[Seminar] MLDS Unit Seminar 2025-6 Ms. Haru Negami, Chiba University

Ms. Haru Negami, Chiba University

Title: Integral transformation of KZ-type equation and construction of representation of braid group

Abstract: The Knizhnik–Zamolodchikov (KZ) equation [1] is a system of linear differential equations in n variables, satisfied by various special functions such as the Appell–Lauricella hypergeometric series. In this talk, we establish a correspondence between two fundamental constructions [2]: the multiplicative middle convolution of KZ-type equations [3], an integral transformation that preserves the class of KZ-type equations, and the Katz–Long–Moody construction [4], an algebraic framework for constructing representations of the braid group B_n. Representations of braid groups occupy a central place in modern mathematics, with deep applications to topology, representation theory, and mathematical physics. Finally, we explore potential applications of this perspective beyond representation theory, including connections to areas such as optimal transport.

References:

[1] Knizhnik, V. and A. Zamolodchikov, Current algebra and Wess-Zumino model in 2 dimensions,  Nucl. Phys. B 247 (1984), 83103.
[2] Negami, H. (2025). Long-Moody construction of braid group representations and Haraoka’s multiplicative middle convolution for KZ-type equations. arXiv preprint arXiv:2503.14840.
[3] Haraoka, Y. (2020). Multiplicative middle convolution for KZ equations. Mathematische Zeitschrift, 294(3), 1787–1839.
[4] Hiroe, K., Negami, H. (2023). Long-Moody construction of braid representations and Katz middle convolution. arXiv preprint arXiv:2303.05770.