6-valent vertex in the $\mathfrak{gl}_N$ web category and its categorification

Abstract

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 $N \geq 6$.