Dualities for spin representations

Let $S$ be the spinor representation of $U_q\mathfrak{so}_N$, for $N$ odd and $q^2$ not a rooot of unity. We show that the commutant of its action on $S^{\otimes n}$ is given by a representation of the nonstandard quantum group $U'_{-q^2}\mathfrak{so}_n$. For $N$ even, an analogous statement also holds for $S=S_+\oplus S_-$ the direct sum of the irreducible spinor representations of $U'_q\mathfrak{so}_N$, with the commutant given by $U'_{-q}\mathfrak{o}_n$, a $\mathbb{Z}/2$-extension of $U'_{-q}\mathfrak{so}_n$. Similar statements also hold for fusion tensor categories with $q$ a root of unity.

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