7
$\begingroup$

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.

$\endgroup$
2
  • 1
    $\begingroup$ If I'm not mistaken, this is written down in sections 3 and 4 of Kazhdan and Wenzl's "Reconstructing Monoidal Categories." $\endgroup$ Feb 3, 2018 at 3:59
  • $\begingroup$ Yes, that does indeed identify the kernel, although it does not give generators like I hoped. If I understand it correctly, the kernel is what I would call the “negligible” elements: $x$ such that a certain trace of $xy$ is always zero. $\endgroup$ Feb 10, 2018 at 6:13

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.