arXiv subjects
Publications and source records attributed to Haihan Wu.
We define a $2π/3$-rotationally invariant 6-valent vertex in the $\mathfrak{gl}_N$ web category. When $N = 4, 5$, we provide a categorification of the 6-valent vertex using $\mathfrak{gl}_N$ foams and decompose the hexagon web into a direct sum of indecomposables. A similar decomposition is conjectured for $N \geq 6$.
We study skein modules using $Sp(2n)$ webs. We define multivariable analogues of Chebyshev polynomials in the Type $C$ setting and use them to construct transparent elements in the skein module at roots of unity. Our arguments are diagrammatic and make use of an explicit braiding formula for $1$ and $k$ labeled strands and an analogue of Kuperberg's tetravalent vertex in the annular setting.
We give a generators and relations presentation for the full monoidal subcategory of representations of the quantum orthogonal group generated by the quantum exterior powers of the defining representation.
We define $2n$-multiwebs on planar graphs and discuss their relation with $\mathrm{Sp}(2n)$-webs. On a planar graph with a symplectic local system we define a matrix whose Pfaffian is the sum of traces of $2n$-multiwebs. As application we generalize Kasteleyn's theorem from dimer covers to $2n$-multiweb covers of planar graphs with $U(n)$ gauge group. For $\mathrm{Sp}(4)$ we relate Kuperberg's ``tetravalent vertex'' to the determinant, and classify reduced $4$-webs on some simple surfaces: the annulus, torus, and pair of pants. We likewise define, for $\mathrm{Sp}(2n)$ and $q=1$, a $2n$-valent vertex corresponding to the determinant, and classify reduced $2n$-webs on an annulus.
We use Kuperberg's diagrammatic description of the space of homomorphisms between fundamental representations of $G_2$ to give explicit recursive formulas for the idempotent projecting to the highest weight irreducible summand in each tensor product of fundamental representations.