Permutation group algebras

Details Permutation group algebras Permutation group algebras

2001-12-18

arxiv.org/abs/math/0112173v2

J. Alg. Combinatorics, 19 (2004), pp. 25--45

Mathematics

Combinatorics

24 pages New version: Improved (and corrected) proof of Prop. 4.2 General corrections and editorial improvements; no new conte

Scientific paper

We consider the permutation group algebra defined by Cameron and show that if the permutation group has no finite orbits, then no homogeneous element of degree one is a zero-divisor of the algebra. We proceed to make a conjecture which would show that the algebra is an integral domain if, in addition, the group is oligomorphic. We go on to show that this conjecture is true in certain special cases, including those of the form H Wr S and H Wr A, and show that in the oligormorphic case, the algebras corresponding to these special groups are polynomial algebras. In the H Wr A case, the algebra is related to the shuffle algebra of free Lie algebra theory. We finish by considering some integer sequences which arise from certain of these groups.

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