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Symmetric functions

By Cali G , Published on Mon 13 January 2020
Category: math / symmetric functions

Background

Symmetric functions are polynomials such that when we permute the indices of the polynomial we get the same polynomial. Symmetric functions have some interesting combinatorics which will be looked out throughout this blog, but for now let's look at the exact definition.

Suppose we have a set $X = \left\{x_1,\, x_2,\, x_3,\, \ldots\right\}$ of infinitely many variables. For some ring $\mbbK$ let $\mbbK\left[X\right]$ be the ring of all polynomials with variables in $X$ and coefficients in $\mbbK$.

Definitions: Let $\sigma$ be a permutation in $\mfS_n$ and $P = x_{i_1} x_{i_2} \ldots x_{i_k}$ be a monomial in $\mbbK\left[X\right]$. We can apply $\sigma$ to the indices of $P$ in the following way: and we call this a permutation of the indices of $P$. A symmetric function $P$ is a polynomial in $\mbbK\left[X\right]$ such that any permutation of the indices keeps $P$ invariant. The set of all symmetric functions together with normal addition and multiplication give us the ring of symmetric functions in variables $X$ which we denote by $\Lambda$.

We can further look at vector subspaces of $\Lambda$. Let $n \in \Z_{\geq 0}$. Then $\Lambda^n$ is the vector subspace of $\Lambda$ whose symmetric functions are functions of degree $n$.

Bases of $\mbox{\Huge\Lambda}$

There are many different bases that can be associated to $\Lambda$. These are normally done through partitions. Recall that a partition $\lambda = (\lambda_1, \lambda_2, \ldots)$ of $n$ is such that $\lambda_i \geq \lambda_{i+1}$. A lot of times, we use Young diagrams to represent a partition.

Monomial Symmetric Functions

Definition: A monomial symmetric function is a symmetric function $m_\lambda(X)$ in $\Lambda$ such that: If there is no ambiguity for $X$, we let $m_{\lambda}$ mean $m_{\lambda}(X)$.

Let us look at some monomial symmetric functions:

Elementary Symmetric Functions

Definition: An elementary symmetric function is the symmetric function $e_n = m_{(1^n)}$.

It turns out there is a nice way to reformulate the elementary symmetric functions

Theorem Let $z$ be some variable.

Proof The proof for this is pretty simple:

Power Symmetric Functions

Definition: A power symmetric function is the symmetric function $p_n = m_{(n)}$.

As an example

Complete Homogeneous Symmetric Functions

Definition: A complete homogeneous symmetric function is the symmetric function $h_n = \sum_{\lambda \vdash n} m_\lambda$. If $\lambda = (\lambda_1, \lambda_2, \ldots)$ is a partition of $n$ then $h_\lambda$ is defined to be the complete homogeneous symmetric function:

As an example, let's look at $h_3$.

Theorem Let $z$ be some variable.

Proof By recalling that $\frac{1}{1-ax} = \sum_{n \geq 0} (ax)^n$ the proof is easy to show.