💁🏽Algebraic Combinatorics Unit 8 – Combinatorial Representation Theory

Key Concepts and Definitions

• Combinatorics studies discrete structures, arrangements, and enumerations
• Representation theory examines abstract algebraic structures by representing their elements as linear transformations of vector spaces
• Partitions are ways of writing a positive integer as a sum of positive integers, ignoring the order of summands
• Young diagrams (Ferrers diagrams) visualize partitions as left-justified arrays of boxes
• Symmetric groups consist of all permutations of a given set, with the group operation being composition
  • Denoted as $S_n$ for a set of size $n$
• Tableaux are numberings of the boxes in a Young diagram that are weakly increasing across rows and strictly increasing down columns
• Characters are functions that assign values to elements of a group, preserving certain properties of the group structure

Fundamental Principles

• Combinatorial representation theory unites combinatorics and representation theory, studying how combinatorial objects can represent algebraic structures
• Fundamental principles include:
  • Examining the action of algebraic structures on combinatorial objects
  • Constructing representations of algebraic structures using combinatorial tools
  • Deriving combinatorial results from representation-theoretic arguments
• Combinatorial objects often have natural symmetries that correspond to algebraic structures
  • Young diagrams are invariant under certain permutations of rows or columns
• Algebraic properties can be translated into combinatorial constraints
  • Multiplication in a group corresponds to combining tableaux in a specific way
• Representation theory provides a powerful lens for understanding the structure and properties of combinatorial objects

Algebraic Structures in Combinatorics

• Groups are fundamental algebraic structures in combinatorics, consisting of a set with a binary operation satisfying certain axioms (closure, associativity, identity, inverses)
• Permutation groups (symmetric groups) are central to combinatorial representation theory
  • Elements are permutations, and the group operation is composition
• Rings generalize groups by adding a second binary operation (usually called multiplication) that distributes over the first operation (addition)
• Algebras are vector spaces equipped with a bilinear multiplication operation
  • Group algebras have basis vectors corresponding to group elements, with multiplication determined by the group operation
• Lie algebras are algebras with an antisymmetric multiplication operation (Lie bracket) satisfying the Jacobi identity
  • Often arise as infinitesimal transformations of geometric objects

Representation Theory Basics

• Representations are functions that assign matrices to elements of an algebraic structure, preserving the structure's operations
  • Matrices must satisfy the same relations as the corresponding elements
• Irreducible representations cannot be decomposed into smaller representations
  • Form the building blocks of all representations
• Characters encode essential information about representations
  • Character of a representation is the trace of its matrices
  • Characters are class functions (constant on conjugacy classes)
• Schur's lemma states that any linear map between irreducible representations is either zero or an isomorphism
  • Implies that irreducible characters are orthonormal under a suitable inner product
• Regular representation has basis vectors corresponding to elements of the structure, with the action given by multiplication
  • Decomposes into irreducible representations

Combinatorial Objects and Their Representations

• Young diagrams and tableaux are fundamental combinatorial objects in representation theory
  • Correspond to representations of symmetric groups
• Permutation modules have basis vectors indexed by tableaux of a given shape
  • Action of the symmetric group permutes the entries of the tableaux
• Specht modules are irreducible representations of symmetric groups
  • Constructed by taking certain linear combinations of tableaux
• Schur functions are symmetric polynomials that serve as characters of irreducible representations of symmetric groups
  • Encode information about the dimensions and multiplicities of representations
• Representations can be constructed using combinatorial algorithms
  • Young symmetrizers project permutation modules onto Specht modules
  • Littlewood-Richardson rule computes tensor products of representations using tableaux

Advanced Techniques and Applications

• Combinatorial representation theory has numerous advanced techniques and applications
• Symmetric functions generalize Schur functions and provide a powerful tool for studying representations
  • Can be defined using various combinatorial objects (tableaux, plane partitions, etc.)
• Hecke algebras are deformations of group algebras that arise in the study of knot invariants and quantum groups
  • Representations are described using analogues of Young diagrams and tableaux
• Crystals are combinatorial objects that encode the structure of representations of Lie algebras
  • Arise in mathematical physics and the study of quantum groups
• Combinatorial Hopf algebras use combinatorial objects to construct Hopf algebras, which have additional structure beyond ordinary algebras
  • Often arise in the study of renormalization in quantum field theory
• Representations of finite groups of Lie type are described using combinatorial objects generalizing Young diagrams and tableaux

Problem-Solving Strategies

• Identify the underlying algebraic structure (group, algebra, etc.) and its relevant properties
• Translate the problem into the language of combinatorial objects (diagrams, tableaux, etc.)
• Use the symmetries and constraints of the combinatorial objects to simplify the problem
  • Exploit the invariance of Young diagrams under permutations of rows or columns
• Construct explicit representations using combinatorial algorithms (Young symmetrizers, Littlewood-Richardson rule, etc.)
• Compute characters and use their properties to gain insights into the structure of representations
  • Orthonormality of irreducible characters can be used to decompose representations
• Relate the combinatorial problem to known results or techniques in representation theory
  • Schur-Weyl duality connects representations of symmetric groups and general linear groups
• Apply representation-theoretic arguments to derive combinatorial identities or enumerate objects

Connections to Other Mathematical Fields

• Combinatorial representation theory has deep connections to various branches of mathematics
• Algebraic geometry: Representations of algebraic groups can be studied using geometric techniques
  • Flag varieties parametrize certain types of representations
• Topology: Representations of braid groups and mapping class groups arise in the study of knots and 3-manifolds
  • Jones polynomial is a knot invariant defined using representations of braid groups
• Mathematical physics: Representations of Lie algebras and quantum groups appear in exactly solvable models and conformal field theory
  • Vertex operators are algebraic objects that encode the structure of certain representations
• Probability theory: Representations of symmetric groups are related to the study of random permutations and partitions
  • Plancherel measure on partitions is a probability distribution that arises naturally from representation theory
• Number theory: Modular forms and automorphic representations are connected to the study of Diophantine equations and arithmetic geometry
  • Combinatorial techniques can be used to construct and study these objects