Representation of equivariant maps

Question

Let $n,m,k$ be positive integers. Consider the action of symmetric group $S^n$ on $\mathbb{R}^{n\times i}$ (for $i\in \{m,k\}$) by permuting rows; i.e. for each $\pi\in S^n$ and every $n\times i$ matrix $X=(x_1,\dots,x_n)^{\top}=(x_i)_{i=1}^{n\,\top}$ we have $$ \pi\cdot X := (x_{\pi(i)})_{i=1}^{n\,\top}. $$ We say that a function $f:\mathbb{R}^{n\times m}\to \mathbb{R}^{n\times k}$ is equivariant (with respect to this action) if: for all $\pi\in S^n$ and every $X\in \mathbb{R}^{n\times m}$ we have $$ f(\pi\cdot X) = \pi\cdot f(X). $$

Is it the case then that $$ f(X) = (g(x_i))_{i=1}^{n,\top} $$ for some function $g\in \mathbb{R}^k\to \mathbb{R}^m$?

Answer 1

Consider $$ (|\det(X)| g(x_i))_{i=1}^{n, \top}. $$

General theorem in this direction can be found e.g. in section 7 of Michor's Topics in differential geometry. The space of equivariant maps is a module over the algebra of $G$-invariant functions.

Theorem 7.13: Let $G$ be a compact Lie group with a finite-dimensional representation $V$ and $\rho_1, \ldots, \rho_k$ be the generators for the algebra $\mathbb{R}^G$ of $G$-invariant polynomials on $V$. If $$ \rho = (\rho_1, \ldots, \rho_k)\colon V \to \mathbb{R}^k, $$ then $$ \rho^*\colon \mathcal{C}^\infty(\mathbb{R}^k) \to \mathcal{C}^\infty(V)^G $$ is surjective and admits a continuous section.

So any smooth $G$-invariant function if expressible as a smooth function in $k$ variables composed with $\rho$.