A poset structure on the alternating group generated by 3-cycles

Mühle, Henri; Nadeau, Philippe

doi:10.5802/alco.83

Mühle, Henri ¹; Nadeau, Philippe ²

¹ Technische Universität Dresden Institut für Algebra Zellescher Weg 12–14 01069 Dresden Germany.
² Univ. Lyon CNRS Université Claude Bernard Lyon 1 UMR5208 Institut Camille Jordan F-69622 Villeurbanne France.

Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1285-1310.

Abstract

We investigate the poset structure on the alternating group that arises when the latter is generated by $3$ -cycles. We study intervals in this poset and give several enumerative results, as well as a complete description of the orbits of the Hurwitz action on maximal chains. Our motivating example is the well-studied absolute order arising when the symmetric group is generated by transpositions, i.e. $2$ -cycles, and we compare our results to this case along the way. In particular, noncrossing partitions arise naturally in both settings.

Received: 2018-03-31
Revised: 2019-01-20
Accepted: 2019-04-28
Published online: 2019-12-03

MR

DOI: 10.5802/alco.83

Classification: 06A07, 05A10, 05E15, 20B35
Keywords: Symmetric group, Alternating group, Noncrossing partitions, Hurwitz action, zeta polynomial

Author's affiliations:

Mühle, Henri ¹; Nadeau, Philippe ²

¹ Technische Universität Dresden Institut für Algebra Zellescher Weg 12–14 01069 Dresden Germany.
² Univ. Lyon CNRS Université Claude Bernard Lyon 1 UMR5208 Institut Camille Jordan F-69622 Villeurbanne France.

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Mühle, Henri; Nadeau, Philippe. A poset structure on the alternating group generated by 3-cycles. Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1285-1310. doi : 10.5802/alco.83. https://alco.centre-mersenne.org/articles/10.5802/alco.83/

References
Cited by

[1] Adin, Ron M.; Roichman, Yuval On maximal chains in the noncrossing partition lattice, J. Comb. Theory, Ser. A, Volume 125 (2014), pp. 18-46 | DOI | Zbl

[2] Armstrong, Drew Generalized noncrossing partitions and combinatorics of Coxeter groups, Memoirs of the American Mathematical Society, 202, American Mathematical Society, 2009 | MR | Zbl

[3] Athanasiadis, Christos A.; Kallipoliti, Myrto The absolute order on the symmetric group, constructible partially ordered sets and Cohen–Macaulay complexes, J. Comb. Theory, Ser. A, Volume 115 (2008), pp. 1286-1295 | DOI | MR | Zbl

[4] Athanasiadis, Christos A.; Roichman, Yuval The absolute order of a permutation representation of a Coxeter group, J. Algebr. Comb., Volume 39 (2014), pp. 75-98 | DOI | MR | Zbl

[5] Baumeister, Barbara; Gobet, Thomas; Roberts, Kieran; Wegener, Patrick On the Hurwitz action in finite Coxeter groups, J. Group Theory, Volume 20 (2017), pp. 103-131 | MR | Zbl

[6] Ben-Itzhak, Tzachi; Teicher, Mina Graph theoretic method for determining Hurwitz equivalence in the symmetric group, Isr. J. Math., Volume 135 (2003), pp. 83-91 | DOI | MR

[7] Bessis, David The dual braid monoid, Ann. Sci. Éc. Norm. Supér. (4), Volume 36 (2003) no. 5, pp. 647-683 | DOI | Numdam | MR | Zbl

[8] Bessis, David Finite complex reflection arrangements are $K (π, 1)$ , Ann. Math., Volume 181 (2015), pp. 809-904 | DOI | MR | Zbl

[9] Biane, Philippe Minimal factorizations of a cycle and central multiplicative functions on the infinite symmetric group, J. Comb. Theory, Ser. A, Volume 76 (1996), pp. 197-212 | DOI | MR | Zbl

[10] Biane, Philippe Some properties of crossings and partitions, Discrete Math., Volume 175 (1997), pp. 41-53 | DOI | MR | Zbl

[11] Björner, Anders Shellable and Cohen–Macaulay partially ordered sets, Trans. Am. Math. Soc., Volume 260 (1980), pp. 159-183 | DOI | MR | Zbl

[12] Brady, Thomas A partial order on the symmetric group and new $K (π, 1)$ ’s for the braid groups, Adv. Math., Volume 161 (2001), pp. 20-40 | DOI | MR | Zbl

[13] Brenti, Francesco; Reiner, Victor; Roichman, Yuval Alternating subgroups of Coxeter groups, J. Comb. Theory, Ser. A, Volume 115 (2008), pp. 845-877 | DOI | MR | Zbl

