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The conjectures of Alon-Tarsi and Rota in dimension prime minus one. (English) Zbl 1227.05095

Summary: A formula for Glynn’s hyperdeterminant \(\det_p (p\) prime) of a square matrix shows that the number of ways to decompose any integral doubly stochastic matrix with row and column sums \(p-1\) into \(p-1\) permutation matrices with even product, minus the number of ways with odd product, is 1 (mod \(p\)). It follows that the number of even Latin squares of order \(p-1\) is not equal to the number of odd Latin squares of that order. Thus Rota’s basis conjecture is true for a vector space of dimension \(p-1\) over any field of characteristic zero or \(p\), and all other characteristics except possibly a finite number.
It is also shown where there is a mistake in a published proof that claimed to multiply the known dimensions by powers of two, and that also claimed that the number of even Latin squares is greater than the number of odd Latin squares. Now, 26 is the smallest unknown case where Rota’s basis conjecture for vector spaces of even dimension over a field is unsolved.

MSC:

05B15 Orthogonal arrays, Latin squares, Room squares
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05B35 Combinatorial aspects of matroids and geometric lattices
05C20 Directed graphs (digraphs), tournaments
11A41 Primes
15A03 Vector spaces, linear dependence, rank, lineability
15A15 Determinants, permanents, traces, other special matrix functions
15A72 Vector and tensor algebra, theory of invariants
15B51 Stochastic matrices
51E20 Combinatorial structures in finite projective spaces
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Online Encyclopedia of Integer Sequences:

The Alon-Tarsi constants AT(n).