The Conjectures of Alon--Tarsi and Rota in Dimension Prime Minus One

A formula for Glynn’s hyperdeterminant detp (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 a published proof that claimed to multiply the known dimensions by powers of two, and also 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. 1. Rota’s Basis Conjecture Rota’s basis conjecture for matroids is the following; see [14]: Conjecture 1.1. Consider any matroid N of rank n. Let A = (aij) be an n×n matrix, such that every element aij is a point of N (an element of rank one), and each row of A is a basis of N . Then the elements in each row of A can be permuted so that in the resulting n× n matrix every column also forms a basis for N . This has been verified for n ≤ 3; see [5]. It is also true for linear matroids of rank p+ 1 (p prime) over fields of all characteristics, except possibly a finite number of prime characteristics; see [8]. There has been a great deal of recent interest in this conjecture as verified by the following papers: [1, 3, 5, 6, 8, 9, 10, 11, 15, 16, 17, 18, 19]. Now let us state the conjecture due to Alon-Tarsi. It is equivalent to another of Huang-White; see [15]: Conjecture 1.2. The number of Latin squares of a given even order of even parity is not equal to the number of Latin squares of that order of odd parity. The definitions for the parities are well-known but we repeat them in Section 2, using a non-standard explanation. From [12, 14, 16], if the Alon-Tarsi conjecture is true for a certain even n, then Rota’s basis conjecture for fields of characteristic 0 and dimension n is true, and this holds also for fields of almost all prime characteristics. Date: 20 January 2010.