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On inversion triples and braid moves. (English) Zbl 1451.05249
Summary: An inversion triple of an element $$w$$ of a simply laced Coxeter group $$W$$ is a set $$\{ \alpha, \beta, \alpha + \beta \}$$, where each element is a positive root sent negative by $$w$$. We say that an inversion triple of $$w$$ is contractible if there is a root sequence for $$w$$ in which the roots of the triple appear consecutively. Such triples arise in the study of the commutation classes of reduced expressions of elements of $$W$$, and have been used to define or characterize certain classes of elements of $$W$$, e.g., the fully commutative elements and the freely braided elements. Also, the study of inversion triples is connected with the representation theory of affine Hecke algebras and double affine Hecke algebras. In this paper, we describe the inversion triples that are contractible, and we give a new, simple characterization of the groups $$W$$ with the property that all inversion triples are contractible. We also study the natural action of $$W$$ on the set of all triples of (not necessarily positive) roots of the form $$\{ \alpha, \beta, \alpha + \beta \}$$. This enables us to prove rather quickly that every triple of positive roots $$\{ \alpha, \beta, \alpha + \beta \}$$ is contractible for some $$w$$ in $$W$$ and, moreover, when $$W$$ is finite, $$w$$ may be taken to be the longest element of $$W$$. At the end of the paper, we pose a problem concerning the aforementioned action.
##### MSC:
 05E16 Combinatorial aspects of groups and algebras 20F55 Reflection and Coxeter groups (group-theoretic aspects)
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##### References:
 [1] Biagioli, R.; Bousquet-Mélou, M.; Jouhet, F.; Nadeau, P., Length enumeration of fully commutative elements in finite and affine Coxeter groups, J. Algebra, 513, 466-515 (2018) · Zbl 1402.20050 [2] R. Biagioli, F. Jouhet, P. Nadeau, Fully commutative elements and lattice walks, In: FPSAC 2013, pp. 145-156. Discrete Mathematics and Theoretical Computer Science, 2013. · Zbl 1294.05012 [3] Biagioli, R.; Jouhet, F.; Nadeau, P., Combinatorics of fully commutative involutions in classical Coxeter groups, Discrete Math, 338, 2242-2259 (2015) · Zbl 1318.05089 [4] Biagioli, R.; Jouhet, F.; Nadeau, P., Fully commutative elements in finite and affine Coxeter groups, Monatsh. Math., 178, 1-37 (2015) · Zbl 1323.05136 [5] Boothby, T.; Burkert, J.; Eichwald, M.; Ernst, DC; Green, RM; Macauley, M., On the cyclically fully commutative elements of Coxeter groups, J Algebraic Combin, 36, 123-148 (2012) · Zbl 1253.20037 [6] Callan, D.; Mansour, T.; Shattuck, M., Twelve subsets of permutations enumerated as maximally clustered permutations, Ann. Math. Inform., 47, 41-74 (2017) · Zbl 1399.05005 [7] I. Cherednik, K. Schneider, Non-gatherable triples for non-affine root systems, SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) 4:079 (2008) 12 pp. · Zbl 1196.17013 [8] Cherednik, I.; Schneider, K., Non-gatherable triples for classical affine root systems, Ann. Comb., 17, 619-654 (2010) · Zbl 1368.17014 [9] Denoncourt, H.; Jones, B., The enumeration of maximally clustered permutations, Ann. Comb., 14, 65-84 (2010) · Zbl 1233.05009 [10] T. Denton, Affine permutations and an affine Catalan monoid, In: Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics, 2014, pp. 339-354. · Zbl 1434.05003 [11] C.K. Fan, A Hecke algebra quotient and properties of commutative elements of a Weyl group, Ph.D. thesis, M.I.T., 1995. [12] Green, RM, On the maximally clustered elements of Coxeter groups, Ann. Comb., 14, 467-478 (2010) · Zbl 1234.20047 [13] Green, RM; Losonczy, J., Freely braided elements in Coxeter groups, Ann. Comb., 6, 337-348 (2002) · Zbl 1052.20028 [14] Green, RM; Losonczy, J., Freely braided elements in Coxeter groups, II, Adv. Appl. Math., 33, 26-39 (2004) · Zbl 1063.20045 [15] Green, RM; Losonczy, J., Schubert varieties and free braidedness, Transform. Groups, 9, 327-336 (2004) · Zbl 1073.14064 [16] Hanusa, C.; Jones, B., The enumeration of fully commutative affine permutations, European J. Combin., 31, 1342-1359 (2010) · Zbl 1230.05028 [17] Hart, S., How many elements of a Coxeter group have a unique reduced expression?, J. Group Theory, 20, 903-910 (2017) · Zbl 06786542 [18] Humphreys, JE, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics (1990), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 0725.20028 [19] Jones, B., Kazhdan-Lusztig polynomials for maximally-clustered hexagon-avoiding permutations, J. Algebra, 322, 3459-3477 (2009) · Zbl 1210.20007 [20] Jouhet, F.; Nadeau, P., Long fully commutative elements in affine Coxeter groups, Integers, 15, #A36 (2015) · Zbl 1336.20041 [21] Losonczy, J., Maximally clustered elements and Schubert varieties, Ann. Comb., 11, 195-212 (2007) · Zbl 1155.20043 [22] Mansour, T., On an open problem of Green and Losonczy: exact enumeration of freely braided elements, Discrete Math. Theor. Comput. Sci., 6, 461-470 (2004) · Zbl 1066.05004 [23] Matsumoto, H., Générateurs et relations des groupes de Weyl généralisés, C.R. Acad. Sci. Paris, 258, 3419-3422 (1964) · Zbl 0128.25202 [24] Nadeau, P., On the length of fully commutative elements, Trans. Amer. Math. Soc., 370, 5705-5724 (2018) · Zbl 1388.05195 [25] P. Papi, Affine permutations and inversion multigraphs, Electron. J. Combin. 4 (1997) #R5. · Zbl 0885.05001 [26] M. Pétréolle, Generating series of cyclically fully commutative elements is rational, 2016. ArXiv Preprint. math.CO/1612.03764. [27] Pétréolle, M., Characterization of cyclically fully commutative elements in finite and affine Coxeter groups, Euro J. Combin., 61, 106-132 (2017) · Zbl 1352.05193 [28] Stembridge, JR, On the fully commutative elements of Coxeter groups, J. Algebraic Combin., 5, 353-385 (1996) · Zbl 0864.20025 [29] Tenner, BE, Reduced decompositions and permutation patterns, J. Algebraic Combin., 24, 263-284 (2006) · Zbl 1101.05003 [30] J. Tits, Le problème des mots dans les groupes de Coxeter, In: Ist. Naz. Alta Mat., 1968, Sympos. Math. 1, Academic Press, London, 1969, pp. 175-185.
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