Let $V:=k[x_0,\ldots,x_n]_d$ be the $k$-vector space of homogeneous polynomials of degree $d$. Let $G:=\mathrm{Gl}(n+1,k)$ act on $V$ induced by the canonical action on the linear forms: For $g=(g_{ij})\in G$, we have $$g.x_j=\sum_{i=0}^n g_{ij} x_i.$$ Now consider the subspace $W:=k[x_1,\ldots,x_j]_d\subseteq V$ of $d$-forms in one less variable. What is the dimension of $G.W$? In particular, is $G.W$ Zariski-dense in $V$? I think this should be well-known, so if someone ran across this before, I'd be glad just to get some pointers to papers or textbooks on this.
Dimension of the GL-orbit of d-forms in one less variable
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representation-theory
lie-groups
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