The Renner Monoids and Cell Decompositions of the Classical Algebraic Monoids

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 Mn (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 Mn ( K )

Conjugacy Classes of Renner Monoids

In this paper we describe conjugacy classes of a Renner monoid $R$ with unit group $W$, the Weyl group. We show that every element in $R$ is conjugate to an element $ue$ where $u\in W$ and $e$ is an idempotent in a cross section lattice. Denote by $W(e)$ and $W_*(e)$ the centralizer and stabilizer of $e\in \Lambda$ in $W$, respectively. Let $W(e)$ act by conjugation on the set of left cosets of $W_*(e)$ in $W$. We find that $ue$ and $ve$ ($u, v\in W$) are conjugate if and only if $uW_*(e)$ and $vW_*(e)$ are in the same orbit. As consequences, there is a one-to-one correspondence between the conjugacy classes of $R$ and the orbits of this action. We then obtain a formula for calculating the number of conjugacy classes of $R$, and describe in detail the conjugacy classes of the Renner monoid of some $\cal J$-irreducible monoids. We then generalize the Munn conjugacy on a rook monoid to any Renner monoid and show that the Munn conjugacy coincides with the semigroup conjugacy, action conjugacy, and character conjugacy. We also show that the number of inequivalent irreducible representations of $R$ over an algebraically closed field of characteristic zero equals the number of the Munn conjugacy classes in $R$.Comment: A reference ([13]) and Corollary 4.5 are added to show the connection between the result in Theorem 4.4 of the previous version and the results in [13]. A paragraph on page 12 is new to show that Theorem 4.4 can also be deduced from the results in [13]. Two necessary concepts from [13] to describe the connection are inserted in Section 2.

Partial mirror symmetry, lattice presentations and algebraic monoids

This is the second in a series of papers that develops the theory of reflection monoids, motivated by the theory of reflection groups. Reflection monoids were first introduced in arXiv:0812.2789. In this paper we study their presentations as abstract monoids. Along the way we also find general presentations for certain join-semilattices (as monoids under join) which we interpret for two special classes of examples: the face lattices of convex polytopes and the geometric lattices, particularly the intersection lattices of hyperplane arrangements. Another spin-off is a general presentation for the Renner monoid of an algebraic monoid, which we illustrate in the special case of the "classical" algebraic monoids.Comment: 41 page