zbMATH — the first resource for mathematics

Dual garside structures and Coxeter sortable elements. (English) Zbl 07239835
Summary: In Artin-Tits groups attached to spherical Coxeter groups, we give a combinatorial formula to express the simple elements of the dual braidmonoids in the classical Artin generators. Every simple dual braid is obtained by lifting an \(S\)-reduced expression of its image in the Coxeter group, in a way which involves Reading’s \(c\)-sortable elements. It has as an immediate consequence that simple dual braids are Mikado braids (the known proofs of this result either require topological realizations of the Artin groups or categorification techniques), and hence that their images in the Iwahori-Hecke algebras have positivity properties. In the classical types, this requires to give an explicit description of the inverse of Reading’s bijection from noncrossing partitions of the Coxeter element \(c\) to \(c\)-sortable elements, which might be of independent interest. The bijections are described in terms of the noncrossing partition models in these types. While the proof of the formula is case-by-case, it is entirely combinatorial and we develop an approach which reduces a uniform proof to uniformly proving a lemma about inversion sets of \(c\)-sortable elements.
20F36 Braid groups; Artin groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
Full Text: DOI
[1] C. A. Athanasiadis and V. Reiner, Noncrossing partitions for the groupDn,SIAM J. Discrete Math.,18(2004), no. 2, 397-417.Zbl 1085.06001 MR 2112514 · Zbl 1085.06001
[2] B. Baumeister, M. Dyer, C. Stump, and P. Wegener, A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements,Proc. Amer. Math. Soc. Ser. B,1 (2014), 149-154.Zbl 1343.20041 MR 3294251 · Zbl 1343.20041
[3] B. Baumeister, T. Gobet, K. Roberts, P. Wegener, On the Hurwitz action in finite Coxeter groups,J. Group Theory,20(2017), no. 1, 103-131.Zbl 1368.20045 MR 3592608 · Zbl 1368.20045
[4] B. Baumeister and T. Gobet, Simple dual braids, noncrossing partitions and Mikado braids of typeDn,Bull. Lond. Math. Soc.,49(2017), no. 6, 1048-1065.Zbl 1379.20010 MR 3743487 · Zbl 1379.20010
[5] D. Bessis, The dual braid monoid,Ann. Sci. École Norm. Sup. (4),36(2003), no. 5, 647-683.Zbl 1064.20039 MR 2032983 · Zbl 1064.20039
[6] D. Bessis, The dual braid monoid,Ann. Sci. École Norm. Sup. (4),36(2003), no. 5, 647-683.Zbl 1064.20039 MR 2032983 · Zbl 1064.20039
[7] D. Bessis, Finite complex reflection arrangements areK.; 1/,Ann. of Math. (2),181 (2015), no. 3, 809-904.Zbl 1372.20036 MR 3296817 · Zbl 1372.20036
[8] D. Bessis, F. Digne, and J. Michel, Springer theory in braid groups and the Birman-Ko-Lee monoid,Pacific J. Math.,205(2002), no. 2, 287-309.Zbl 1056.20023 MR 1922736 · Zbl 1056.20023
[9] P. Biane, Some properties of crossings and partitions,Discrete Math.,175(1997), no. 1-3, 41-53.Zbl 0892.05006 MR 1475837 · Zbl 0892.05006
[10] J. Birman, K. H. Ko, and S. J. Lee, A new approach to the word and conjugacy problems in the braid groups,Adv. Math.,139(1998), no. 2, 322-353.Zbl 0937.20016 MR 1654165 · Zbl 0937.20016
[11] A. Björner and F. Brenti,Combinatorics of Coxeter groups, Graduate Texts in Mathematics, 231, Springer, New York, 2005.Zbl 1110.05001 MR 2133266 · Zbl 1110.05001
[12] N. Bourbaki,Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, 1337, Hermann, Paris, 1968.Zbl 0186.33001 MR 240238 · Zbl 0186.33001
[13] T. Brady, A. Kenny, and C. Watt, Climbing elements in finite Coxeter groups,Electron. J. Combin.,17(2010), no. 1, Research Paper 156, 11pp.Zbl 1211.20034 MR 2745709 · Zbl 1211.20034
[14] N. Brady, J. P. McCammond, B. Mühlherr, and W. D. Neumann, Rigidity of Coxeter groups and Artin groups. Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000),Geom. Dedicata,94(2002), 91-109.Zbl 1031.20035 MR 1950875 · Zbl 1031.20035
[15] T. Brady and C. Watt, Non-crossing partition lattices in finite real reflection groups,Trans. Amer. Math. Soc.,360(2008), no. 4, 1983-2005.Zbl 1187.20051 MR 2366971 · Zbl 1187.20051
[16] T. Brady and C. Watt, A partial order on the orthogonal group,Comm. Algebra,30(2002), no. 8, 3749-3754.Zbl 1018.20040 MR 1922309 · Zbl 1018.20040
[17] R. W. Carter, Conjugacy classes in the Weyl group,Compositio Math.,25(1972), 1-59. Zbl 0254.17005 MR 318337 · Zbl 0254.17005
[18] P. Dehornoy, Three-dimensional realizations of braids,J. London Math. Soc. (2),60(1999), no. 1, 108-132.Zbl 0940.20044 MR 1721819 · Zbl 0940.20044
[19] P. Dehornoy, F. Digne, D. Krammer, E. Godelle, and J. Michel.Foundations of Garside theory, EMS Tracts in Mathematics, 22, European Mathematical Society (EMS), Zürich, 2015.Zbl 1370.20001 MR 3362691 · Zbl 1370.20001
[20] F. Digne, Présentations duales des groupes de tresses de type affineAz,Comment. Math. Helv.,81(2006), no. 1, 23-47.Zbl 1143.20020 MR 2208796 · Zbl 1143.20020
[21] F. Digne, Garside presentation for Artin-Tits groups of typeCzn,Ann. Inst. Fourier (Grenoble),62(2012), no. 2, 641-666.Zbl 1260.20056 MR 2985512 · Zbl 1260.20056
[22] F. Digne and T. Gobet, Dual braid monoids, Mikado braids and positivity in Hecke algebras,Math. Z.,285(2017), no. 1-2, 215-238.Zbl 1400.20029 MR 3598810 · Zbl 1400.20029
[23] M. J. Dyer, Hecke algebras and reflections in Coxeter groups, Ph. D. thesis, University of Sydney, 1987.
