The Schur functions can also be defined algebraically in terms of the **antisymmetric functions**. A polynomial function $f(x_1,\ldots,x_n)$ is *antisymmetric* if interchanging two of the variables results in the *negative* of the function. In general, then, a permutation of the variables will change the function by the sign of the permutation.

For instance, $3x^2y-3y^2x$ is antisymmetric, since interchanging $x$ and $y$ negates the function.

In three variables, the function

$$x^4y^2z-x^4z^2y-y^4x^2z+y^4z^2x+z^4x^2y-z^4y^2x$$ is also antisymmetric. This is an example of a **monomial antisymmetric function**, constructed by starting with a monomial, in this case $x^4y^2z$, and adding to it every monomial formed by permuting its variables and multiplying by the sign of the permutation. We let $a_\nu$ denote the monomial antisymmetric function with the partition $\nu$ as its sequence of exponents; the function above is $a_{(4,2,1)}$.

Now, suppose we started with the monomial $x^2y^2z$. Because of the repeating exponent $2$, this term cancels with the term $-y^2x^2z$, and so on, leaving $a_{(2,2,1)}=0$. In general, in order for $a_{\nu}$ to be nonzero, it cannot have any repeating exponents, including exponents of $0$.

Thus, the nonzero monomial antisymmetric functions $a_\nu$ are precisely those for which $\nu=\lambda+\delta$ where $\lambda$ is any partition and $\delta=(n-1,n-2,\ldots,1,0)$. (For instance, $(4,2,1)=(2,1,1)+(2,1,0)$.) And, just as with the monomial symmetric functions, it is easy to see that every antisymmetric function can be written as a linear combination of $a_{\lambda+\delta}$`s.

Let’s consider the simplest of these, $a_{\delta}$, and say we are working with three variables $x,y,z$. Then

$$a_\delta=x^2y-x^2z-y^2x+y^2z+z^2x-z^2y=(x-y)(x-z)(y-z).$$ Notice the nice factorization! In general, $a_\delta$ in $n$ variables $x_1,\ldots,x_n$ factors as $\prod_{i<j}(x_i-x_j)$. (It is a fun challenge to prove this – it is related to the Vandermonde determinant.)

Because of this factorization, $a_\delta$ divides every antisymmetric polynomial: Given any antisymmetric function $f$, it must be divisible by $x_i-x_j$, because setting $x_i=x_j$ makes the polynomial identically zero. So by symmetry, $f$ must be divisible by all of $a_\delta$!

Finally, notice that if we multiply an antisymmetric function $a$ by a symmetric function $s$, the resulting function $sa$ is also antisymmetric. Similarly, the quotient of two antisymmetric functions, if it is a polynomial, is symmetric. We can therefore take the natural monomial basis of the antisymmetric functions, $\{a_{\lambda+\delta}\}$, and turn it into a basis of the symmetric functions simply by dividing by $a_{\delta}$.

So, we define:

$$s_\lambda=\frac{a_{\lambda+\delta}}{a_{\delta}}$$ And indeed, these are the same Schur functions we saw on the previous page! They are algebraically natural, fairly easy to describe combinatorially, and are essential to representation theory. They are not an obvious basis of the symmetric functions, but they are a useful basis nontheless.

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. ðŸ™‚

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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