TITLE: Schur polynomial, change of variable
QUESTION [7 upvotes]: Let $k=(k_1,k_2,k_3,k_4)\in \mathbb{N}^4$ and let $s_k(x_1,x_2,x_3,x_4)$ be the Schur polynomial on $GL_4$.
Question 1: If I replace $x_3$ with $x_1$ and $x_4$ with $x_2$, can $s_k(x_1,x_2,x_1,x_2)$ be written as Schur polynomials on $GL_2$ in variables $x_1$ and $x_2$? Is there any clean formula for such reduction?
Question 2: Can $s_k(x_1,x_1^{-1},x_3,x_3^{-1})$ be written in Schur polynomial of $(x_1,x_1^{-1})$ and Schur polynomial of $(x_3,x_3^{-1})$?

REPLY [7 votes]: To expand a bit on the answer of Matt Samuel, let
$x$ and $y$ be two sets of variables (of any
length) and $\lambda$ any partition. Then
  \begin{eqnarray*} s_\lambda(x,y) & = & \sum_{\mu\subseteq\lambda}
  s_\mu(x)s_{\lambda/\mu}(y)\\ & = &
   \sum_{\mu,\nu} c_{\mu\nu}^\lambda s_\mu(x)s_\nu(y).
   \end{eqnarray*}
See for instance equation (7.66) of Enumerative Combinatorics,
vol. 2. Now put $y=x$ to get
  $$ s_\lambda(x,x) = \sum_{\mu,\nu,\rho}c_{\mu\nu}^\lambda
           c_{\mu\nu}^\rho s_\rho(x). $$