I came across the following algebraic structure when working on a seemingly unrelated problem and am unable to find a name for it.
Let $R$ be a commutative ring with identity. Given indeterminates $x^{\alpha}_{i},$ where $1 \leq \alpha \leq m,$ $1 \leq i \leq n,$ (Note: $\alpha$ is superscript, not an exponent), consider the $R$-module $W_q$ of of defined by, $$ W_q =\left\{ \sum_{p \leq q} S^{\, i_1 \, i_2 \, \dots i_p}_{\alpha_1\alpha_2 \dots \alpha_p} \, x^{\alpha_1}_{i_1} x^{\alpha_2}_{i_1} \dots x^{\alpha_p}_{i_p} \ \ | \ \ S^{\, i_1 \, i_2 \, \dots i_p}_{\alpha_1\alpha_2 \dots \alpha_p} \in R , \ \text{skew-symm. in lower and upper indices} \right\}, $$ where the Einstein summation convention is in effect here.
Clearly $W_0=R$, $W_1=\{ a + b^i_\alpha x^\alpha_i \ : \ a,b^i_\alpha \in R\}$ and $W_{r}=W_{\max\{m,n\}}$ whenever $r \geq \max\{m,n\}.$
Also, $W_q$ is an $R$-submodule of $R[x^1_1,x^1_2,\dots,x^1_n,x^2_1,x^2_2,\dots, x_{m}^n].$ Moreover, $W_q$ is a submodule of the of the symmetric polynomials.
My specific question is this: Is there a generally accepted name for the set $W_q$, or least an $R$-module isomorphism between $W_q$ and some well recognized structure?