Abstract
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.
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