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A bijective proof of Macdonald’s reduced word formula. (English) Zbl 1409.05024
Summary: We give a bijective proof of Macdonald’s reduced word identity using pipe dreams and Little’s bumping algorithm. This proof extends to a principal specialization due to S. Fomin and R. P. Stanley [Adv. Math. 103, No. 2, 196–207 (1994; Zbl 0809.05091)]. Such a proof has been sought for over 20 years. Our bijective tools also allow us to solve a problem posed by S. Fomin and A. N. Kirillov [J. Algebr. Comb. 6, No. 4, 311–319 (1997; Zbl 0882.05010)] using work of Wachs, Lenart, Serrano and Stump. These results extend earlier work by the third author [“A Markov growth process for Macdonald’s distribution on reduced words”, Preprint, arXiv:1409.7714] on a Markov process for reduced words of the longest permutation.

MSC:
05A17 Combinatorial aspects of partitions of integers
05A15 Exact enumeration problems, generating functions
05E10 Combinatorial aspects of representation theory
05E05 Symmetric functions and generalizations
14M15 Grassmannians, Schubert varieties, flag manifolds
60J99 Markov processes
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[1] Bergeron, Nantel; Billey, Sara C., RC-graphs and Schubert polynomials, Exp. Math., 2, 4, 257-269, (1993) · Zbl 0803.05054
[2] Billey, Sara C.; Hamaker, Zachary; Roberts, Austin; Young, Benjamin, Coxeter-Knuth graphs and a signed Little map for type B reduced words, Electron. J. Comb., 21, 4, 39 p. pp., (2014) · Zbl 1298.05006
[3] Billey, Sara C.; Jockusch, William; Stanley, Richard P., Some Combinatorial Properties of Schubert Polynomials, J. Algebr. Comb., 2, 4, 345-374, (1993) · Zbl 0790.05093
[4] Bressoud, David M., Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture, xv+274 p. pp., (1999), Cambridge University Press · Zbl 0944.05001
[5] Carlitz, Leonard, A combinatorial property of \(q\)-Eulerian numbers, Am. Math. Mon., 82, 51-54, (1975) · Zbl 0296.05007
[6] Chevalley, Claude, Algebraic Groups and their generalizations: Classical methods (University Park, PA, 1991), 56, Part 1, Sur les décompositions cellulaires des espaces \({G}/{B}\), 1-23, (1994), American Mathematical Society · Zbl 0824.14042
[7] Edelman, Paul; Greene, Curtis, Balanced tableaux, Adv. Math., 63, 1, 42-99, (1987) · Zbl 0616.05005
[8] Foata, Dominique, On the Netto inversion number of a sequence, Proc. Am. Math. Soc., 19, 236-240, (1968) · Zbl 0157.03403
[9] Fomin, Sergey; Kirillov, Anatol N., Grothendieck polynomials and the Yang-Baxter equation, Proceedings of the Sixth Conference in Formal Power Series and Algebraic Combinatorics, 183-190, (1994), American Mathematical Society
[10] Fomin, Sergey; Kirillov, Anatol N., Yang-Baxter Equation, Symmetric Functions, and Schubert Polynomials, Discrete Math., 153, 1-3, 123-143, (1996) · Zbl 0852.05078
[11] Fomin, Sergey; Kirillov, Anatol N., Reduced words and plane partitions, J. Algebr. Comb., 6, 4, 311-319, (1997) · Zbl 0882.05010
[12] Fomin, Sergey; Stanley, Richard P., Schubert Polynomials and the NilCoxeter Algebra, Adv. Math., 103, 2, 196-207, (1994) · Zbl 0809.05091
[13] Garsia, Adriano, The Saga of Reduced Factorizations of Elements of the Symmetric Group, 29, (2002), Laboratoire de combinatoire et d’informatique mathématique
[14] Gessel, Ira M.; Viennot, Xavier, Determinants, paths, and plane partitions, (1989)
[15] Gupta, Hansraj, A new look at the permutations of the first \(n\) natural numbers, Indian J. Pure Appl. Math., 9, 6, 600-631, (1978) · Zbl 0386.05005
[16] Haglund, James; Loehr, Nicholas; Remmel, Jeffrey B., Statistics on wreath products, perfect matchings, and signed words, Eur. J. Comb., 26, 6, 835-868, (2005) · Zbl 1063.05009
[17] Hamaker, Zachary; Marberg, Eric; Pawlowski, Brendan, Transition formulas for involution Schubert polynomials, (2016) · Zbl 1452.20002
[18] Hamaker, Zachary; Young, Benjamin, Relating Edelman-Greene insertion to the Little map, J. Algebr. Comb., 40, 3, 693-710, (2014) · Zbl 1309.05007
