Fully commutative elements in finite and affine Coxeter groups

An element of a Coxeter group $$W$$W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge, in particular in the finite case. They index naturally a basis of the generalized Temperley–Lieb algebra. In this work we deal with any finite or affine Coxeter group $$W$$W, and we give explicit descriptions of fully commutative elements. Using our characterizations we then enumerate these elements according to their Coxeter length, and find in particular that the corrresponding growth sequence is ultimately periodic in each type. When the sequence is infinite, this implies that the associated Temperley–Lieb algebra has linear growth.

[1]  T. Lenagan,et al.  Gelfand-Kirillov dimension of algebras , 1999 .

[2]  R. M. Green.  ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En , 2007, 0704.0283.

[3]  R. Carter.  REFLECTION GROUPS AND COXETER GROUPS (Cambridge Studies in Advanced Mathematics 29) , 1991 .

[4]  C. Fan,et al.  Structure of a Hecke algebra quotient , 1997 .

[5]  Y. Samoĭlenko,et al.  Growth of generalized Temperley–Lieb algebras connected with simple graphs , 2009 .

[6]  Richard P. Stanley,et al.  Some Combinatorial Properties of Schubert Polynomials , 1993 .

[7]  GENERALIZED TEMPERLEY–LIEB ALGEBRAS AND DECORATED TANGLES , 1997, q-alg/9712018.

[8]  John R. Stembridge,et al.  Some combinatorial aspects of reduced words in finite Coxeter groups , 1997 .

[9]  Cellular algebras arising from Hecke algebras of type $H_n$ , 1997, q-alg/9712019.

[10]  V. A. Ufnarovskij.  Combinatorial and Asymptotic Methods in Algebra , 1995 .

[11]  R. Green,et al.  Fully commutative Kazhdan-Lusztig cells , 2001, math/0102003.

[12]  D. Ernst.  Diagram calculus for a type affine C Temperley–Lieb algebra, II , 2009, Journal of Pure and Applied Algebra.

[13]  Mireille Bousquet-Mélou,et al.  A method for the enumeration of various classes of column-convex polygons , 1996, Discret. Math..

[14]  Alberto Del Lungo,et al.  Some permutations with forbidden subsequences and their inversion number , 2001, Discret. Math..

[15]  Vaughan F. R. Jones,et al.  Hecke algebra representations of braid groups and link polynomials , 1987 .

[16]  On the cyclically fully commutative elements of Coxeter groups , 2012, 1202.6657.

[17]  Elliott H Lieb,et al.  Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[18]  Jian-yi Shi.  Fully commutative elements and Kazhdan-Lusztig cells in the finite and affine coxeter groups , 2003 .

[19]  John R. Stembridge,et al.  On the Fully Commutative Elements of Coxeter Groups , 1996 .

[20]  R. M. Green,et al.  On the Affine Temperley–Lieb Algebras , 1997, Journal of the London Mathematical Society.

[21]  Manabu Hagiwara.  Minuscule Heaps over Dynkin Diagrams of Type à , 2004, Electron. J. Comb..

[22]  Alexander Postnikov.  Affine approach to quantum Schubert calculus , 2002 .

[23]  J. Humphreys.  Reflection groups and coxeter groups , 1990 .

[24]  John R. Stembridge.  The Enumeration of Fully Commutative Elements of Coxeter Groups , 1998 .

[25]  John R. Stembridge,et al.  Minuscule elements of Weyl groups , 2001 .

[26]  C. Kenneth Fan,et al.  A Hecke algebra quotient and properties of commutative elements of a Weyl group , 1995 .

[27]  Philippe Nadeau,et al.  Long Fully Commutative Elements in Affine Coxeter Groups , 2015, Integers.

[28]  Sergey Fomin,et al.  Noncommutative schur functions and their applications , 2006, Discret. Math..

[29]  Victor Reiner.  Combinatorics of minuscule representations (Cambridge Tracts in Mathematics 199) By R. M. Green , 2015 .

[30]  Christopher R. H. Hanusa,et al.  The enumeration of fully commutative affine permutations , 2009, Eur. J. Comb..

[31]  R. M. Green.  Combinatorics of Minuscule Representations , 2013 .

[32]  R. M. Green.  On 321-Avoiding Permutations in Affine Weyl Groups , 2001 .

[33]  George Lusztig,et al.  Some examples of square integrable representations of semisimple p-adic groups , 1983 .