Simply laced Coxeter groups and groups generated by symplectic transvections (Q5954576)

Let \(N_n^0\) denote the semialgebraic set of all unipotent upper-triangular \(n\times n\) matrices \(x\) with real entries such that, for every \(k=1,\dots,n-1\), the minor of \(x\) with rows \(1,\dots,k\) and columns \(n-k+1,\dots,n\) is nonzero. Then the number \(\#_n\) of connected components of \(N_n^0\) is given as follows: \(\#_2=2\), \(\#_3=6\), \(\#_4=20\), \(\#_5=52\), and \(\#_n=3\cdot 2^{n-1}\) for \(n\geq 6\). These results were obtained by \textit{B.~Shapiro, M.~Shapiro} and \textit{A.~Vainshtein} [in Int. Math. Res. Not. 1997, No. 10, 469-493 (1997; Zbl 0902.14035) and ibid. 1998, No. 11, 563-588 (1998; Zbl 0911.14025)].NEWLINENEWLINENEWLINEIn this paper the authors present the following far-reaching generalization of these results. Let \(W\) be an arbitrary Coxeter group of simply laced type. Let \(u\) and \(v\) be any two elements in \(W\), and let \(\mathbf i\) be a reduced word (of length \(m=l(u)+l(v)\)) for the pair \((u,v)\) in the Coxeter group \(W\times W\). The authors associate to \(\mathbf i\) a subgroup \(\Gamma_{\mathbf i}\) in \(\text{GL}_m(\mathbb{Z})\) generated by symplectic transvections. The authors prove that the subgroups corresponding to different reduced words for the same pair \((u,v)\) are conjugate to each other inside \(\text{GL}_m(\mathbb{Z})\). To recover the group \(\Gamma_n\) from this general construction, one needs several specializations and reductions: take \(W\) to be the symmetric group \(S_n\); take \((u,v)=(w_0,e)\), where \(w_0\) is the longest permutation in \(S_n\) and \(e\) is the identity permutation; take \(\mathbf i\) to be the lexicographically minimal reduced word \(1,2,1,\dots,n-1,n-2,\dots,1\) for \(w_0\); and take the group \(\Gamma_{\mathbf i}(\mathbb{F}_2)\) obtained from \(\Gamma_{\mathbf i}\) by reducing the linear transformations from \(\mathbb{Z}\) to \(\mathbb{F}_2\). The authors generalize the enumeration result [from loc. cit.] by showing that under certain assumptions on \(u\) and \(v\), the number of \(\Gamma_{\mathbf i}(\mathbb{F}_2)\)-orbits in \(\mathbb{F}_2^m\) is equal to \(3\cdot 2^s\), where \(s\) is the number of simple reflections in \(W\) that appear in a reduced decomposition for \(u\) or \(v\).