Latin square determinants II

The theory of latin square determinants may be regarded as a direct continuation of the line of research which led Frobenius to introduce group characters. A previous paper introduced the basic ideas and indicated how the theory relates to quasigroup character theory. The work here sets out further developments. The linear factors of a latin square determinant are characterised. Results on a lower bound for the number of irreducible factors are obtained, and methods to factorise determinants with various kinds of symmetries are given, as well as determinants arising as extensions. A 'Molien series' for a latin square is defined, generalising that arising in group invariant theory. A determinant arising out of a pair of squares is discussed, and when the pair of squares is an orthogonal pair arising from a finite field it is shown that this determinant has a special property. Further examples have been calculated using symbolic manipulation packages.