I'm trying to prove that the Hecke operators $T_m$ and $T_n$ commute for any $m, n\in\mathbb{Z}$. I know that $T_m$ is just a polynomial in the $T_{p_i^{r_i}}$, for $m = \Pi_i p_i^{r_i}$ so I only need to prove that $T_{p^r}$ and $T_{q^s}$ commute for primes $p,q$ and $r,s\in\mathbb{N}$.
I know how to prove that they commute for $r=s=1$ but I don't really know how to do it in this further case.
By the recursive formula $$ T_{p^{r+1}} = T_{p^r}T_p - \langle p \rangle p^{k-1}T_{p^{r-1}} $$ the Hecke operator $T_{p^r}$ can be expressed as a polynomial in $T_p$ and $\langle p \rangle$, so it suffices to prove that $T_p$, $T_q$, $\langle p \rangle$ and $\langle q \rangle$ commute. You already know that $T_p$ and $T_q$ commute, so it suffices to prove that the operators of the kind $T$ commute with the diamond operators, which can be done directly.
