Post 18: Symmetric Functions

27 May 2025

In this post, I’ll discuss symmetric functions. This post will lay the groundwork for future posts on symmetric functions in other areas of mathematics.

Introduction

Throughout the past (academic) year, I have become aware of the wide range of uses of symmetric functions in representation theory and K-theory. My goal is to share some of these, without getting into too much detail. However, before I can discuss the applications, I must lay the groundwork. That is what I will do in this post.

I have tried to write most of this in as elementary a manner as possible. Someone who knows some basics about groups and rings should certainly understand most of what I say. However, towards the end, I unfortunately have to use some more “fancy” stuff – namely graded rings and inverse limits – in an essential way. I have attempted to explain these notions.

Symmetric Polynomials

I will begin by remarking that I will not use “symmetric function” and “symmetric polynomial” interchangeably. I will justify this later. I must first discuss symmetric polynomials.

Fix a (commutative unital) ring $R$, which will serve as the coefficient ring for our polynomials/functions. I will usually be thinking of $R=\Z$, but other coefficient rings are needed later on. As a side note, any time I say “ring”, I will mean “commutative unital ring”.

Given a polynomial $f(x_1,\dots,x_n)$ in $n$ variables, we can obtain new polynomials by swapping the variables around. If I get the same polynomial no matter how I swap the variables around, then the polynomial is said to be symmetric. In other words, the symmetric group $S_n$ acts on $R[x_1,\dots,x_n]$ by permuting variables, and the fixed points of this action are the symmetric polynomials. Symmetric polynomials in $n$ variables form a ring, denoted $\Lambda_{n,R}$ or $R\otimes \Lambda_n$. For simplicity, I will just denote these by $\Lambda_n$.

Elementary Symmetric Polynomials

One of the most fundamental families of symmetric polynomials are the so-called elementary symmetric polynomials $e_i$. To avoid confusion, I will temporarily use the notation $e_i^{(n)}$ to denote the $i$th elementary symmetric function in $n$ variables. There are $n$ of these. The first one is $e_1^{(n)}(x_1,\dots,x_n)=x_1+\cdots+x_n$. As demonstrated above, the fact that this is symmetric essentially comes down to commutativity of addition. Similarly, the last one is $e_n^{(n)}(x_1,\dots,x_n)=x_1\cdots x_n$, which is symmetric by commutativity of multiplication. The other elementary symmetric polynomials are sort of interpolations of adding and multiplying the variables in all possible combinations. Explicitly, $e_k^{(n)}(x_1,\dots,x_n)$ is the sum of all products that take $k$ distinct variables at a time. For instance, we have $e_2^{(3)}(x_1,x_2,x_3)=x_1x_2+x_1x_3+x_2x_3$.

One may recognize these elementary symmetric polynomials from the context of Vieta’s formulas, which relate the roots of a polynomial to its coefficients. In particular, $(t+x_1)\cdots(t+x_n)=t^n+e_1^{(n)}(x_1,\dots,x_n)t^{n-1}+\cdots+e_n^{(n)}(x_1,\dots,x_n)$. Note that the expression $(t+x_1)\cdots(t+x_n)$ is invariant under permutations of the $x_i$, so the coefficients of its expansion into powers of $t$ must also be invariant under permutations of the $x_i$. This gives a nice way to define the elementary symmetric polynomials that immediately implies they are symmetric.

One of the most fundamental results in this subject, often dubbed the “fundamental theorem of symmetric polynomials”, is that any symmetric polynomial in $n$ variables can be expressed uniquely as a polynomial in the $e_1^{(n)},\dots,e_n^{(n)}$. In particular, $\Lambda_n\cong R[e_1^{(n)},\dots,e_n^{(n)}]$. For example, the symmetric polynomial $x_1^2+x_2^2$ in two variables can be written as $e_1^{(2)}(x_1,x_2)^2-2e_2^{(2)}(x_1,x_2)$. I think this is a bit surprising: we took the invariants of $S_n$ acting on $R[x_1,\dots,x_n]$ and ended up with an isomorphic ring.

