Padraig O'Cathain (Worcester Polytechnic Institute)

will speak on

Complex Hadamard matrices, minimal polynomials and morphisms

Time: 2:00PM
Date: Thu 12th March 2020
Location: Seminar Room SCN 1.25 [map]

Further information

Abstract: Let M be a matrix with complex entries of unit norm. A well-known theorem of Hadamard bounds the magnitude of the determinant of $M$ as a function of its dimension, and $M$ is a (complex) Hadamard matrix if $M$ meets Hadamard's bound with equality.

The Hadamard conjecture concerns the existence of real Hadamard matrices (that is, with entries in $\{\pm 1\}$. While existence of real Hadamard matrices has been studied for over 100 years, (complex) Hadamard matrices with entries in some extension field of $\mathbb{Q}$ have not received the same attention. In fact, apart from matrices with entries $\{\pm 1, \pm i\}$, existence and non-existence of complex Hadamard matrices is poorly understood.

In this talk, I will discuss relations between the minimal polynomial of a complex Hadamard matrix and tensor-product-like constructions for Hadamard matrices with entries in a smaller field. These results generalise theorems of Turyn and Compton-Craigen-de Launey which give real Hadamard matrices from certain complex Hadamard matrices. Applying these techniques to certain biquadratic extensions of $\mathbb{Q}$ recovers a construction of Mukhopadhyay-Seberry for skew-Hadamard matrices, and time permitting, I will report on techniques for controlling the minimal polynomial of a Hadamard matrix. I will finish the talk with some open questions and directions for future research.

Joint work with Ronan Egan, Phillip Heikoop, Guillermo Nunez Ponasso, Jack Pugmire and Eric Swartz.

(This talk is part of the Algebra and Number Theory series.)

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