Algebra
Symmetric Functions
Цели урока
- Master the four bases of Lambda: e_k, h_k, p_k, s_lambda and transitions between them
- Understand the Jacobi-Trudi identity: s_lambda = det(h_{lambda_i-i+j})
- Know the Hall inner product and orthonormality of Schur functions
- Apply the Littlewood-Richardson rule for products of s_lambda
Предварительные знания
- Representation theory of Lie algebras
- Young diagrams and partitions
- Linear algebra
Newton in 1707 proves the fundamental theorem on symmetric polynomials. Three hundred years later it governs string theory, knot invariants, and algorithms in enumerative combinatorics. Four numbers - e_k, h_k, p_k, s_lambda - are the vocabulary for describing symmetry in any dimension.
- Particle physics: Littlewood-Richardson numbers compute tensor product decompositions of SU(n)-representations in quark models
- AdS/CFT correspondence: Schur functions appear in one-loop correction formulas in supersymmetric field theory
- Algorithms: computing the number of SYT via hook length formula runs in O(n log n) using Jacobi-Trudi
- Coding theory: symmetric functions over finite fields are used in constructing algebraic-geometry codes
From Newton to Macdonald
Isaac Newton in 1707 proved the representation theorem via elementary symmetric polynomials. Augustin-Louis Cauchy in 1815 introduced Schur functions via the bialternant formula. Issai Schur in 1901 connected them to GL(n) representation theory. Dudley Littlewood and Archibald Richardson in the 1930s discovered the multiplication rule. Ian Macdonald in 1979 wrote the monograph 'Symmetric Functions and Hall Polynomials' - the standard reference for five generations of mathematicians.
Bases of the Ring of Symmetric Functions
1707. Isaac Newton proves the theorem on symmetric polynomials. Any symmetric polynomial in x_1,...,x_n is expressible in e_1,...,e_n. One of the few theorems where the structure is absolutely complete. Three hundred years later Schur functions appear in string theory, quantum chromodynamics, and combinatorics of diagrams.
The four bases of Lambda are four 'languages' for one ring: e_k (squarefree monomials), h_k (all monomials), p_k (power sums), s_lambda (Schur functions). Each is convenient for different tasks. Transitions between them are the core of Macdonald's book.
The Jacobi-Trudi identity for the single-row partition lambda=(k) gives:
Jacobi-Trudi: s_lambda = det(h_{lambda_i-i+j}). For lambda=(k): a 1x1 matrix = (h_k). Determinant = h_k.
Schur Orthogonality and the Hall Inner Product
Schur functions are orthonormal. Not in some vague sense - in a precisely defined inner product. This turns Lambda into an infinite-dimensional Hilbert space (with appropriate topology). And the expansion in Schur functions is a Fourier decomposition in algebra.
Product s_(2) * s_(1)
Computation via the LR rule
s_(2) * s_(1) = s_(3) + s_(2,1). LR numbers: c^{(3)}_{(2),(1)} = 1, c^{(2,1)}_{(2),(1)} = 1. Physically: GL(n)-representation V_{(2)} tensor V_{(1)} = V_{(3)} + V_{(2,1)} (symmetric times fundamental).
Littlewood-Richardson coefficients c^nu_{lambda,mu} are always nonneg. This is not obvious from the algebraic definition. The combinatorial proof via LR words is one of the deep results of algebraic combinatorics.
In the Hall inner product, Schur functions s_lambda:
<s_lambda, s_mu> = delta_{lambda,mu} - the key property of Schur functions. This follows from them being characters of GL(n)-modules and the generalized Weyl character formula.
Schur Functions in Representation Theory and Physics
Schur functions are not abstractions. They are specific numbers, computable algorithmically. And they appear where least expected: in supersymmetry, in topological field theory, in Euler-Poincare numbers of flag manifolds. One object - fifty applications.
| Object | Formula via s_lambda | Application |
|---|---|---|
| Character of GL(n)-module V_lambda | s_lambda(x_1,...,x_n) | Representation theory of GL(n) |
| Decomposition V_lambda tensor V_mu | s_lambda * s_mu = sum c^nu s_nu | Littlewood-Richardson numbers |
| Dimension of V_lambda (GL(n)) | s_lambda(1,...,1) via Weyl formula | Multiplicities in physical models |
| Superspace | s_lambda(x|y) - super Schur | AdS/CFT, supersymmetry |
The Schur function s_lambda(x_1,...,x_n) is the character of:
s_lambda(x_1,...,x_n) = ch(V_lambda^{GL(n)}) - the theorem connecting combinatorics of partitions and representation theory of GL(n).
Plethysm and Specializations
Plethysm is an operation on symmetric functions discovered by Littlewood in 1950. It describes representations arising from composition: if V is a GL(m)-module, then S^k(V) is a GL(m)-module. What is its character? The answer is plethysm s_k[s_lambda].
The specialization s_lambda(x_1,...,x_n) at x_i = 1 for all i gives:
s_lambda(1^n) = dim V_lambda^{GL(n)} via the Weyl formula: product over cells (lambda_i - lambda_j + j - i) / product (j-i).
Connections to other topics
The ring of symmetric functions is a central object connecting algebra, combinatorics, and geometry.
- alg-29-alg-comb — extends
Итоги
- Lambda = Z[e_1,...] = Z[h_1,...] = Q[p_1,...] - three algebraic descriptions of one ring
- Schur functions {s_lambda} form an orthonormal basis of Lambda: <s_lambda, s_mu> = delta_{lambda,mu}
- Jacobi-Trudi: s_lambda = det(h_{lambda_i-i+j}) via h, or s_lambda = det(e_{lambda'_i-i+j}) via e
- LR rule: s_lambda * s_mu = sum c^nu_{lambda,mu} s_nu with nonneg LR coefficients
- Representation theory link: s_lambda(x_1,...,x_n) = ch(V_lambda^{GL(n)})
Вопросы для размышления
- Why does the Jacobi-Trudi identity express the Schur function as a determinant of h_k values?
- How does the orthonormality of Schur functions in the Hall product connect to orthogonality of characters?
- What do the Littlewood-Richardson numbers c^nu_{lambda,mu} mean physically for decomposing SU(3)-representations?
Связанные уроки
- alg-27-lie-repr — Schur functions are characters of GL(n)-representations
- alg-29-alg-comb — RSK and Kostka numbers encode the combinatorics of Schur functions
- alg-26-quantum-groups — Kazhdan-Lusztig polynomials are q-analogues of Schur functions