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.