Generating finite Coxeter groups with elements of the same order
Abstract: Supposing G is a group and k a natural number, d k (G) is defined to be the minimal number of elements of G of order k which generate G (setting d k (G) = 0 if G has no such generating sets). This paper investigates d k (G) when G is a finite Coxeter group either of type B n or D n , or of exceptional type. Together with the work of Garzoni and Yu, this determines d k (G) for all finite irreducible Coxeter groups G when 2 ≤ k ≤ rank(G) (rank(G) + 1 when G is of type A n ).
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Root cycles in Coxeter groups
For an element 𝑤 of a Coxeter group 𝑊, there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on Φ, the root system of 𝑊. This paper investigates the interaction between these two features of 𝑤, introducing the notion of the crossing number of 𝑤, κ ( w ) \kappa(w) . Writing w = c 1 ⋯ c r w=c_{1}\cdots c_{r} as a product of disjoint cycles, we associate to each cycle c i c_{i} a “crossing number” κ ( c i ) \kappa(c_{i}) , which is the number of positive roots 𝛼 in c i c_{i} for which w ⋅ α w\cdot\alpha is negative. Let Seq κ ( w ) {\mathrm{Seq}}_{\kappa}({w}) be the sequence of κ ( c i ) \kappa(c_{i}) written in increasing order, and let κ ( w ) = max Seq κ ( w ) \kappa(w)=\max{\mathrm{Seq}}_{\kappa}({w}) . The length of 𝑤 can be retrieved from this sequence, but Seq κ ( w ) {\mathrm{Seq}}_{\kappa}({w}) provides much more information. For a conjugacy class 𝑋 of 𝑊, let κ min ( X ) = min { κ ( w ) ∣ w ∈ X } \kappa_{\min}(X)=\min\{\kappa(w)\mid w\in X\} and let κ ( W ) \kappa(W) be the maximum value of κ min \kappa_{\min} across all conjugacy classes of 𝑊. We call κ ( w ) \kappa(w) and κ ( W ) \kappa(W) , respectively, the crossing numbers of 𝑤 and 𝑊. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups, if 𝑢 and 𝑣 are two elements of minimal length in the same conjugacy class 𝑋, then Seq κ ( u ) = Seq κ ( v ) {\mathrm{Seq}}_{\kappa}({u})={\mathrm{Seq}}_{\kappa}({v}) and κ min ( X ) = κ ( u ) = κ ( v ) \kappa_{\min}(X)=\kappa(u)=\kappa(v) .
Root cycles in Coxeter groups
For an element 𝑤 of a Coxeter group 𝑊, there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on Φ, the root system of 𝑊. This paper investigates the interaction between these two features of 𝑤, introducing the notion of the crossing number of 𝑤, κ ( w ) \kappa(w) . Writing w = c 1 ⋯ c r w=c_{1}\cdots c_{r} as a product of disjoint cycles, we associate to each cycle c i c_{i} a “crossing number” κ ( c i ) \kappa(c_{i}) , which is the number of positive roots 𝛼 in c i c_{i} for which w ⋅ α w\cdot\alpha is negative. Let Seq κ ( w ) {\mathrm{Seq}}_{\kappa}({w}) be the sequence of κ ( c i ) \kappa(c_{i}) written in increasing order, and let κ ( w ) = max Seq κ ( w ) \kappa(w)=\max{\mathrm{Seq}}_{\kappa}({w}) . The length of 𝑤 can be retrieved from this sequence, but Seq κ ( w ) {\mathrm{Seq}}_{\kappa}({w}) provides much more information. For a conjugacy class 𝑋 of 𝑊, let κ min ( X ) = min { κ ( w ) ∣ w ∈ X } \kappa_{\min}(X)=\min\{\kappa(w)\mid w\in X\} and let κ ( W ) \kappa(W) be the maximum value of κ min \kappa_{\min} across all conjugacy classes of 𝑊. We call κ ( w ) \kappa(w) and κ ( W ) \kappa(W) , respectively, the crossing numbers of 𝑤 and 𝑊. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups, if 𝑢 and 𝑣 are two elements of minimal length in the same conjugacy class 𝑋, then Seq κ ( u ) = Seq κ ( v ) {\mathrm{Seq}}_{\kappa}({u})={\mathrm{Seq}}_{\kappa}({v}) and κ min ( X ) = κ ( u ) = κ ( v ) \kappa_{\min}(X)=\kappa(u)=\kappa(v) .
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