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!)

Hi Maria,

As I’m sure you’re aware there are other ways of defining the Schur functions. One characterization as that they are the unique integral orthonormal basis of $Lambda$, up to sign. Specifically, ${s_lambda : lambda vdash n}$ is obtained from ${h_lambda : lambda vdash n}$ by applying Gram-Schmidt for $lambda$ in decreasing lexicographic order. As a definition this is clunky, because it relies on properties of the inner product on symmetric functions, but I think it provides an enlightening perspective on the representation theory of $S_n$. The upper-triangular system which expresses the tabloid modules in terms of the (irreducible) Specht modules shows that the characters ${chi_lambda : lambda vdash n}$ of the Specht modules are obtained from the characters ${xi_lambda : lambda vdash n}$ of the tabloid modules by applying Gram-Schmidt for $lambda$ in decreasing lexicographic order. Using this observation we can construct an isometry which takes $xi_lambda$ to $h_lambda$ and $chi_lambda$ to $s_lambda$. All the standard facts about $chi_lambda$ then follow from facts about symmetric functions. This is the approach Fulton takes in his book Young Tableaux. Have fun studying!

Hi Steven,

Indeed! I believe the isometry you’re referring to is the Frobenius map, which I actually intend to post about next weekend. You’re ahead of the game. 🙂

The Hall inner product is a strange one, so I’ve been attempting to explain symmetric function theory without having to go through the tedium of defining it and discussing all its properties. It remains to be seen if I can get through next week’s post without it.

Fulton’s book is a good one – I’ve read it too. These facts are explained in even more detail in Enumerative Combinatorics Vol. 2 and in Bruce Sagan’s book “The Symmetric Group”, if you’re interested in further reading.

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