res(r,c)=c-rmod p and in the p-restricted labelling of the simple modules…
3 definitions, 4 lemmas, 2 propositions, 5 theorems, 1 corollary, 1 remarkExamples & counterexamples →This page develops the induction product and restriction coproduct on the graded character ring of the symmetric groups.
3 definitions, 4 lemmas, 1 proposition, 5 theorems, 2 corollariesExamples & 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 →This page defines the Kronecker coefficients of the symmetric group, transports them to the internal product of symmetric functions, proves the exact character, reduction and…
4 definitions, 3 lemmas, 2 propositions, 1 theorem, 1 corollary, 1 remarkExamples & counterexamples →This page develops the rim-hook (border-strip) calculus of symmetric-group characters and the Murnaghan–Nakayama recursion that computes every character value from it.
1 definition, 4 lemmas, 1 proposition, 3 theorems, 2 corollariesExamples & counterexamples →This page develops the FI-module machinery of Church–Ellenberg–Farb and its character-polynomial consequences over a field of characteristic 0.
8 definitions, 7 lemmas, 4 propositions, 3 theorems, 1 corollaryExamples & counterexamples →This page develops the abacus calculus of partitions and applies it to classify the blocks of the q-Schur algebras in the stable range d≥ n, the type-A Hecke algebras, and the modular symmetric groups by e-cores.
2 definitions, 2 lemmas, 2 propositions, 5 theorems, 1 corollary, 1 remarkExamples & counterexamples →- Murnaghan Stability via FI-Modules15 results
This page proves Murnaghan's stabilization theorem for Kronecker coefficients, develops the reduced coefficients through the lambda-ring and vertex-operator calculus, and…
4 definitions, 6 lemmas, 1 proposition, 4 theoremsExamples & counterexamples →