Permutation Groups, a Related Algebra and a Conjecture of Cameron |
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Authors: | Julian D Gilbey |
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Institution: | (1) School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK |
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Abstract: | 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. |
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Keywords: | oligomorphic permutation groups integral domains shuffle algebras random graph random tournament Lyndon words |
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