Geodesic
Groups acting on necklaces and sandpile groups
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIII, Tome 421 (2014), pp. 81-93 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We introduce a group naturally acting on aperiodic necklaces of length $n$ with two colours using the 1–1 correspondences between such necklaces and irreducible polynomials of degree $n$ over the field $\mathbb F_2$ of two elements. We notice that this group is isomorphic to the quotient group of non-degenerate circulant matrices of size $n$ over that field modulo a natural cyclic subgroup. Our groups turn out to be isomorphic to the sandpile groups for a special sequence of directed graphs. 
@article{ZNSL_2014_421_a6,
     author = {S. V. Duzhin and D. V. Pasechnik},
     title = {Groups acting on necklaces and sandpile groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {81--93},
     year = {2014},
     volume = {421},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_421_a6/}
}
TY  - JOUR
AU  - S. V. Duzhin
AU  - D. V. Pasechnik
TI  - Groups acting on necklaces and sandpile groups
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2014
SP  - 81
EP  - 93
VL  - 421
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2014_421_a6/
LA  - en
ID  - ZNSL_2014_421_a6
ER  - 
%0 Journal Article
%A S. V. Duzhin
%A D. V. Pasechnik
%T Groups acting on necklaces and sandpile groups
%J Zapiski Nauchnykh Seminarov POMI
%D 2014
%P 81-93
%V 421
%U http://geodesic.mathdoc.fr/item/ZNSL_2014_421_a6/
%G en
%F ZNSL_2014_421_a6
S. V. Duzhin; D. V. Pasechnik. Groups acting on necklaces and sandpile groups. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIII, Tome 421 (2014), pp. 81-93. http://geodesic.mathdoc.fr/item/ZNSL_2014_421_a6/

[1] T. van Aardenne-Ehrenfest, N. G. de Bruijn, “Circuits and trees in oriented linear graphs”, Simon Stevin, 28 (1951), 203–217 | MR | Zbl

[2] V. I. Arnold, From Hilbert's superposition problem to dynamical dystems, Transcript of a Lecture given at Fields Institute in June 1997. Online at http://www.pdmi.ras.ru/~arnsem/Arnold/arnlect1.ps.gz

[3] Chan Swee Hong, Bachelor's Thesis, NTU, Singapore, 2012

[4] S. H. Chan, H. D. L. Hollmann, D. V. Pasechnik, “Critical groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields”, Proc. EuroComb (2013), The Seventh European Conference on Combinatorics, Graph Theory and Applications, CRM Series, eds. J. Nešetřil, M. Pellegrini, Springer, 2013

[5] S. Chmutov, S. Duzhin, J. Mostovoy, Introduction to Vassiliev knot invariants, Cambridge Univ. Press, Cambridge, 2012 ; Draft version online at http://www.pdmi.ras.ru/~duzhin/papers/cdbook | MR | Zbl

[6] S. Duzhin, An explicit expansion of the Drinfeld associator up to degree 12, Web publication http://www.pdmi.ras.ru/~arnsem/dataprog

[7] S. W. Golomb, “Irreducible polynomials, sy nchronization codes, primitive necklaces and the cyclotomic algebra”, Comb. Math. Appl., Proc. Conf. Univ. North Carolina, 1969, 358–370 | MR | Zbl

[8] Michael E. Hoffman, “The algebra of multiple harmonic series”, J. Algebra, 194 (1997), 477–495 | DOI | MR | Zbl

[9] L. Levine, “Sandpile groups and spanning trees of directed line graphs”, J. Combin. Theory Ser. A, 118:2 (2011), 350–364 | DOI | MR | Zbl

[10] L. Levine, J. Propp, “What is $\dots$ a sandpile?”, Notices Amer. Math. Soc., 57:8 (2010), 976–979 | MR | Zbl

[11] R. Lidl, H. Niederreiter, Finite Fields, Cambridge Univ. Press, Cambridge, 2008 | MR | Zbl

[12] Ch. Reutenauer, Free Lie Algebras, The Clarendon Press, 1993 | MR | Zbl

[13] Computer algebra system Sage, http://www.sagemath.org

[14] N. J. A. Sloane, Online encyclopaedia of integer sequences, http://oeis.org/

[15] E. Witt, “Treue Darstellung Liescher Ringe”, J. reine angew. Math., 177 (1937), 152–160 | Zbl