[14] Deligne, Pierre Letter to Eduard Looijenga, 1974 (Available at http://homepage.univie.ac.at/christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf)

[15] Deutsch, Emeric; Feretíc, Svjetlan; Noy, Marc Diagonally convex directed polyominoes and even trees: a bijection and related issues, Discrete Math., Volume 256 (2002), pp. 645-654 | DOI | MR | Zbl

[16] Garside, Frank A. The braid group and other groups, Q. J. Math., Volume 20 (1969), pp. 235-254 | DOI | MR | Zbl

[17] Gould, Henry W. Some generalizations of Vandermonde’s convolution, Am. Math. Mon., Volume 63 (1956), pp. 84-91 | DOI | MR | Zbl

[18] Goulden, Ian P.; Jackson, David M. The combinatorial relationship between trees, cacti and certain connection coefficients for the symmetric group, Eur. J. Comb., Volume 13 (1992), pp. 357-365 | DOI | MR | Zbl

[19] Goulden, Ian P.; Jackson, David M. Transitive factorizations in the symmetric group, and combinatorial aspects of singularity theory, Eur. J. Comb., Volume 21 (2000), pp. 1001-1016 | DOI | MR | Zbl

[20] Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren Concrete mathematics, Addison-Wesley, 1994 | Zbl

[21] Heller, Julia; Schwer, Petra Generalized non-crossing partitions and buildings, Electron. J. Comb., Volume 25 (2018), Paper no. P1.24, 29 pages | MR | Zbl

[22] Herzog, Marcel; Reid, Kenneth B. Representation of permutations as products of cycles of fixed length, Journal of the Australian Mathematical Society (Series A), Volume 22 (1976), pp. 321-331 | DOI | MR | Zbl

[23] Hou, Xiang-dong Hurwitz equivalence in tuples of generalized quaternion groups and dihedral groups, Electron. J. Comb., Volume 15 (2008), Paper no. R80, 10 pages | MR | Zbl

[24] Huang, Jia; Lewis, Joel B.; Reiner, Victor Absolute order in general linear groups, J. Lond. Math. Soc., Volume 95 (2017), pp. 223-247 | DOI | MR | Zbl

[25] Hurwitz, Adolf Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., Volume 39 (1891), pp. 1-60 | DOI | Zbl

[26] Krattenthaler, Christian; Müller, Thomas W. Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions, Trans. Am. Math. Soc., Volume 362 (2010), pp. 2723-2787 | DOI | MR | Zbl

[27] Kreweras, Germain Sur les partitions non croisées d’un cycle, Discrete Math., Volume 1 (1972), pp. 333-350 | DOI | Zbl

[28] Lando, Sergei K.; Zvonkin, Alexander K. Graphs on surfaces and their applications, Springer, Berlin, 2004 | MR | Zbl

[29] Mitsuhashi, Hideo The $q$ -analogue of the alternating group and its representations, J. Algebra, Volume 240 (2001), pp. 535-558 | DOI | MR | Zbl

[30] Mühle, Henri; Nadeau, Philippe; Williams, Nathan $k$ -indivisible noncrossing partitions (2019) (Available at https://arxiv.org/abs/1904.05573)

[31] Mühle, Henri; Ripoll, Vivien Connectivity properties of factorization posets in generated groups (2017) (Available at https://arxiv.org/abs/1710.02063) | Zbl

[32] Regev, Amitai; Roichman, Yuval Permutation statistics on the alternating group, Adv. Appl. Math., Volume 33 (2004), pp. 676-709 | DOI | MR | Zbl

[33] Rotbart, Aviv Generator sets for the alternating group, Sémin. Lothar. Comb., Volume 65 (2011), Paper no. B65b, 16 pages | MR | Zbl

[34] Sia, Charmaine Hurwitz equivalence in tuples of dihedral groups, dicyclic groups, and semidihedral groups, Electron. J. Comb., Volume 16 (2009), Paper no. R95, 17 pages | MR | Zbl

[35] Simion, Rodica Noncrossing partitions, Discrete Math., Volume 217 (2000), pp. 397-409 | MR | Zbl

[36] Sloane, Neil J. A. The Online Encyclopedia of Integer Sequences (http://www.oeis.org)

[37] Stanley, Richard P. Enumerative combinatorics, Vol. 2, Cambridge University Press, Cambridge, 2001 | Zbl

[38] Stanley, Richard P. Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 2012 (2nd edition) | Zbl

[39] Wegener, Patrick On the Hurwitz action in affine Coxeter groups (2017) (Available at https://arxiv.org/abs/1710.06694)

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