The Role of the Number of Variables

That superscript notation on the elementary symmetric polynomials is certainly cumbersome, eh? Thankfully, the elementary symmetric functions satisfy a nice “consistency” property: if $k\le n$, then $e_k^{(n+1)}(x_1,\cdots,x_n,0)=e_k^{(n)}(x_1,\cdots,x_n)$, and $e_{n+1}^{(n+1)}(x_1,\cdots,x_n,0)=0$. The second condition allows us to say $e_k^{(n)}=0$ for $k>n$. You can prove these by using the expansion of $(t+x_1)\cdots(t+x_{n+1})$ and $(t+x_1)\cdots(t+x_n)$. This consistency allows us to rather safely drop the superscript; if I write $e_4$, but you and I choose different number of variables to use, we can still ultimately agree by setting some variables to 0 (effectively getting rid of them).

On first glance, that last sentence I said might seem… “bad”. You might think “surely you lose information when you do that”. However, all of the important symmetric polynomial identities do not depend on the number of variables. For example, recall the earlier example I gave: $x_1^2+x_2^2=e_1^{(2)}(x_1,x_2)^2-2e_2^{(2)}(x_1,x_2)$. This generalizes to the identity $p_2 = e_1^2-2e_2$, where $p_2(x_1,\dots,x_n)=x_1^2+\cdots+x_n^2$ for any number of variables (this also has the “consistency” property).

One interesting phenomenon that appears here is that any given $e_k$, despite technically making sense for any number of variables, only “contributes” something non-trivial when the number of variables is at least $k$. In particular, if you have $n$ variables, then you can’t “access” any $e_k$ for $k>n$. Maybe you add in a few more variables, but there will still be infinitely many $e_k$ that are “out of reach”.

You might think that we can remedy this issue by taking infinitely many variables, and while you would be on the right track, it’s not as simple as it sounds. Certainly, $R[x_1,x_2,\dots]$ is a perfectly valid ring, but… $e_1 = x_1+x_2+\cdots$ would not be an element of this ring! Polynomials, by definition, are finite sums. In fact, any element of $R[x_1,x_2,\dots]$ necessarily only uses finitely many variables, so it cannot be symmetric. In order to properly get a handle of an infinite number of variables, we need the notion of an inverse limit.

Symmetric Functions

I will now define the ring $\Lambda$ of symmetric functions. There are two ways to do this, and I will discuss both. I note now that, for example, $e_1=x_1+x_2+\cdots$ will be an element of this ring, and since this is not a polynomial, we instead call it a function. The definitions of $\Lambda$ will use the notions of graded rings and inverse limits. If you do not know these, please read the appendix to this post first.

Both definitions of $\Lambda$ make use of the “restriction” ring homomorphisms $\rho_{m,n}:\Lambda_m\to\Lambda_n$ for $m\ge n$, defined by setting $x_{n+1},\dots,x_m$ to $0$. We saw this used when discussing the consistency property of the $e_i$. The rings $\Lambda_m$ are graded rings, with the natural grading by degree of polynomials. The $\rho_{m,n}$ respect the grading, and give an inverse system of graded rings. We can then define $\Lambda$ as the inverse limit of this system, in the category of graded rings. Somewhat surprisingly, if you take the inverse limit in the category of rings, you do not get the same ring – check that $\prod_{i=1}^\infty(1+x_i)$ is in the inverse limit in the category of rings, but not in $\Lambda$.

Now for the second definition, let $\Lambda_n^{(k)}$ be the subgroup of homogeneous degree $k$ polynomials in $\Lambda_n$. The graded ring homomorphisms $\rho_{m,n}$ restrict to group homomorphisms $\rho_{m,n}^{(k)}:\Lambda_m^{(k)}\to\Lambda_n^{(k)}$, and these form an inverse system of abelian groups. Let $\Lambda^{(k)}$ be the inverse limit of this system, and then let $\Lambda = \bigoplus_{k=1}^\infty \Lambda^{(k)}$. A priori, this does not have a ring structure. Let $\rho_n^{(k)}:\Lambda^{(k)}\to \Lambda_n^{(k)}$ be the projection maps, and let $\rho_n=\bigoplus_{k=1}^\infty \rho_n^{(k)}:\Lambda\to\Lambda_n$. Then there is a unique graded ring structure on $\Lambda$ such that the $\rho_n$ are ring homomorphisms. Explicitly, if $f\in\Lambda^{(j)}$ and $g\in\Lambda^{(k)}$, then we define $fg\in\Lambda^{(j+k)}$ by specifying its projections to the $\Lambda^{(j+k)}_n$; namely, $\rho_n^{(j+k)}(fg)=\rho_n^{(j)}(f)\rho_n^{(k)}(g)$. Then we can multiply elements of $\Lambda$ by extending the multiplciations on the $\Lambda^{(k)}$ linearly. Check that this gives the same ring as before.

