On the Borel Submonoid of a Symplectic Monoid

Mahir Bilen Can; Hayden Houser; Corey Wolfe

paper

On the Borel Submonoid of a Symplectic Monoid

arXiv:2006.00929

Abstract

In this article, we study the Bruhat-Chevalley-Renner order on the complex symplectic monoid $M S p_{n}$ . 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 $M S p_{n}$ . 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 $M S p_{n}$ . We show that, contrary to the case of $M S p_{n}$ , the nilpotent subsemigroup of the Borel submonoid of $M_{n}$ is irreducible.