I continue to be amazed at the multitude of different contexts in which the Schur functions naturally appear.

In a previous post, I defined the Schur symmetric functions combinatorially, via the formula $$s_\lambda=\sum_{|\mu|=|\lambda|}K_{\lambda\mu} m_\mu$$ where $K_{\lambda\mu}$ is the number of semistandard Young tableaux of shape $\lambda$ and content $\mu$ and $m_\mu$ is the monomial symmetric function of shape $\lambda$. I also defined them as the ratio $$s_\lambda=\frac{a_{\lambda+\delta}}{a_\lambda}$$ where $a_\lambda$ is the elementary antisymmetric function.

And, in another post, I pointed out that the Frobenius map sends the irreducible characters of the symmetric group $S_n$ to the Schur functions $s_\lambda$. This can be taken as a definition of the Schur functions:

The Schur functions $s_\lambda$, for $|\lambda|=n$, are the images of the irreducible representations of $S_n$ under the Frobenius map.

Today, I’d like to introduce an equally natural representation-theoretic definition of the Schur functions:

The Schur functions $s_\lambda$, for $\lambda$ having at most $n$ parts, are the characters of the polynomial representations of the general linear group $GL_n(\mathbb{C})$.

I recently read about this in Fulton’s book on Young Tableaux, while preparing to give a talk on symmetric function theory in this term’s new seminar on Macdonald polynomials. Here is a summary of the basic ideas (turn to page 2):