# Article

Full entry | PDF   (0.2 MB)
Keywords:
Latin square; quasigroup; automorphism; $\lambda$-ring
Summary:
The automorphisms of a quasigroup or Latin square are permutations of the set of entries of the square, and thus belong to conjugacy classes in symmetric groups. These conjugacy classes may be recognized as being annihilated by symmetric group class functions that belong to a $\lambda$-ideal of the special $\lambda$-ring of symmetric group class functions.
References:
[1] Atiyah M.F., Tall D.O.: Group representations, $\lambda$-rings, and the $J$-homomorphism. Topology 8 (1969), 253–297. DOI 10.1016/0040-9383(69)90015-9 | MR 0244387 | Zbl 0159.53301
[2] Bryant D., Buchanan M., Wanless I.M.: The spectrum for quasigroups with cyclic automorphisms and additional symmetries. Discrete Math. 309 (2009), 821–833. DOI 10.1016/j.disc.2008.01.020 | MR 2502191
[3] tom Dieck T.: Transformation Groups and Representation Theory. Springer, Berlin, 1979. MR 0551743 | Zbl 0529.57020
[4] Falcón R.M.: Cycle structures of autotopisms of the Latin squares of order up to $11$. arXiv:0709.2973v2 [math.CO], 2009; to appear in Ars Combinatoria.
[5] Falcón R.M., Martín-Morales J.: Gröbner bases and the number of Latin squares related to autotopisms of order $\leq 7$. J. Symbolic Comput. 42 (2007), 1142–1154. DOI 10.1016/j.jsc.2007.07.004 | MR 2368076
[6] Knutson D.: $\lambda$-rings and the Representation Theory of the Symmetric Group. Springer, Berlin, 1973. MR 0364425 | Zbl 0272.20008
[7] McKay B.D., Meynert A., Myrvold W.: Small Latin squares, quasigroups and loops. J. Combin. Designs 15 (2007), 98–119. DOI 10.1002/jcd.20105 | MR 2291523 | Zbl 1112.05018
[8] Smith J.D.H.: An Introduction to Quasigroups and Their Representations. Chapman and Hall/CRC, Boca Raton, FL, 2007. MR 2268350 | Zbl 1122.20035
[9] Smith J.D.H., Romanowska A.B.: Post-Modern Algebra. Wiley, New York, NY, 1999. MR 1673047 | Zbl 0946.00001
[10] Wanless I.M.: Diagonally cyclic latin squares. European J. Combin. 25 (2004), 393–413. DOI 10.1016/j.ejc.2003.09.014 | MR 2036476 | Zbl 1047.05007

Partner of