Cell Decompositions of the Special Orthogonal Algebraic Monoids

Abstract

Let MSO _n (n is even) be the special orthogonal algebraic monoid, B a Borel subgroup of its unit group. We take a more direct definition of cells for MSO _n in terms of B × B-orbits and explicitly determine the cell decompositions of MSO _n . It turns out that each cell is the intersection of MSO _n with a cell of M _n (K), the monoid of all n×n matrices over an algebraically closed field K.

Keywords:

• Algebraic monoids
• Cell decompositions
• Renner monoids
• One-parameter subgroups
• G_m actions

Acknowledgment

The author thanks Professor Lex Renner for his advice. Supported by Lex Renner's grant from NSERC.