# Yet another definition of the Schur functions

A polynomial representation of $GL_n(\mathbb{C})$ is a homomorphism to $GL_m(\mathbb{C})$ for some $m$ such that each coordinate function is a polynomial function of the $n^2$ entries of the original matrices. For instance, the determinant representation, which sends each matrix $$X=\left(x_{ij}\right)_{i,j=1}^n$$ to the $1\times 1$ matrix $$\left(\det(X)\right)$$ is a polynomial representation because the determinant is a polynomial in the variables $x_{ij}$.

The character of a polynomial representation, then, is defined to be the trace of the action of the diagonal matrix
$$\left(\begin{array}{cccc} x_1 & 0 & \cdots & 0 \\ 0 & x_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & x_n \end{array}\right).$$ That is, the character is the trace of the image of this matrix in $GL_m$, which must be a polynomial in the variables $x_1,\ldots,x_n$. Magically, these turn out to be the Schur functions in the case of the irreducible representations of $GL_n$.

The irreducible polynomial representations of $GL_n$ can be constructed in several ways. One is an explicit construction similar to that of the Specht module construction for the irreducible representations of $S_n$. The modules in the case of $GL_n(\mathbb{C})$ are called Schur modules, and they can be defined as follows.

First, let $V=\mathbb{C}^n$ be the standard representation of $GL_n$, and consider the induced $GL_n$-module $$\Lambda^{\lambda_1^\ast}V\otimes \Lambda^{\lambda_2^\ast}V \otimes \cdots \otimes \Lambda^{\lambda_l^\ast}V,$$ where $\lambda^\ast$ is the conjugate partition of $\lambda$ and has $l$ rows. We can think of this as the space of Young tableaux $T$ of shape $\lambda$ with entries in $V$, such that, if $T’$ is formed from $T$ by interchanging two elements of the same column, then $T=-T’$.

Given a Young tableau of shape $\lambda$ whose box labels are elements of $V$, a column exchange is given by choosing two columns, choosing $k$ entries from each column for some $k$, and interchanging them, preserving their vertical ordering. For instance, we show below a transfer of two entries chosen from each of the first and second columns:

Finally, define the Schur module $V^\lambda$ to be the quotient
$$V^\lambda=\Lambda^{\lambda_1^\ast}V\otimes \Lambda^{\lambda_2^\ast}V \otimes \cdots \otimes \Lambda^{\lambda_l^\ast}V/Q,$$ where $Q$ is the submodule generated by the relations $T-\sum U$, where $U$ ranges over the tableaux formed from $T$ by all possible exchanges between two fixed columns in $T$, where the boxes on the right column are fixed as well. Then this module $V^\lambda$, and all other such modules, turn out to be precisely the irreducible polynomial representations of $GL_n$!

It takes some work to prove this and show that the Schur functions are their characters. Nevertheless, I feel that this particular construction of the Schur modules $V^\lambda$ gives insight into how the Schur functions are related, given their combinatorial definition in terms of tableaux.