Idempotent-conjugate monoids with nilpotent unit groups. (Q2491192)

An important result in elementary linear algebra is that idempotent matrices of the same rank are similar. A monoid \(M\) with unit group \(G\) is called an idempotent-conjugate monoid (or an IC-monoid) if for any \(\mathcal J\)-related idempotents \(e\) and \(f\), there exists \(x\in G\) such that \(xex^{-1}=f\). This conjugacy property is an important ingredient in the author's theory of monoids of Lie type [see \textit{M. S. Putcha}, J. Algebra 120, No. 1, 139-169 (1989; Zbl 0683.20051); ibid. 163, No. 3, 636-662 (1994; Zbl 0801.20051)]. IC-monoids also occur naturally in group representation theory. Of course if \(M\) is an inverse monoid then the complex monoid algebra \(\mathbb{C} M\) is semisimple [see \textit{A. H. Clifford} and \textit{G. B. Preston}, Algebraic theory of semigroups. Vol. I, Math. Surv. 7. Providence: AMS (1961; Zbl 0111.03403)]. For what groups \(G\) is this the only way that the complex monoid algebra \(\mathbb{C} M\) can be semisimple? If \(M\) is a finite IC-monoid with a nilpotent unit group such that \(\mathbb{C} M\) is semisimple then \(M\) is an inverse monoid. Conversely, let \(G\) be a finite group such that for any finite IC-monoid \(M\) with unit group \(G\), \(\mathbb{C} M\) semisimple implies that \(M\) is an inverse monoid. Then \(G\) be a nilpotent group. There are given some examples.