[24] M. J. Dyer, Reflection subgroups of Coxeter systems,J. Algebra,135(1990), no. 1, 57-73. Zbl 0712.20026 MR 1076077 · Zbl 0712.20026
[25] M. J. Dyer, Modules for the dual nil Hecke ring. Available at: http://www3.nd.edu/ dyer/papers/nilhecke.pdf
[26] M. J. Dyer, On minimal lengths of expressions of Coxeter group elements as products of reflections,Proc. Amer. Math. Soc.,129(2001), no. 9, 2591-2595.Zbl 0980.20028 MR 1838781 · Zbl 0980.20028
[27] M. J. Dyer and G. I. Lehrer, On positivity in Hecke algebras,Geom. Dedicata,35(1990), no. 1-3, 115-125.Zbl 0717.20027 MR 1066561 · Zbl 0717.20027
[28] M. Geck, G. Hiss, F. Luebeck, G. Malle, and G. Pfeiffer, CHEVIE - a system for computing and processing generic character tables. Computational methods in Lie theory (Essen, 1994),Appl. Algebra Engrg. Comm. Comput.,7(1996), no. 3, 175-210.Zbl 0847.20006 MR 1486215 · Zbl 0847.20006
[29] T. Gobet, Noncrossing partitions, fully commutative elements and bases of the Temperley-Lieb algebra,J. Knot Theory Ramifications,25(2016), no. 6, 1650035, 27pp. Zbl 1365.57016 MR 3498138 · Zbl 1365.57016
[30] T. Gobet, Twisted filtrations of Soergel bimodules and linear Rouquier complexes,J. Algebra,484(2017), 275-309.Zbl 1386.20024 MR 3656723 · Zbl 1386.20024
[31] T. Gobet, On cycle decompositions in Coxeter groups,Sém. Lothar. Combin.,78B(2017), Art. 45, 12pp.Zbl 1384.20037 MR 3678627 · Zbl 1384.20037
[32] T. Gobet and N. Williams, Noncrossing partitions and Bruhat order,European J. Combin., 53(2016), 8-34.Zbl 1328.05014 MR 3434422 · Zbl 1328.05014
[33] A. Hubery and H. Krause, A categorification of non-crossing partitions,J. Eur. Math. Soc. (JEMS),18(2016), no. 10, 2273-2313.Zbl 1381.16006 MR 3551191 · Zbl 1381.16006
[34] J. Humphreys,Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, Cambridge, 1990.Zbl 0725.20028 MR 1066460 · Zbl 0725.20028
[35] L. T. Jensen, The 2-braid group and Garside normal form,Math. Z.,286(2017), no. 1-2, 491-520.Zbl 1423.20031 MR 3648506 · Zbl 1423.20031
[36] T. Licata and H. Queffelec, Braid groups of type ADE, Garside structures, and the categorified root lattice, 2017.arXiv:1703.06011
[37] J. Michel, The development version of the CHEVIE package of GAP3,J. Algebra,435 (2015), 308-336.Zbl 1322.20002 MR 3343221 · Zbl 1322.20002
[38] L. Paris, Artin monoids inject in their groups,Comment. Math. Helv.,77(2002), no. 3, 609-637.Zbl 1020.20026 MR 1933791 · Zbl 1020.20026
[39] N. Reading, Clusters, Coxeter-sortable elements and noncrossing partitions,Trans. Amer. Math. Soc.,359(2007), no. 12, 5931-5958.Zbl 1189.05022 MR 2336311 · Zbl 1189.05022
[40] N. Reading, Noncrossing partitions and the shard intersection order,J. Algebraic Combin., 33(2011), no. 4, 483-530.Zbl 1290.05163 MR 2781960 · Zbl 1290.05163
[41] N. Reading and D. E. Speyer, Sortable elements in infinite Coxeter groups,Trans. Amer. Math. Soc.,363(2011), no. 2, 699-761.Zbl 1231.20036 MR 2728584 · Zbl 1231.20036
[42] V.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.