Question concerning a property of polynomial functions on $\Gamma:=\text{GL}_n(K)$ and the Schur algebra

Question

I'm reading Green's book ''Polynomial Representations of GL_n. with an Appendix on Schensted Correspondence and Littelmann Paths''.

Consider theorem 2.4b) part (i) on page 14:

Consider the map $e : K\Gamma \rightarrow S_K(n,r)$, defines as follows: For each $g\in \Gamma$ we define the element $e_g\in S_K(n,r)$ by $e_g(c):=c(g)$ for all $c\in A_K(n,r)$. We extend the map $g \mapsto e_g$ linearly and get a map $e : K\Gamma \rightarrow S_K(n,r)$ which is a morphism of $K$-algebras. Any function $f\in K^{\Gamma}$ has a unique extension to a linear map $f : K\Gamma → K$. With this convention, the image under $e$ of an element $\kappa=\sum \kappa_gg\in K\Gamma$ is ''evaluation at $\kappa$''; i.e. $e(\kappa) : c \mapsto c(\kappa)$, for all $c\in A_K(n,r)$.

Question: Why is $e$ surjective? Assume the contrary. Then I don't know, why there would exist some $0\neq c\in A_K(n,r)$, such that $e_g(c)=c(g)=0\ \forall\ g\in \Gamma$, if $\text{Im}(e)$ were a proper subspace of $S_K(n,r)={A_K(n,r)}^{*}$.

Any hints would be appreciated.

Thanks for the help!

Answer 1

The proposition is mostly a result about finite-dimensional vector spaces, and has very little to do with the Schur algebra itself. Let $V$ be a finite-dimensional vector space over some fixed field $K$, and let $A \subset V^*$ be a proper subspace. Then we can choose a basis $\{v_1, \dots, v_n\}$ of $V$ such that $v_1^*, \dots, v_m^*$ are a basis of $A$ for $m = \dim A$, where the $v_i^*$ are the dual elements $v_i^*(v_j) = \delta_{ij}$. In particular, there exists some nonzero $v\in V$ such that $f(v) = 0$ for all $f\in A$.

Thus if $e(K\Gamma)$ is a proper subspace of $S_K(n, r)$, then we can find some nonzero $f\in A_K(n, r)$ such that $f(g) = 0$ for all $g\in \Gamma$. But $K$ is infinite, so any such $f$ must vanish.