Long-Moody construction of braid group representations and Haraoka's multiplicative middle convolution for KZ-type equations

2025/03/19 by Haru Negami, Negami, Haru · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2503.14840

openalex publication_date 2025/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We establish a correspondence between the algebraic and analytic approaches to constructing representations of the braid group Bn: the Katz--Long--Moody construction and the multiplicative middle convolution for Knizhnik--Zamolodchikov (KZ)-type equations, respectively. Furthermore, we show that this construction preserves the unitarity of representations relative to a Hermitian matrix. We present an algorithm for determining the signature of this matrix, and show that the signature is well defined for arbitrary parameters lambda satisfying |lambda| = 1 and lambda != 1 by continuity.

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