Abstract
Let S be an irreducible linear algebraic semigroup over an algebraically closed field k. We analyze the Rees theorem for a regular \(\mathcal{J}\)-class J of S. We define the support of J to be \(\mathbb{X} = J/\mathcal{H}\). We show that \(\mathbb{X}\) is a quasi-projective variety that is isomorphic to the direct product of two geometric quotients of algebraic group actions. When J is completely simple, \(\mathbb{X}\) is an affine variety, while in reductive monoids, \(\mathbb{X}\) is always a projective variety. We define the support of S to be that of the maximum regular \(\mathcal{J}\)-class of S. We study closed irreducible regular subsemigroups S of the full linear monoid M n (k) with projective supports \(\mathbb{X}\). We determine the possible \(\mathbb{X}\) and for a given \(\mathbb{X}\), all the possible S. Along the way, we pose some open problems, the chief among which is the conjecture that any irreducible regular linear algebraic semigroup S with zero has projective support. We prove this in the simplest case of when S is a completely 0-simple semigroup.
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Acknowledgements
I would like to thank Lex Renner for many useful discussions on topics related to this paper. I would also like to than Michel Brion and the referee for their suggestions. Finally I would like to thank Zhenheng Li, Mahir Can, Ben Steinberg and Qiang Wang for organizing this workshop.
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Putcha, M.S. (2014). Rees Theorem and Quotients in Linear Algebraic Semigroups. In: Can, M., Li, Z., Steinberg, B., Wang, Q. (eds) Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics. Fields Institute Communications, vol 71. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0938-4_3
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DOI: https://doi.org/10.1007/978-1-4939-0938-4_3
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