Generic quantum Schur–Weyl mutual centralizers over Q(q)

Suppose first $N\ge 2$ and $n\ge 1$. By [F2] and the commuting inclusions [F1], ${\dim }_{\mathbb{Q}}{A}_0\le {\dim }_F{A}_F\le {\dim }_F{\mathrm{End}}_{B_F}(M)\le {\dim }_{\mathbb{Q}}{A}_0$, so every inequality is equality and $A_F={\mathrm{End}}_{B_F}(M)$ because the included finite-dimensional subspaces have equal dimension. The other sandwich is ${\dim }_{\mathbb{Q}}{B}_0\le {\dim }_F{B}_F\le {\dim }_F{\mathrm{End}}_{A_F}(M)\le {\dim }_{\mathbb{Q}}{B}_0$, which gives $B_F={\mathrm{End}}_{A_F}(M)$ by the same inclusion-and-dimension argument. This establishes both mutual centralizer equalities without generic semisimplicity or a character formula.

Assume $N\ge n\ge 1$. On $v=e_1\otimes \cdots \otimes e_n$, the orbit vectors ${\rho }_n(t_w)v$ have entries regular at $q=1$; their specializations are the $n!$ distinct tensors obtained by permuting the distinct basis labels. Their vectorized column matrix therefore has an $n!\times n!$ minor equal to $1$ at $q=1$, which remains nonzero over $F$ by [F4]. Thus the orbit vectors are linearly independent over $F$. If $h={\sum }_{w\in S_n}{a}_w{t}_w$ lies in the kernel, the standard basis [F3] gives this expansion and $0={\rho }_n(h)v={\sum }_w{a}_w{\rho }_n(t_w)v$ forces every $a_w=0$. Hence the abstract action is faithful.

If $n=0$, then $M=F$ and both unital images and both commutants are $F$; the unique unital map from the abstract algebra $H_0=F$ is the identity, so this action is faithful. If $N=1$, $M$ is one-dimensional; the quantum image contains the scalars and every Hecke generator acts by $q$, so both images and both commutants are again $F$. These direct cases cover the parameters excluded from step 1.1.

In the range $N<n$ the equalities in step 1.1 remain image equalities and do not identify $B_F$ with the abstract Hecke algebra. The distinction is real: for $N=1,n=2$, the abstract algebra has basis $1,t_1$ by [F3], while its one-dimensional tensor-space action sends $t_1$ to $q$; the nonzero element $t_1-q$ is in the kernel. No claim of faithfulness in the full range $N<n$ is used in either centralizer equality.