Signed symmetric group of finite degree is a Coxeter group

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Statement

Suppose n is a nonnegative integer and G is the signed symmetric group of degree n + 1. Then, G is isomorphic to a Coxeter group with generators s_1,s_2,\dots,s_n,t = s_{n+1}, the m_{ij} are as follows:

  • m_{i(i+1)} = 3 for 1 \le i \le n - 1
  • m_{n(n+1)} = 4
  • m_{ij} = 2 if |i - j| > 1

Under the isomorphism, s_i is idnetified with the transposition (i,i+1) and t = s_{n+1} is the diagonal matrix with the last entry -1 and all remaining entries equal to 1.