1. Determinants of Seidel Tournament Matrices
- Author
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Klanderman, Sarah, Montee, MurphyKate, Piotrowski, Andrzej, Rice, Alex, and Shader, Bryan
- Subjects
Mathematics - Combinatorics ,53C20, 15A15 - Abstract
The Seidel matrix of a tournament on $n$ players is an $n\times n$ skew-symmetric matrix with entries in $\{0, 1, -1\}$ that encapsulates the outcomes of the games in the given tournament. It is known that the determinant of an $n\times n$ Seidel matrix is $0$ if $n$ is odd, and is an odd perfect square if $n$ is even. This leads to the study of the set \[ \mathcal{D}(n)= \{ \sqrt{\det S}: \mbox{ $S$ is an $n\times n$ Seidel matrix}\}. \] This paper studies various questions about $\mathcal{D}(n)$. It is shown that $\mathcal{D}(n)$ is a proper subset of $\mathcal{D}(n+2)$ for every positive even integer, and every odd integer in the interval $[1, 1+n^2/2]$ is in $\mathcal{D}(n)$ for $n$ even. The expected value and variance of $\det S$ over the $n\times n$ Seidel matrices chosen uniformly at random is determined, and upper bounds on $\max \mathcal{D}(n)$ are given, and related to the Hadamard conjecture. Finally, it is shown that for infinitely many $n$, $\mathcal{D}(n)$ contains a gap (that is, there are odd integers $k<\ell
- Published
- 2024