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Hecke algebra representations from the Katz-Long-Moody construction
Authors:
Haru Negami
Abstract:
We study the Katz-Long-Moody (KLM) construction and classify exactly when the resulting braid group representations factor through Hecke algebras, for scalar braid part and semisimple free-group part, over an algebraically closed field of characteristic zero and at every convolution parameter lambda different from 1. We show that, except for one exceptional two-strand family, this property then de…
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We study the Katz-Long-Moody (KLM) construction and classify exactly when the resulting braid group representations factor through Hecke algebras, for scalar braid part and semisimple free-group part, over an algebraically closed field of characteristic zero and at every convolution parameter lambda different from 1. We show that, except for one exceptional two-strand family, this property then depends only on the eigenvalues of the free-group part and is independent of lambda. Semisimplicity is a genuine hypothesis: we exhibit non-semisimple inputs, namely g = I + N with N nonzero and N^2 = 0, whose KLM quotient is nonetheless a Hecke module, realizing the permutation representation of the symmetric group. We also give an exact criterion for when the resulting representations factor through Temperley-Lieb algebras.
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Submitted 4 August, 2026; v1 submitted 3 August, 2026;
originally announced August 2026.
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Long-Moody construction of braid group representations and Haraoka's multiplicative middle convolution for KZ-type equations
Authors:
Haru Negami
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…
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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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Submitted 13 July, 2026; v1 submitted 18 March, 2025;
originally announced March 2025.
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Long-Moody construction of braid representations and Katz middle convolution
Authors:
Kazuki Hiroe,
Haru Negami
Abstract:
The Long-Moody construction is a method to obtain representations of braid groups introduced by Long and Moody. Also the Katz middle convolution is known to be a method to construct local systems on $\mathbb{C}\backslash\{n\text{-points}\}$ introduced by Katz. In this paper, we explain that these two methods are naturally unified and define a new functor which we call the Katz-Long-Moody functor.…
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The Long-Moody construction is a method to obtain representations of braid groups introduced by Long and Moody. Also the Katz middle convolution is known to be a method to construct local systems on $\mathbb{C}\backslash\{n\text{-points}\}$ introduced by Katz. In this paper, we explain that these two methods are naturally unified and define a new functor which we call the Katz-Long-Moody functor. This functor extends the framework of Katz algorithm to categories of local systems on various topological spaces, for example, $B_{n}$-bundles associated with simple Weierstrass polynomials, complements of hyperplane arrangements of fiber-type, link complements in the solid torus, and so on.
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Submitted 10 March, 2023;
originally announced March 2023.