The Iwahori-Hecke algebra $H_n(q)$ acts on the $n$th tensor power of the standard representation of $U_q(\mathfrak{sl}_m)$. What is the kernel of this action? Does anyone know a reference?
I'm happy to assume $q$ is not a root of unity. If $m > n$ then I think the kernel is trivial. If $m \le n$ then I think it's the two sided ideal generated by $\sum_{w \in S_m} \mathop{sgn}(w) T_w$, where $T_w$ is the word in the generators of $H_n(q)$ that corresponds to a shortest word for $w$ in the generators of $S_m$. However I have not been able to find a reference for either of these claims.