Fully commutative elements in finite and affine Coxeter groups

An element w of a Coxeter group W is said to be fully commutative, if any reduced expression of w can be obtained from any other by transposing adjacent pairs of generators. These elements were described in 1996 by Stembridge in the case of finite irreducible groups, and more recently by Biagioli, Jouhet and Nadeau (BJN) in the affine cases. We focus here on the length enumeration of these elements. Using a recursive description, BJN established for the associated generating functions systems of non-linear q-equations. Here, we show that an alternative recursive description leads to explicit expressions for these generating functions.

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

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

[3]  Diagram calculus for a type affine C Temperley–Lieb algebra, II , 2018, Journal of Pure and Applied Algebra.

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

[5]  G. Viennot.  Heaps of Pieces, I: Basic Definitions and Combinatorial Lemmas , 1989 .

[6]  On the length of fully commutative elements , 2015, 1511.08788.

[7]  Sergei A. Abramov.  Rational solutions of linear difference and q-difference equations with polynomial coefficients , 1995, ISSAC '95.

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

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

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

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

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

[13]  Riccardo Biagioli,et al.  Combinatorics of fully commutative involutions in classical Coxeter groups , 2015, Discret. Math..

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

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

[16]  Svjetlan Feretic,et al.  An alternative method for q-counting directed column-convex polyominoes , 2000, Discret. Math..

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

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

[19]  Mathias Pétréolle,et al.  Characterization of cyclically fully commutative elements in finite and affine Coxeter groups , 2014, Eur. J. Comb..

[20]  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.

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

[22]  T. Prellberg,et al.  Critical exponents from nonlinear functional equations for partially directed cluster models , 1995 .

[23]  Nicolas Bourbaki,et al.  Eléments de Mathématique , 1964 .

[24]  Ronald L. Rivest,et al.  Asymptotic bounds for the number of convex n-ominoes , 1974, Discret. Math..

[25]  Pierre Cartier,et al.  Problemes combinatoires de commutation et rearrangements , 1969 .

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

[27]  Jean-Marc Fedou,et al.  Enumeration of skew Ferrers diagrams , 1993, Discret. Math..

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

[29]  C. KENNETH FAN,et al.  STRUCTURE OF A HECKE ALGEBRA QUOTIENT , 1996 .

[30]  C. Krattenthaler,et al.  THE THEORY OF HEAPS AND THE CARTIER–FOATA MONOID , 2022 .

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

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

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

[34]  A. Owczarek,et al.  Exact solution of the discrete (1+1)-dimensional SOS model with field and surface interactions , 1993 .

[35]  J. Stembridge.  On the fully commutative elements of Coxeter groups , 1996 .

[36]  Patxi Laborde-Zubieta,et al.  Periodic parallelogram polyominoes , 2017, Electron. Notes Discret. Math..

[37]  I. Goulden,et al.  Combinatorial Enumeration , 2004 .

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

[39]  Svjetlan Feretic.  A q-enumeration of convex polyominoes by the festoon approach , 2004, Theor. Comput. Sci..

[40]  Riccardo Biagioli,et al.  321-avoiding Affine Permutations, Heaps, and Periodic Parallelogram Polyominoes , 2017, Electron. Notes Discret. Math..

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

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

[43]  Mireille Bousquet-Mélou,et al.  Codage des polyominos convexes et équations pour l'énumération suivant l'aire , 1994, Discret. Appl. Math..

[44]  Patxi Laborde-Zubieta,et al.  Generating series of Periodic Parallelogram polyominoes , 2016 .

[45]  Sabrina Hirsch,et al.  Reflection Groups And Coxeter Groups , 2016 .

[46]  A. Björner,et al.  Combinatorics of Coxeter Groups , 2005 .

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

[48]  Mireille Bousquet-Mélou,et al.  Stacking of segments and q -enumeration of convex directed polyominoes , 1992 .

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

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

[51]  Riccardo Biagioli,et al.  Fully commutative elements in finite and affine Coxeter groups , 2014 .

[52]  H. Temperley.  Combinatorial Problems Suggested by the Statistical Mechanics of Domains and of Rubber-Like Molecules , 1956 .

[53]  Bruno Salvy,et al.  Non-Commutative Elimination in Ore Algebras Proves Multivariate Identities , 1998, J. Symb. Comput..

[54]  Mireille Bousquet-Mélou,et al.  The generating function of convex polyominoes: The resolution of a q-differential system , 1995, Discret. Math..

[55]  Mireille Bousquet-Mélou,et al.  Bijection of convex polyominoes and equations for enumerating them according to area , 1994 .

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

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