Determinants of Seidel tournament matrices

The Seidel matrix of a tournament on n players is an n×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×n Seidel matrix is 0 if n is odd, and is an odd perfect square if n is even. This lead...

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Bibliographic Details
Published in:Linear algebra and its applications 2025-02, Vol.707, p.126-151
Main Authors: Klanderman, Sarah, Montee, MurphyKate, Piotrowski, Andrzej, Rice, Alex, Shader, Bryan
Format: Article
Language:English
Subjects:
Determinants
Pfaffian
Seidel matrix
Skew-symmetric matrix
Tournaments
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Summary:The Seidel matrix of a tournament on n players is an n×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×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, D(n), of square roots of determinants of n×n Seidel matrices. It is shown that D(n) is a proper subset of D(n+2) for every positive even integer, and every odd integer in the interval [1,1+n2/2] is in D(n) for n even. The expected value and variance of det⁡S over the n×n Seidel matrices chosen uniformly at random is determined, and upper bounds on max⁡D(n) are given, and related to the Hadamard conjecture. Finally, it is shown that for infinitely many n, D(n) contains a gap (that is, there are odd integers k
ISSN:0024-3795
DOI:10.1016/j.laa.2024.11.011