THE SYMPLECTIC MONOID

摘要

In this paper, we describe explicitly the symplectic monoid M and its Renner monoid R using elementary methods. We refine the Bruhat–Renner decomposition of M and analyze in detail the length function on R. We then show that every element of M has a unique canonical form decomposition, which is an analogue of the canonical form of elements in Chevalley groups. We also compute the order of M over a finite field, and as a consequence we obtain a new combinatorial identity.