A poset structure on the alternating group generated by 3-cycles
Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1285-1310.

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: 2018-04-01
Revised: 2019-01-21
Accepted: 2019-04-29
Published online: 2019-12-04
DOI: https://doi.org/10.5802/alco.83
Classification: 06A07,  05A10,  05E15,  20B35
Keywords: Symmetric group, Alternating group, Noncrossing partitions, Hurwitz action, zeta polynomial
@article{ALCO_2019__2_6_1285_0,
     author = {M\"uhle, Henri and Nadeau, Philippe},
     title = {A poset structure on the alternating group generated by 3-cycles},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {2},
     number = {6},
     year = {2019},
     pages = {1285-1310},
     doi = {10.5802/alco.83},
     mrnumber = {4049847},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2019__2_6_1285_0/}
}
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/item/ALCO_2019__2_6_1285_0/

[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 | Article | Zbl 1295.05266

[2] Armstrong, Drew Generalized noncrossing partitions and combinatorics of Coxeter groups, Memoirs of the American Mathematical Society, Volume 202, American Mathematical Society, 2009 | MR 2561274 | Zbl 1191.05095

[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 | Article | MR 2450345 | Zbl 1189.05173

[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 | Article | MR 3144393 | Zbl 1294.05166

[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 3592608 | Zbl 1368.20045

[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 | Article | MR 1996396

[7] Bessis, David The dual braid monoid, Ann. Sci. Éc. Norm. Supér. (4), Volume 36 (2003) no. 5, pp. 647-683 | Article | Numdam | MR 2032983 | Zbl 1064.20039

[8] Bessis, David Finite complex reflection arrangements are K(π,1), Ann. Math., Volume 181 (2015), pp. 809-904 | Article | MR 3296817 | Zbl 1372.20036

[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 | Article | MR 1416014 | Zbl 0861.05001

[10] Biane, Philippe Some properties of crossings and partitions, Discrete Math., Volume 175 (1997), pp. 41-53 | Article | MR 1475837 | Zbl 0892.05006

[11] Björner, Anders Shellable and Cohen–Macaulay partially ordered sets, Trans. Am. Math. Soc., Volume 260 (1980), pp. 159-183 | Article | MR 570784 | Zbl 0441.06002

[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 | Article | MR 1857934 | Zbl 1011.20040

[13] Brenti, Francesco; Reiner, Victor; Roichman, Yuval Alternating subgroups of Coxeter groups, J. Comb. Theory, Ser. A, Volume 115 (2008), pp. 845-877 | Article | MR 2417024 | Zbl 1211.20035

[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 | Article | MR 1935781 | Zbl 1012.05049

[16] Garside, Frank A. The braid group and other groups, Q. J. Math., Volume 20 (1969), pp. 235-254 | Article | MR 248801 | Zbl 0194.03303

[17] Gould, Henry W. Some generalizations of Vandermonde’s convolution, Am. Math. Mon., Volume 63 (1956), pp. 84-91 | Article | MR 75170 | Zbl 0072.00702

[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 | Article | MR 1181077 | Zbl 0804.05023

[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 | Article | MR 1797682 | Zbl 0965.05003

[20] Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren Concrete mathematics, Addison-Wesley, 1994 | Zbl 0836.00001

[21] Heller, Julia; Schwer, Petra Generalized non-crossing partitions and buildings, Electron. J. Comb., Volume 25 (2018), 29 pages | MR 3785003 | Zbl 1380.05198

[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 | Article | MR 430034 | Zbl 0349.05007

[23] Hou, Xiang-dong Hurwitz equivalence in tuples of generalized quaternion groups and dihedral groups, Electron. J. Comb., Volume 15 (2008), 10 pages | MR 2411457 | Zbl 1188.20032

[24] Huang, Jia; Lewis, Joel B.; Reiner, Victor Absolute order in general linear groups, J. Lond. Math. Soc., Volume 95 (2017), pp. 223-247 | Article | MR 3653091 | Zbl 06775076

[25] Hurwitz, Adolf Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., Volume 39 (1891), pp. 1-60 | Article | Zbl 23.0429.01

[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 | Article | MR 2584617 | Zbl 1228.05306

[27] Kreweras, Germain Sur les partitions non croisées d’un cycle, Discrete Math., Volume 1 (1972), pp. 333-350 | Article | Zbl 0231.05014

[28] Lando, Sergei K.; Zvonkin, Alexander K. Graphs on surfaces and their applications, Springer, Berlin, 2004 | MR 2036721 | Zbl 1040.05001

[29] Mitsuhashi, Hideo The q-analogue of the alternating group and its representations, J. Algebra, Volume 240 (2001), pp. 535-558 | Article | MR 1841346 | Zbl 1046.20010

[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 07054649

[32] Regev, Amitai; Roichman, Yuval Permutation statistics on the alternating group, Adv. Appl. Math., Volume 33 (2004), pp. 676-709 | Article | MR 2095861 | Zbl 1057.05004

[33] Rotbart, Aviv Generator sets for the alternating group, Sémin. Lothar. Comb., Volume 65 (2011), 16 pages | MR 2800980 | Zbl 1295.05269

[34] Sia, Charmaine Hurwitz equivalence in tuples of dihedral groups, dicyclic groups, and semidihedral groups, Electron. J. Comb., Volume 16 (2009), 17 pages | MR 2529804 | Zbl 1191.20035

[35] Simion, Rodica Noncrossing partitions, Discrete Math., Volume 217 (2000), pp. 397-409 | MR 1766277 | Zbl 0959.05009

[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 0978.05002

[38] Stanley, Richard P. Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 2012 (2nd edition) | Zbl 1247.05003

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