TY - JOUR

T1 - Latin square determinants II

AU - Johnson, K. W.

N1 - Funding Information:
The author is indebtedt o E.Kaltofen for his help in the factorisationo f squares of higher orders. Thanks are also due R. Beck for his help in the use of MACSYMA, and to G. Andrews for permissiont o use an experimentalv ersion of SCRATCHPAD availablet o him. Much of the computationw as carried out using a Research Initiation Grant from The PennsylvaniaS tate University, and thanks are due to the undergraduatesD . Raymond, M. Skurla and K. Musike, who did various parts of the work. Thanks are also due to J.D.H. Smith who pointed out that the proof of Theorem 6.1 could be completed by using the resultso n Coates graphs.

PY - 1992/8/14

Y1 - 1992/8/14

N2 - 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.

AB - 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.

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U2 - 10.1016/0012-365X(92)90136-4

DO - 10.1016/0012-365X(92)90136-4

M3 - Article

AN - SCOPUS:38249008204

SN - 0012-365X

VL - 105

SP - 111

EP - 130

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 1-3

ER -