7 - Symmetric Functions

Summary

Symmetric Functions in General

The theory of symmetric functions has many applications to enumerative combinatorics, as well as to such other branches of mathematics as group theory, Lie algebras, and algebraic geometry. Our aim in this chapter is to develop the basic combinatorial properties of symmetric functions; the connections with algebra will only be hinted at in Sections 7.18 and 7.24, Appendix 2, and in some exercises.

Let x = (x₁, x₂,…) be a set of indeterminates, and let n ∈ ℕ. A homogeneous symmetric function of degree n over a commutative ring R (with identity) is a formal power series

where (a) α ranges over all weak compositions α = (α₁, α₂, …) of n (of infinite length), (b) c_α ∈ R, (c) x^α stands for the monomial x^α1₁, x^α2₂,… and (d) f (x_ω(1)x_ω(2), …) = f (x₁, x₂,…) for every permutation ω of the positive integers ℙ. (A symmetric function of degree 0 is just an element of R.) Note that the term “symmetric function” is something of a misnomer; f(x) is not regarded as a function but rather as a formal power series. Nevertheless, for historical reasons we adhere to the above terminology.

The set of all homogeneous symmetric functions of degree n over R is denoted Λⁿ_R. Clearly if f, g ∈ Λⁿ_R and a, b ∈ R, then a f + b g ∈ Λⁿ_R in other words, Λⁿ_R is an R-module. For our purposes it will suffice to take R = ℚ (or sometimes ℚ with some indeterminates adjoined), so Λⁿ_ℚ is a ℚ-vector space.