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Mathematics, Algebraic Geometry, Representation Theory
Abstract:
Let $\mathbb X\subset\mathbb P(V)$ be a projective variety, which is not contained in a hyperplane. Then every vector $v$ in $V$ can be written as a sum of vectors from the affine cone $X$ over $\mathbb X$. The minimal number of summands in such a sum is called the rank of $v$. In this paper, we classify all equivariantly embedded homogeneous projective varieties $\mathbb X\subset\mathbb P(V)$ whose rank function is lower semi-continuous. Classical examples are: the variety of rank one matrices (Segre variety with two factors) and the variety of rank one quadratic forms (quadratic Veronese variety). In the general setting, $\mathbb X$ is the orbit in $\mathbb P(V)$ of a highest weight line in an irreducible representation $V$ of a reductive algebraic group $G$. Thus, our result is a list of all irreducible representations of reductive groups, for which the corresponding rank function is lower semi-continuous.
