Algebraic Graph Theory Seminar - Guillermo Nunez Ponasso

Title: Hadamard’s Maximal Determinant Problem and Generalisations

Abstract:  Any matrix $M$ of order $n$ with entries taken from the complex unit disk satisfies Hadamard’s determinantal inequality $|\det M|\leq n^{n/2}$. Matrices meeting this bound with equality have pairwise orthogonal rows and columns. Such matrices are known as Hadamard matrices, and character tables of finite abelian groups give examples at every order. However if we restrict the entries of $M$ to a finite subset of the unit circle, such as $+1$ and $-1$, then the Hadamard bound is not always achieved – It is interesting then to find the maximal determinant for matrices with restricted entries. In this talk we will consider the “classical” Maximal Determinant Problem which concerns $\pm 1$ matrices, and generalisations of this problem to the $m$-th roots of unity. Motivated by some refinements of the Hadamard bound in the $\pm 1$ case we focus mainly on the cases $m$=2,3,4 and 6.