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March 15th, 2014, 09:57 AM  #1 
Newbie Joined: Mar 2014 Posts: 1 Thanks: 0  A Group Structure for Unlabeled Necklaces 8beads 8 colors?
[attachment=0:nahm4rcz]Unlabeled Necklaces 6beads6colors.png[/attachment:nahm4rcz]Dear, Let's consider the finite set of the unlabeled necklaces with 8 beads and possibly up to 8 colors UN(8,. They are known as unlabeled necklaces, invariant by permutation of the beads'color and by rotation. There are 544 different such necklaces, named Ni, 0<i<543. Below the necklaces Ni are sorted by their minimum equivalent values in (0,8^ : N0 : (00000000)=(11111111)=(...)=(77777777) : the only necklace with 1 color N1 : (00000001)=(10000000) rotation=(66666664) color permutation N2 : (00000011) N3 : (00000012) N4 : (00000101) ... ... N543 : (01234567) = the only necklace with 8 different colors Question : Can a group structure by associated to this set? I did my very best to sort the necklaces on a torus so that the torus look the most regular as possible, with the idea to use the group defined by the squarerootn of unity on the complex plane, but I failed. For illustration, I have attached an image of UN(6,6) and of a torus made from necklace...in jewelry The bibliography is quite difficult* : The padic representation and the Witt vector theory seem to be good candidate for doing so... (maybe 7adic here, because the first pearl of each collar is known (0) and because all the necklace but the latest have a maximum of 7 colors). N. Metropolis and C.C. Rota, Witt vectors and the Algebra of Necklaces, Adv. in Math. 50 (1983), 95–125 Cristian Lenart, Formal GroupTheoretic Generalizations of the Necklace Algebra, Including a qDeformation, Journal of Algebra DOI:10.1006/jabr.1997.7203 Any experience or suggestion for further reading? Thank 

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8beads, colors, group, necklaces, structure, unlabeled 
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