\(\mathrm{SL}_2\)-factorizations of Chevalley groups. (Q354785)

The authors' aim is to obtain short factorizations of Chevalley groups over fields and nice rings in terms of \(\mathrm{SL}(2)\)'s. In this paper, they prove that over any field \(K\), and for any irreducible, reduced root system \(\Phi\), the simply connected Chevalley group \(G(\Phi,K)\) is the product of at most \(3|\Phi^+|\) copies of the fundamental \(\mathrm{SL}(2,K)\). The proof is very simple and, apart from Bruhat decomposition, it depends on the following simple, but key, observation: Let \(T=\{t_1,t_2,\dots,t_N\}\) be the set of root reflections of the Weyl group \(W=W(\Phi)\), written in an arbitrary (but fixed) order. Then, each element \(w\in W\) can be expressed as a subword of \(t_1t_2\cdots t_N\). In the case of Bezout domains \(R\) (integral domains where every finitely generated ideal is principal), the authors show that \(\mathrm{SL}(n,R)\) is the product of at most \(2|\Phi^+|=n^2-n\) copies of the fundamental \(\mathrm{SL}(2)\). They believe that similar results hold for all Chevalley groups including the twisted ones. But, the proof is expected to be technically much harder for the twisted cases. In the last few years, proofs of many results of this type seem to have got simpler!