THE RENNER MONOIDS AND CELL DECOMPOSITIONS OF THE SYMPLECTIC ALGEBRAIC MONOIDS
2003 ◽
Vol 13
(02)
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pp. 111-132
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Keyword(s):
A Cell
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In this paper we explicitly determine the Renner monoid ℛ and the cross section lattice Λ of the symplectic algebraic monoid MSpn in terms of the Weyl group and the concept of admissible sets; it turns out that ℛ is a submonoid of ℛn, the Renner monoid of the whole matrix monoid Mn, and that Λ is a sublattice of Λn, the cross section lattice of Mn. Cell decompositions in algebraic geometry are usually obtained by the method of [1]. We give a more direct definition of cells for MSpn in terms of the B × B-orbits, where B is a Borel subgroup of the unit group G of MSpn. Each cell turns out to be the intersection of MSpn with a cell of Mn. We also show how to obtain these cells using a carefully chosen one parameter subgroup.