Web Calculus and Tilting Modules in Type $C_2$

Abstract

Using Kuperberg's B2/C2 webs, and following Elias and Libedinsky, we describe a "light leaves" algorithm to construct a basis of morphisms between arbitrary tensor products of fundamental representations for so5 ∼ = sp 4 (and the associated quantum group). Our argument has very little dependence on the base field. As a result, we prove that when [2]q = 0, the Karoubi envelope of the C2 web category is equivalent to the category of tilting modules for the divided powers quantum group U Z q (sp 4 ).

Type $C$ Webs

“…One can check that after the appropriate rescaling, the coefficient of the identity on the right hand side is now 1. This should be compared with the "'elementary neutral ladders" from [10,5]. Finally, the second equation on the third line of the relations in [26,Definition 2.1] is the analogue of our (3.10e) (and is thus redundant).…”

“…However, all of these useful tools will be provided in the sequel to this paper [6]. In particular, we will construct a double ladders basis of morphism spaces in Web(sp 2n ) that is analogous to the basis constructed in type A by the second-named author in [10], and which extends the C 2 double ladders basis constructed by the first-named author [5].…”

Triple Clasp Formulas for $\mathfrak{g}_2$

“…When ξ is a root of unity in C, we can study the relation between C ⊗ q=ξ Web q (g 2 ), and the category of tilting modules of C ⊗ q=ξ U q (g 2 ). It is possible to adapt the approach from [1], which itself is based on [8], to prove that the Karoubi envelope of C ⊗ q=ξ Web q (g 2 ) is equivalent to the category of tilting modules as long as [2] ξ , [3] ξ = 0. The same result, but using a slightly different approach is also work in progress of Victor Ostrik and Noah Snyder.…”

“…The recursive formula in Equation ( 1) has proven useful in link homology [6], Soergel bimodules [7], and the theory of subfactors and planar algebras [28]. The present work is concerned with generalizing Equation (1) to from sl 2 to the Lie algebra g 2 . However, many things we say in the introduction make sense for all semisimple Lie algebras.…”

Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sp}_4$

“…He gave a diagrammatic interpretation of categories of finite-dimensional representations of their quantum groups. The diagrammatic category of U q (sp 4 ) was further studied by Bodish [Bod20,Bod22], and that for U q (sp 2n ) by [BERT21]. On the other hand, the canonical cluster K 2 -structure on the moduli space A G,Σ of decorated twisted G-local system on a marked surface Σ [FG06a] has been constructed by Le [Le19] (for classical types, and particular cluster charts) and [GS19] (for all semisimple types, and a more general class of cluster charts).…”