Light ladders and clasp conjectures
Light ladders and clasp conjectures
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光梯和卡扣猜想
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发表时间：
2015
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通讯作者：
Ben Elias
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作者：
Ben Elias
用CAUTIS-Kamnitzer-Morison的sl(N)网描述了sl(N)的量子群基本表示的张量积之间的态射。利用这些网，我们给出了一个具有基本表示的不可约表示的张量积的每个和所附的局部交形的显式根论公式。我们证明了这个公式对n至多为4，并猜想它对所有n都成立。
给定两个基本权重序列，它们之和为相同的主权重，则扣是相应张量积之间的态射，该张量积投射到顶部不可分解的和。利用我们对交集形式的计算，我们给出了一个递归的、“三钩展开”的扣环公式。
此外，我们还描述了sl(N)-webs上的元胞结构，并证明了sl(N)-webs是量子群倾斜模的积分形式。
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.