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Dualities for spin representations
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abstract
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.
Forward citations
Cited by 3 Pith papers
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Type $B$ Webs
Type B webs give a complete diagrammatic presentation of the subcategory of U_q(so_{2n+1})-representations generated by the fundamental representations.
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The quantum spin Brauer category
The quantum spin Brauer category is a braided diagrammatic category whose Karoubi completion covers all finite-dimensional U_q(so(N)) and U_q(o(N)) modules, including spin modules.
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A Kohno--Drinfeld Theorem for iquantum Weyl groups
For the split symmetric pair so_m ⊂ sl_m, the monodromy of the boundary Casimir connection is isomorphic to the iota-quantum Weyl group representation.
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