Light ladders and clasp conjectures

Abstract

Morphisms between tensor products of fundamental representations of the quantum group of sl(n) are described by the sl(n)-webs of Cautis-Kamnitzer-Morrison. Using these webs, we provide an explicit, root-theoretic formula for the local intersection forms attached to each summand of the tensor product of an irreducible representation with a fundamental representation. We prove this formula for n at most 4, and conjecture that it holds for all n. Given two sequences of fundamental weights which sum to the same dominant weight, the clasp is the morphism between the corresponding tensor products which projects to the top indecomposable summand. Using our computation of intersection forms, we provide a recursive, ``triple-clasp expansion'' formula for clasps. In addition, we describe the cellular structure on sl(n)-webs, and prove that sl(n)-webs are an integral form for tilting modules of quantum groups.

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References in corpus (2)

• A Diagrammatic Category for the Representation Theory of U_q(sl_n)
• A Formula for the Jones-Wenzl Projections

Cited by in corpus (11)

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• Semisimplification of the category of tilting modules for GL_n
• Quantum K-theoretic geometric Satake
• SL2 tilting modules in the mixed case
• Quivers for SL(2) tilting modules
• Modular Valenced Temperley-Lieb Algebras
• On webs in quantum type $C$
• Web invariants for flamingo Specht modules
• Planar algebra presentations of ${\text{URep}}_{\mathbb{C}}({\mathbb{C}}^{+})$ and ${\text{URep}}_{{\mathbb{F}}_p}({\mathbb{F}}_p^{+})$
• On rank one 2-representations of web categories
• Triple Clasp Formulas for $G_2$