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Consider a rational representation of an algebraic torus $T$ on a vector space $V$. Suppose that $\{f_1,\cdots ,f_p\}$ is a homogeneous minimal generating set for the ring of invariants, $\mathbf{k}[V{]}^T$. New upper bounds are derived for the number $N_{V,T}:=\max \{\mathrm{deg}{f}_i\}$. These bounds are expressed in terms of the volume of the convex hull of the weights of $V$ and other geometric data. Also an algorithm is described for constructing an (essentially unique) partial set of generators $\{f_1,\cdots ,f_s\}$ consisting of monomials and such that $\mathbf{k}[V{]}^T$ is integral over $k[f_1,\cdots ,f_s]$.

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...

The arc graph $\delta \left(G\right)$ of a digraph $G$ is the digraph with the set of arcs of $G$ as vertex-set, where the arcs of $\delta \left(G\right)$ join consecutive arcs of $G$. In 1981, S. Poljak and V. Rödl characterized the chromatic number of $\delta \left(G\right)$ in terms of the chromatic number of $G$ when $G$ is symmetric (i.e., undirected). In contrast, directed graphs with equal chromatic numbers can have arc graphs with distinct chromatic numbers. Even though the arc graph of a symmetric graph is not symmetric, we show that the chromatic number of the...

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.