On the Borel Submonoid of a Symplectic Monoid

TL;DR

This paper explores the structure of the Borel submonoid within the complex symplectic monoid, introduces new type B set partitions, and analyzes their properties and counts, revealing differences in nilpotent subsemigroup irreducibility.

Contribution

It establishes the relationship between Bruhat-Chevalley-Renner orders on symplectic and matrix monoids, introduces type B set partitions, and analyzes nilpotent subsemigroups in these contexts.

Findings

The Bruhat-Chevalley-Renner order on $MSp_n$ is determined by that on $M_n$.

New type B set partitions are introduced and counted.

The nilpotent subsemigroup of the Borel submonoid of $M_n$ is irreducible, unlike in $MSp_n$.

Abstract

In this article, we study the Bruhat-Chevalley-Renner order on the complex symplectic monoid $MSp_n$. After showing that this order is completely determined by the Bruhat-Chevalley-Renner order on the linear algebraic monoid of $n\times n$ matrices $M_n$, we focus on the Borel submonoid of $MSp_n$. By using this submonoid, we introduce a new set of type B set partitions. We determine their count by using the ``folding'' and ``unfolding'' operators that we introduce. We show that the Borel submonoid of a rationally smooth reductive monoid with zero is rationally smooth. Finally, we analyze the nilpotent subsemigroups of the Borel semigroups of $M_n$ and $MSp_n$. We show that, contrary to the case of $MSp_n$, the nilpotent subsemigroup of the Borel submonoid of $M_n$ is irreducible.