A SLANT ON THE TWISTED DETERMINANTS THEOREM DAVID G. GLYNN Abstract. Using finite Abelian groups instead of a cyclic construc- tion we generalise Cohen’s theorem to construct sums of n determi- nants of order n that are identically zero. The proof is also simplified. There are connections with pairs of “amicable” Latin squares, and quantum sets of lines in space which are equivalent to self-dual quan- tum codes. A seemingly unrelated identity with only three square matrices of side four is exhibited. We also show how to calculate the invariant X 4 , or equivalently a certain symmetric polynomial in the roots of a sextic polynomial in one variable over any field of even characteristic using only six multiplications. 1. Introduction In 1964 Cohen [2] found an identity involving n determinants of order n each having the same n 2 variables. For example in the case n = 3 there was:       a b c d e f g h i       +       a b c f d e h i g       +       a b c e f d i g h       = 0. (1) His “twisted determinants” theorem was this: Theorem 1.1. If A 1 is a general n ⇥ n matrix over a commutative ring, then one can construct matrices A i , i =2,...,n, by “twisting” cyclically one row of A 1 incrementally in one direction, and twisting another row incrementally in the other direction. The resulting matrices then satisfy |A 1 | + ··· + |A n | =0. Given any square matrix, if we transpose it, or switch two rows or columns, or use a sequence of such operations, we can obtain other ma- trices with the same determinant, perhaps multiplied by minus one. We Key words and phrases. determinant, sum, identity, amicable latin squares, self-dual quantum code, abelian group, symmetric function, invariant. 2010 Mathematics Subject Classification. 11C20 15A15 05B15 14N20 94B05. 1