The Renner Monoids and Cell Decompositions of the Classical Algebraic Monoids

Author

Zhenheng Li, University of Western Ontario

Date of Award

2001

Degree Type

Thesis

Degree Name

Doctor of Philosophy

Program

Mathematics

Supervisor

Lea Rennor

Abstract

The Renner monoids, cross section lattices and cell decompositions of the classical algebraic monoids are studied.

The Renner monoid is extremely important in the theory of reductive algebraic monoids. It is well know that the Renner monoid [Special characters omitted.] of M_n (K ) is the monoid of all zero-one matrices which have at most one entry equal to one in each row and column, i.e., [Special characters omitted.] consists of injective partial maps on a set of n elements. We obtain that the Renner monoids of the symplectic algebraic monoids and special orthogonal algebraic monoids turn out to be submonoids of [Special characters omitted.] consisting of symplectic and special orthogonal 1-1 partial maps, respectively. The cardinalities of the Renner monoids are obtained, as well.

The cross section lattice is another very important concept in the theory of irreducible algebraic monoids. The cross section lattices of the symplectic and special orthogonal algebraic monoids are explicitly characterized.

The cell decompositions of symplectic algebraic monoids and special orthogonal monoids are explicitly determined. Each cell here turns out to be an intersection of the monoid with some cell of M_n ( K ).

Recommended Citation

Li, Zhenheng, "The Renner Monoids and Cell Decompositions of the Classical Algebraic Monoids" (2001). Digitized Theses. 3206.
https://ir.lib.uwo.ca/digitizedtheses/3206