In the last few posts (see here and here), I’ve been talking about various bases for the symmetric functions: the monomial symmetric functions $m_\lambda$, the elementary symmetric functions $e_\lambda$, the power sum symmetric functions $p_\lambda$, and the homogeneous symmetric functions $h_\lambda$. As some of you aptly pointed out in the comments, there is one more important basis to discuss: the Schur functions!

When I first came across the Schur functions, I had no idea why they were what they were, why every symmetric function can be expressed in terms of them, or why they were useful or interesting. I first saw them defined using a simple, but rather arbitrary-sounding, combinatorial approach:

First, define a **semistandard Young tableau** (SSYT) to be a way of filling in the squares of a partition diagram (Young diagram) with numbers such that they are *nondecreasing* across rows and *strictly increasing* down columns. For instance, the Young diagram of $(5,3,1,1)$ is:

and one possible SSYT of this shape is:

(Fun fact: The plural of tableau is *tableaux*, pronounced exactly the same as the singular, but with an x.)

Now, given a SSYT $T$ with numbers of size at most $n$, let $\alpha_i$ be the number of $i$`s written in the tableau. Given variables $x_1,\ldots,x_n$, we can define the monomial $x^T=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$. Then the **Schur function** $s_\lambda$ is defined to be the sum of all monomials $x^T$ where $T$ is a SSYT of shape $\lambda$.

For instance, if $\lambda=(2,1)$, then the possible SSYT’s of shape $\lambda$ with numbers of size at most $2$ are:

So the Schur function $s_{(2,1)}$ in $2$ variables $x$ and $y$ is $x^2y+y^2x$.

This combinatorial definition seemed rather out-of-the-blue when I first saw it. Even more astonishing is that the Schur functions have an abundance of nice properties. To name a few:

**The Schur functions are symmetric.**Interchanging any two of the variables results in the same polynomial.**The Schur functions form a basis for the symmetric functions,**Like the elementary symmetric functions, every symmetric polynomial can be expressed uniquely as a linear combination of $s_\lambda$`s.**The Schur functions arise as the characters of the irreducible polynomial representations of the general linear group.**This was proven by Isaac Schur and was the first context in which the Schur functions were defined. Here, a polynomial representation is a matrix representation in which the entries are given by polynomials in the entries of the elements of $GL_n$.**The transition matrix between the power sum symmetric functions and the Schur functions is precisely the character table of the symmetric group $S_n$.**This fact is essential in proving the Murnaghan-Nakayama rule that I mentioned in a previous post.

All of this is quite remarkable – but why is it true? It is not even clear that they are symmetric, let alone a basis for the symmetric functions.

After studying the Schur functions for a few weeks, I realized that while this combinatorial definition is very useful for quickly writing down a given $s_\lambda$, there is an equivalent algebraic definition that is perhaps more natural in terms of understanding its role in symmetric function theory. (Turn to page 2!)