So what is an element $f$ of $\Lambda$, i.e. what is a symmetric function? It is a sequence of symmetric polynomials $f_n\in\Lambda_n$ such that each $f_{n+1}$ reduces to $f_n$ upon setting $x_{n+1}=0$. Thus, all the elementary symmetric polynomials give elementary symmetric functions. Similarly, the polynomials $p_2(x_1,\dots,x_n)=x_1^2+\cdots+x_n^2$ give an element $p_2\in\Lambda$, and the identity $p_2=e_1^2-2e_2$ holds in $\Lambda$. In fact, just as $\Lambda_n\cong R[e_1,\dots,e_n]$ for all $n$, we have $\Lambda_n\cong R[e_1,e_2,\dots]$.

Appendix

I will now explain the notions of graded rings and inverse limits.

Graded Rings

A (non-negatively) graded ring is a ring $S$ with a direct sum decomposition $S=\bigoplus_{i=0}^\infty S^{(i)}$, where the $S^{(i)}$ are subgroups of $S$ with respect to addition, and where $s_is_j\in S^{(i+j)}$ for $s_i\in S^{(i)}$ and $s_j\in S^{(j)}$. There is also a notion of a graded ring homomorphism. If $S,T$ are graded rings, then a graded ring homomorphism $f:S\to T$ is a ring homomorphism with the additional property that $f(S^{(i)})\subseteq T^{(i)}$ for each $i$.

The basic example of a graded ring, luckily for us, is that of a polynomial ring. $S=R[x_1,\dots,x_n]$ has a natural grading by degree. In other words, $S^{(i)}$ is the additive subgroup consisting of homogeneous polynomials of degree $i$ (and the polynomial $0$, which has ambiguous degree). This induces a grading on $\Lambda_n$; the additive subgroup $\Lambda_n^{(i)}$ consists of homogeneous symmetric polynomials of degree $i$.

Inverse Limits

Suppose we have a sequence of “objects” (e.g. sets, groups, rings) $A_1,A_2,\dots$ and “appropriate” maps (e.g. functions, group homomorphisms, ring homomorphisms) between them $f_{m,n}:A_m\to A_n$ whenever $m\ge n$. Furthermore, suppose that $f_{n,n}$ is the identity map, and whenever $k\ge m\ge n$, we have $f_{m,n}\circ f_{k,m}=f_{k,n}$. The second equation can be thought of as saying that the maps are “compatible”. We then say that the $A_n$ and the $f_{m,n}$ form an inverse system.

There are two ways to define the inverse limit of this system. The first is very explicit. It is the “object” $A$ whose elements $t$ are sequences of elements $t_n\in A_n$ satisfying the “consistency” or “compatibility” condition that $f_{m,n}(t_m)=t_n$ whenever $m\ge n$. If the “objects” are groups or rings, then this $A$ will have a natural group or ring structure that makes the maps $t\mapsto t_n$ group or ring homomorphisms. In particular, the operations of addition and multiplication are defined on the sequences, e.g. $s+t$ is the sequence of elements $s_n+t_n$.

The second way to define the inverse limit of an inverse system is less explicit, but is more easily generalized to other situations. An inverse limit of the inverse system is an “object” $A$ with maps $f_n:A\to A_n$ satisfying the following two conditions. The first condition is that $f_{m,n}\circ f_m = f_n$ whenever $m\ge n$. This is a “compatibility” condition. The second condition is what’s known as a universal property. If $B$ is another “object” with maps $g_n:B\to A_n$ such that $f_{m,n}\circ g_m = g_n$ for $m\ge n$, then there is a unique map $g:B\to A$ such that $g_n = f_n\circ g$. This condition shows that an inverse limit is “unique up to unique isomorphism”, which lets us say “the” inverse limit instead of “an” inverse limit. As an exercise, check that the explicitly defined inverse limit above satisfies this universal property.