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Determinants of Seidel Tournament MatricesSarah Klanderman, MurphyKate Montee, Andrzej Piotrowski, Alex Rice, Bryan Shaderhttps://arxiv.org/abs/2406.09697
Determinants of Seidel Tournament MatricesThe 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}(…