Conjugacy decomposition of reductive monoids. (Q2574919)

An algebraic monoid \(M\) is a monoid and an affine variety over an algebraically closed field \(K\), for which the product map is a morphism of varieties. An algebraic monoid \(M\) is linear, i.e., \(M\) is isomorphic to a closed submonoid of some \(M_n(K)\). \(M\) is reductive if its unit group \(G\) is so. Let \(T\) be a maximal torus of \(G\). The finite monoid \(R=\overline{N_G(T)}/T\) (\(=E(\overline T)W\)) is referred to as the Renner monoid of \(M\), where \(E(X)=\{x\in X\mid x^2=x\}\) and \(W\) is the Weyl group of \(G\). Thus \(W\) is the unit group of the Renner monoid \(R\). Now assume \(M\) is reductive with \(B\supset T\) a Borel subgroup of \(G\). Then \(M=\coprod_{\sigma\in R}B\sigma B\) (the Bruhat-Renner decomposition) and the Bruhat-Chevalley order extends to \(R\) as \(\sigma\leq\theta\) if \(B\sigma B\subseteq\overline{B\theta B}\). The author defines for each \(\sigma\in R\), \(X(\sigma):=\bigcup_{x\in G}x^{-1}B\sigma Bx\) and \(Y(\sigma):=\overline{X(\sigma)}\). He then defines \(\approx\) and \(\preceq\) on \(R\) by: \(\sigma\approx\sigma'\) if \(Y(\sigma)=Y(\sigma')\); \(\sigma\preceq\sigma'\) if \(Y(\sigma)\subseteq Y(\sigma')\). Then consider the subset of Gauss-Jordan elements in \(R\): \(\text{GJ}:=\{\sigma\in R\mid B\sigma\subseteq\sigma B\}\) and \(\widetilde R=\text{GJ}/\!\!\sim\), where \(\sim\) is the conjugacy relation. Denote by \([\sigma]\) the conjugacy class of \(\sigma\). In \(\text{GJ}\subset R\), it is shown that \(\sigma\sim\theta\Leftrightarrow\sigma\approx\theta\Leftrightarrow X(\sigma)=X(\theta)\), and that \(\sigma\preceq\theta\Rightarrow[\sigma]\leq [\theta]\). The author then shows that \(M=\coprod_{[\sigma]\in\widetilde R}X(\sigma)\), and that if \([\sigma]\in\widetilde R\) then \(Y(\sigma)=\coprod_{[\theta]\leq [\sigma]}X(\theta)\).