This element has many expressions. It is characterised up to a scalar multiple by the property that $\sigma_iA_n=-q^{-1}A_n$ for $i=1,2,\ldots ,n-1$. It also satisfies $A_n\sigma_i=-q^{-1}A_n$ for $i=1,2,\ldots ,n-1$. In particular it is central.
Using this property you can calculate $A_n^2$. Note that $\sigma_\pi A_n=(-q)^{\ell(\pi)}= sgn(\pi)q^{\ell(\pi)}$A_n=(-q)^{-\ell(\pi)}= sgn(\pi)q^{-\ell(\pi)}$. So $$A_n.\sum_\pi sgn(\pi)q^{\ell(\pi)}\sigma_\pi=A_n.\sum_\pi q^{2\ell(\pi)}$$sgn(\pi)q^{-\ell(\pi)}\sigma_\pi=A_n.\sum_\pi q^{-2\ell(\pi)}$$You fix the scalar factor by the condition A_n^2=A_n. Alternatively in the one dimensional representation \sigma_i \mapsto -q^{-1} you require A_n \mapsto 1. 1 This element has many expressions. It is characterised up to a scalar multiple by the property that \sigma_iA_n=-q^{-1}A_n for i=1,2,\ldots ,n-1. It also satisfies A_n\sigma_i=-q^{-1}A_n for i=1,2,\ldots ,n-1. In particular it is central. Using this property you can calculate A_n^2. Note that \sigma_\pi A_n=(-q)^{\ell(\pi)}= sgn(\pi)q^{\ell(\pi)}. So$$A_n.\sum_\pi sgn(\pi)q^{\ell(\pi)}\sigma_\pi=A_n.\sum_\pi q^{2\ell(\pi)}$$You fix the scalar factor by the condition$A_n^2=A_n$. Alternatively in the one dimensional representation$\sigma_i \mapsto -q^{-1}$you require$A_n \mapsto 1\$.