arXiv subjects

Haru Negami

Publications and source records attributed to Haru Negami.

4 recordsLinked to original sources

arXiv2026

A closed signature formula for the Katz-Long-Moody Hermitian form

The Katz-Long-Moody construction associates to a representation of the semidirect product of a free group and a braid group, defined by the Artin action, and a nonzero parameter a new representation of the same group. On the pure braid group it corresponds to Haraoka's multiplicative middle convolution for KZ-type equations. For unitary input and a convolution parameter on the unit circle other than one, the construction equips the quotient representation with a canonical non-degenerate invariant Hermitian form. We give a closed formula for its signature in terms of the eigenangles of the input local monodromies, the eigenangles of their ordered product, and the convolution parameter. The formula accounts for the kernel of the form before passage to the quotient and for signature changes at resonant parameters. It determines precisely when the induced form is definite, answering the definiteness problem posed in the companion paper. Definiteness implies unitarizability of the output representation; the converse holds when that representation is irreducible. The proof uses elementary linear algebra: a determinant identity, explicit block-pivot formulas, and an inertia formula for sums of Cayley transforms of unitary matrices. As applications, we compare the rank-one construction explicitly with Haraoka's invariant form for Pochhammer systems, recover the Gauss case of the Beukers-Heckman interlacing criterion, and determine the definite parameter intervals for the Hecke and Temperley-Lieb specializations.

math-ph↗

arXiv2026

Hecke algebra representations from the Katz-Long-Moody construction

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.

math.RT↗

arXiv2025

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

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.

math-ph↗

arXiv2023

Long-Moody construction of braid representations and Katz middle convolution

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.

math.GT↗