💁🏽Algebraic Combinatorics Unit 4 – Symmetric Group and Functions

Key Concepts and Definitions

• Symmetric group $S_n$ the group of all permutations of a set with $n$ elements
• Permutation a bijective function from a set to itself, rearranging the order of elements
• Cycle a permutation that moves elements in a circular fashion (e.g., $(1 2 3)$)
• Transposition a permutation that swaps two elements and fixes the rest (e.g., $(1 2)$)
• Symmetric function a function invariant under permutations of its variables
• Partition a way of writing a positive integer as a sum of positive integers (e.g., $5 = 3 + 2$)
• Monomial symmetric function sum of all distinct monomials obtained by permuting the variables
• Elementary symmetric function sum of all products of $k$ distinct variables

Symmetric Group Fundamentals

• Composition of permutations applying one permutation followed by another
• Identity permutation the permutation that leaves all elements in their original positions
• Inverse permutation a permutation that undoes the effect of another permutation
• Generators a set of permutations that can generate the entire symmetric group through composition
  • Transpositions are generators of the symmetric group
• Order of a permutation the smallest positive integer $k$ such that applying the permutation $k$ times yields the identity
• Parity of a permutation even if it can be expressed as a product of an even number of transpositions, odd otherwise
• Conjugacy classes permutations with the same cycle structure belong to the same conjugacy class

Properties of Symmetric Functions

• Invariance under permutations a symmetric function remains unchanged when the variables are permuted
• Degree the highest total degree of the monomials in a symmetric function
• Homogeneity a symmetric function is homogeneous if all its monomials have the same total degree
• Algebraic independence a set of symmetric functions is algebraically independent if no function can be expressed as a polynomial in the others
• Generating function a formal power series that encodes the coefficients of a symmetric function
• Specialization setting the variables of a symmetric function to specific values
• Stability property the coefficients of a symmetric function stabilize as the number of variables increases

Types of Symmetric Functions

• Power sum symmetric function sum of the $k$-th powers of the variables
• Complete homogeneous symmetric function sum of all monomials of degree $k$
• Schur function a symmetric function that forms a basis for the space of symmetric functions
  • Schur functions are indexed by partitions and have deep connections to representation theory
• Monomial basis the basis of the space of symmetric functions consisting of monomial symmetric functions
• Elementary basis the basis of the space of symmetric functions consisting of elementary symmetric functions
• Forgotten symmetric function a symmetric function that is dual to the elementary symmetric function

Algebraic Structures and Operations

• Ring of symmetric functions the set of all symmetric functions forms a commutative ring under addition and multiplication
• Basis a linearly independent set that spans the space of symmetric functions (e.g., Schur functions)
• Scalar product an inner product on the space of symmetric functions that makes the Schur functions orthonormal
• Plethysm a composition operation on symmetric functions that generalizes substitution of variables
• Kronecker product a bilinear operation on symmetric functions related to the tensor product of representations
• Hopf algebra the ring of symmetric functions has a compatible coalgebra structure, making it a Hopf algebra
• Frobenius characteristic a ring homomorphism that connects the representation theory of the symmetric group to symmetric functions

Applications in Combinatorics

• Partition identities many identities involving partitions can be proved using symmetric functions
• Enumeration of permutations symmetric functions can be used to count permutations with certain properties
• Representation theory of the symmetric group Schur functions are the characters of irreducible representations of $S_n$
• Combinatorial Hopf algebras many combinatorial structures (e.g., graphs, posets) give rise to Hopf algebras related to symmetric functions
• Macdonald polynomials a generalization of Schur functions with connections to combinatorics and representation theory
• Symmetric function analogs many combinatorial objects have symmetric function analogs (e.g., quasisymmetric functions, noncommutative symmetric functions)

Problem-Solving Techniques

• Generating function methods using generating functions to derive identities and solve enumeration problems
• Algebraic manipulation manipulating symmetric functions using their algebraic properties and relations
• Combinatorial interpretations interpreting coefficients and identities in terms of combinatorial objects
• Representation-theoretic arguments using the connection between symmetric functions and representations of $S_n$
• Specialization arguments setting variables to specific values to obtain identities or evaluate expressions
• Induction proving statements about symmetric functions by induction on the degree or number of variables
• Bijective proofs establishing identities by constructing explicit bijections between combinatorial objects

Connections to Other Areas

• Algebraic geometry symmetric functions appear in the study of flag varieties and Schubert calculus
• Invariant theory the ring of symmetric functions is isomorphic to the ring of polynomial invariants of the general linear group
• Representation theory symmetric functions are deeply connected to the representation theory of the symmetric group and the general linear group
• Quantum groups certain quantum groups have a natural action on the ring of symmetric functions
• Integrable systems some integrable systems (e.g., the KP hierarchy) have solutions that can be expressed in terms of symmetric functions
• Random matrix theory moments of random matrices can be expressed using symmetric functions
• Enumerative combinatorics symmetric functions are a powerful tool for solving enumeration problems in combinatorics