Nonorthogonal geometric realizations of Coxeter groups. (Q2925688)

The author introduces a Coxeter datum and investigates how this relates to established similar concepts.NEWLINENEWLINE Let \(S\) be an arbitrary set in which each unordered pair \(\{s,t\}\) of elements \(s,t\) in \(S\) is assigned some \(m_{s,t}\in\mathbb Z\cup\{\infty\}\), subject to the condition \(m_{ss}=1\) for all \(s\) in \(S\), and \(m_{st}\geq 2\) for all distinct \(s,t\) in \(S\). Suppose that \(V_1\) and \(V_2\) are vector spaces over the real field \(\mathbb R\), and suppose that there is a bilinear map \(\langle\;,\;\rangle\colon V_1 \times V_2\to\mathbb R\) and sets \(\Pi_1=\{\alpha_s\mid s\in S \}\subseteq V_1\) and \(\Pi_2=\{\beta_s\mid s\in S\}\subseteq V_2\).NEWLINENEWLINE The author calls \(\mathcal C=(S,V_1,V_2,\Pi_1,\Pi_2,\langle\;,\;\rangle)\) a Coxeter datum, if certain conditions are satisfied. The \(m_{st}\) where \(s,t\) are elements in \(S\) are called the Coxeter parameters of \(\mathcal C\).NEWLINENEWLINE For each \(s\in S\) define \(\rho_{V_1}(s)(x)=x-2\langle x,\beta_s\rangle\alpha_s\) for all \(x\in V_1\), and \(\rho_{V_2}(s)(y)=y-2\langle\alpha_s,y\rangle\beta_s\) for all \(y\in V_2\). Let \(W_i(\mathcal C)\), \(i\in\{1,2\}\), be the subgroup of \(\text{GL}(V_i)\) generated by \(\{\varrho_{V_i}(s)\mid s\in S\}\).NEWLINENEWLINE Let \((W,R)\) be a Coxeter system where \(R=\{r_s\mid s\in S\}\) is a set of involutions generating \(W\) such that the order of the product \(r_sr_t\) is \(m_{st}\) whenever \(s,t\) are in \(S\) with \(m_{st}\neq\infty\). The author shows that then there are group homomorphisms \(f_1\colon W\to W_1(\mathcal C)\) and \(f_2\colon W\to W_2(\mathcal C)\) such that \(f_1(r_s)=\rho_{V_1}(s)\) and \(f_2(r_s)=\rho_{V_2}(s)\) for all \(s\in S\).NEWLINENEWLINE One of the main results that the author obtains is that \(f_1\) and \(f_2\) are isomorphisms, that is, \((W_1(\mathcal C),R_1(\mathcal C))\) and \((W_2(\mathcal C),R_2(\mathcal C))\) are both Coxeter systems isomorphic to \((W,R)\).