show/hide this revision's text 2 Fixed the signs.

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 $A_n \mapsto 1$.

show/hide this revision's text 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 $A_n \mapsto 1$.