Hyperplane arrangements and tensor product invariants

2016/11/07 by Belkale, Prakash, Brosnan, Patrick, Mukhopadhyay, Swarnava · 2 citations
#14H60 #32S22 Secondary 32G34 #81T40 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 17B67 #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1611.01861

Abstract

In the first part of this paper, we consider, in the context of an arbitrary hyperplane arrangement, the map between compactly supported cohomology to the usual cohomology of a local system. A formula (i.e., an explicit algebraic de Rham representative) for a generalized version of this map is obtained. These results are applied in the second part to invariant theory: Schechtman and Varchenko connect invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements. The first part of this paper is then used, following and completing arguments of Looijenga, to determine the image of invariants in cohomology. In suitable cases (e.g., corresponding to positive integral levels), the space of invariants is shown to acquire a mixed Hodge structure over a cyclotomic field. We investigate the Hodge filtration on the space of invariants, and characterize the subspace of conformal blocks in Hodge theoretic terms.

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Related

• Hodge spectrum of hyperplane arrangements
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• Rank one local systems on complements of hyperplanes and Aomoto complexes
• Cohomology of semisimple local systems and the Decomposition theorem
• Arrangements and local systems