On the Borel Submonoid of a Symplectic Monoid

Abstract

In this article, we study the Bruhat-Chevalley-Renner order on the complex symplectic monoid M Sp n . After showing that this order is completely determined by the Bruhat-Chevalley-Renner order on the linear algebraic monoid of n × n matrices M n , we focus on the Borel submonoid of M Sp 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 Sp n . We show that, contrary to the case of M Sp n , the nilpotent subsemigroup of the Borel submonoid of M n is irreducible.