Abstract
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.
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Notes
In the work of Cherednik and Schneider, non-contractible inversion triples are called “non-gatherable triangle triples.”
References
R. Biagioli, M. Bousquet-Mélou, F. Jouhet, P. Nadeau, Length enumeration of fully commutative elements in finite and affine Coxeter groups, J. Algebra 513 (2018) 466-515.
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.
R. Biagioli, F. Jouhet, P. Nadeau, Combinatorics of fully commutative involutions in classical Coxeter groups, Discrete Math. 338 (2015) 2242–2259.
R. Biagioli, F. Jouhet, P. Nadeau, Fully commutative elements in finite and affine Coxeter groups, Monatsh. Math. 178 (2015) 1–37.
T. Boothby, J. Burkert, M. Eichwald, D.C. Ernst, R.M. Green, M. Macauley, On the cyclically fully commutative elements of Coxeter groups, J. Algebraic Combin. 36 (2012) 123–148.
D. Callan, T. Mansour, M. Shattuck, Twelve subsets of permutations enumerated as maximally clustered permutations, Ann. Math. Inform. 47 (2017) 41–74.
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.
I. Cherednik, K. Schneider, Non-gatherable triples for classical affine root systems, Ann. Comb. 17 (2010) 619–654.
H. Denoncourt, B. Jones, The enumeration of maximally clustered permutations, Ann. Comb. 14 (2010) 65–84.
T. Denton, Affine permutations and an affine Catalan monoid, In: Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics, 2014, pp. 339–354.
C.K. Fan, A Hecke algebra quotient and properties of commutative elements of a Weyl group, Ph.D. thesis, M.I.T., 1995.
R.M. Green, On the maximally clustered elements of Coxeter groups, Ann. Comb. 14 (2010) 467–478.
R.M. Green, J. Losonczy, Freely braided elements in Coxeter groups, Ann. Comb. 6 (2002) 337–348.
R.M. Green, J. Losonczy, Freely braided elements in Coxeter groups, II, Adv. Appl. Math. 33 (2004) 26–39.
R.M. Green, J. Losonczy, Schubert varieties and free braidedness, Transform. Groups 9 (2004) 327–336.
C. Hanusa, B. Jones, The enumeration of fully commutative affine permutations, European J. Combin. 31 (2010) 1342–1359.
S. Hart, How many elements of a Coxeter group have a unique reduced expression?, J. Group Theory 20 (2017) 903–910.
J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge Univ. Press, Cambridge, 1990.
B. Jones, Kazhdan–Lusztig polynomials for maximally-clustered hexagon-avoiding permutations, J. Algebra 322 (2009) 3459–3477.
F. Jouhet, P. Nadeau, Long fully commutative elements in affine Coxeter groups, Integers 15 (2015) A36.
J. Losonczy, Maximally clustered elements and Schubert varieties, Ann. Comb. 11 (2007) 195–212.
T. Mansour, On an open problem of Green and Losonczy: exact enumeration of freely braided elements, Discrete Math. Theor. Comput. Sci. 6 (2004) 461–470.
H. Matsumoto, Générateurs et relations des groupes de Weyl généralisés, CR Acad. Sci. Paris 258 (1964) 3419–3422.
P. Nadeau, On the length of fully commutative elements, Trans. Amer. Math. Soc. 370 (2018) 5705–5724.
P. Papi, Affine permutations and inversion multigraphs, Electron. J. Combin. 4 (1997) #R5.
M. Pétréolle, Generating series of cyclically fully commutative elements is rational, 2016. ArXiv Preprint. math.CO/1612.03764.
M. Pétréolle, Characterization of cyclically fully commutative elements in finite and affine Coxeter groups, European J. Combin. 61 (2017) 106–132.
J.R. Stembridge, On the fully commutative elements of Coxeter groups, J. Algebraic Combin. 5 (1996) 353–385.
B.E. Tenner, Reduced decompositions and permutation patterns, J. Algebraic Combin. 24 (2006) 263–284.
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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The author would like to thank the anonymous referee for offering some very helpful comments and suggestions.
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Losonczy, J. On Inversion Triples and Braid Moves. Ann. Comb. 24, 531–547 (2020). https://doi.org/10.1007/s00026-020-00501-8
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DOI: https://doi.org/10.1007/s00026-020-00501-8