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Let $𝔽$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,...,V_p$ of $C_p$ defined over $𝔽$. Thus if $V$ is any finite dimensional $C_p$-representation there are non-negative integers $0\le n_1,n_2,...,n_k\le p-1$ such that $V\cong {\oplus }_{i=1}^k{V}_{n_i+1}$. It is also well-known there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of $S{L}_2\left(ℂ\right)$ given by its action on the space $R_d$ of $d$ forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring...

We consider problems in invariant theory related to the classification of four vector subspaces of an $n$-dimensional complex vector space. We use castling techniques to quickly recover results of Howe and Huang on invariants. We further obtain information about principal isotropy groups, equidimensionality and the modules of covariants.