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Register a new userCategorification of level two representations of quantum sl(N) via generalized arc rings
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Yanfeng Chen
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We build extensions of the arc rings, relate their centers to the cohomology rings of the Springer varieties, and categorify all level two representations of quantum sl(N).
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We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum sl(2).
Motion groups of links in the three sphere $mathbb{S}^3$ are generalizations of the braid groups, which are motion groups of points in the disk $mathbb{D}^2$. Representations of motion groups can be used to model statistics of extended objects such as closed strings in physics. Each $1$-extended $(3+1)$-topological quantum field theory (TQFT) will provide representations of motion groups, but it is difficult to compute such representations explicitly in general. In this paper, we compute representations of the motion groups of links in $mathbb{S}^3$ with generalized axes from Dijkgraaf-Witten (DW) TQFTs inspired by dimension reduction. A succinct way to state our result is as a step toward a twisted generalization (Conjecture ref{mainconjecture}) of a conjecture for DW theories of dimension reduction from $(3+1)$ to $(2+1)$: $textrm{DW}^{3+1}_G cong oplus_{[g]in [G]} textrm{DW}^{2+1}_{C(g)}$, where the sum runs over all conjugacy classes $[g]in [G]$ of $G$ and $C(g)$ the centralizer of any element $gin [g]$. We prove a version of Conjecture ref{mainconjecture} for the mapping class groups of closed manifolds and the case of torus links labeled by pure fluxes.
We endow a non-semisimple category of modules of unrolled quantum sl(2) with a Hermitian structure. We also prove that the TQFT constructed in arXiv:1202.3553 using this category is Hermitian. This gives rise to projective representations of the mapping class group in the group of indefinite unitary matrices.
In arXiv:1910.12059 Liu, Palcoux and Wu proved a remarkable necessary condition for a fusion ring to admit a unitary categorification, by constructing invariants of the fusion ring that have to be positive if it is unitarily categorifiable. The main goal of this note is to provide a somewhat more direct proof of this result. In the last subsection we discuss integrality properties of the Liu-Palcoux-Wu invariants.
Bernstein, Frenkel and Khovanov have constructed a categorification of tensor products of the standard representation of $mathfrak{sl}_2$ using singular blocks of category $mathcal{O}$ for $mathfrak{sl}_n$. In earlier work, we construct a positive characteristic analogue using blocks of representations of $mathfrak{sl}_n$ over a field $textbf{k}$ of characteristic $p > n$, with zero Frobenius character, and singular Harish-Chandra character. In the present paper, we extend these results and construct a categorical $mathfrak{sl}_k$-action, following Sussans approach, by considering more singular blocks of modular representations of $mathfrak{sl}_n$. We consider both zero and non-zero Frobenius central character. In the former setting, we construct a graded lift of these categorifications which are equivalent to a geometric construction of Cautis, Kamnitzer and Licata. We establish a Koszul duality between two geometric categorificatons constructed in their work, and resolve a conjecture of theirs. For non-zero Frobenius central characters, we show that the geometric approach to categorical symmetric Howe duality by Cautis and Kamnitzer can be used to construct a graded lift of our categorification using singular blocks of modular representations of $mathfrak{sl}_n$.
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