[19] Kasteleyn, Pieter W., The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice, Physica, 27, 12, 1209-1225, (1961) · Zbl 1244.82014
[20] Knutson, Allen, Schubert polynomials and symmetric functions; notes for the Lisbon combinatorics summer school 2012, (2012)
[21] Knutson, Allen; Miller, Ezra, Gröbner geometry of Schubert polynomials, Ann. Math., 161, 3, 1245-1318, (2005) · Zbl 1089.14007
[22] Kogan, Mikhail, Schubert geometry of Flag Varieties and Gelfand-Cetlin theory, (2000)
[23] Koike, Kazuhiko; Terada, Itaru, Young-diagrammatic methods for the representation theory of the classical groups of type \({B}_n, {C}_n, {D}_n\), J. Algebra, 107, 2, 466-511, (1987) · Zbl 0622.20033
[24] Lam, Thomas; Lapointe, Luc; Morse, Jennifer; Schilling, Anne; Shimozono, Mark; Zabrocki, Mike, k-Schur Functions and Affine Schubert Calculus, 33, (2014), The Fields Institute for Research in the Mathematical Sciences · Zbl 1360.14004
[25] Lam, Thomas; Shimozono, Mark, A Little bijection for affine Stanley symmetric functions, Sémin. Lothar. Comb., 54A, 12 p. pp., (2005) · Zbl 1178.05009
[26] Lascoux, Alain, Proceedings of the Nagoya 2000 International Workshop, Transition on Grothendieck polynomials, 164-179, (2000), World Scientific
[27] Lascoux, Alain; Schützenberger, Marcel-Paul, Polynômes de Schubert, C. R. Acad. Sci., Paris, Sér. I, 294, 447-450, (1982) · Zbl 0495.14031
[28] Lascoux, Alain; Schützenberger, Marcel-Paul, Invariant theory (Montecatini, 1982), 996, Symmetry and flag manifolds, 118-144, (1983), Springer · Zbl 0542.14031
[29] Lascoux, Alain; Schützenberger, Marcel-Paul, Schubert Polynomials and the Littlewood-Richardson Rule, Lett. Math. Phys., 10, 111-124, (1985) · Zbl 0586.20007
[30] Lenart, Cristian, A unified approach to combinatorial formulas for Schubert polynomials, J. Algebr. Comb., 20, 3, 263-299, (2004) · Zbl 1056.05146
[31] Little, David P., Combinatorial aspects of the Lascoux-Schützenberger tree, Adv. Math., 174, 2, 236-253, (2003) · Zbl 1018.05102
[32] Little, David P., Factorization of the Robinson-Schensted-Knuth correspondence, J. Comb. Theory, Ser. A, 110, 1, 147-168, (2005) · Zbl 1059.05106
[33] Macdonald, Ian G., Notes on Schubert Polynomials, 6, (1991), Université du Québec
[34] Manivel, Laurent, Symmetric Functions, Schubert Polynomials and Degeneracy Loci, 6, (2001), American Mathematical Society · Zbl 0998.14023
[35] Monks Gillespie, Maria, A combinatorial approach to the \(q,t\)-symmetry relation in Macdonald polynomials, Electron. J. Comb., 23, 2, 64 p. pp., (2016) · Zbl 1337.05109
[36] Novick, Mordechai, A bijective proof of a major index theorem of Garsia and Gessel, Electron. J. Comb., 17, 1, 12 p. pp., (2010) · Zbl 1189.05010
[37] Postnikov, Alexander; Stanley, Richard P., Chains in the Bruhat order, J. Algebr. Comb., 29, 2, 133-174, (2009) · Zbl 1238.14036
[38] Proctor, Robert A., Odd symplectic groups, Invent. Math., 92, 2, 307-332, (1988) · Zbl 0621.22009
[39] Proctor, Robert A., New symmetric plane partition identities from invariant theory work of De Concini and Procesi, Eur. J. Comb., 11, 3, 289-300, (1990) · Zbl 0726.05008
[40] Reiner, Victor; Tenner, Bridget Eileen; Yong, Alexander, Poset edge densities, nearly reduced words, and barely set-valued tableaux, (2016) · Zbl 1391.05269
[41] Serrano, Luis; Stump, Christian, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Generalized triangulations, pipe dreams, and simplicial spheres, 885-896, (2011), The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS) · Zbl 1355.05078
[42] Serrano, Luis; Stump, Christian, Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials, Electron. J. Comb., 19, 1, 18 p. pp., (2012) · Zbl 1244.05239
[43] Skandera, Mark, An Eulerian partner for inversions, Sémin. Lothar. Comb., 46, 19 p. pp., (2001) · Zbl 0982.05006
[44] Stanley, Richard P., On the Number of Reduced Decompositions of Elements of Coxeter Groups, Eur. J. Comb., 5, 359-372, (1984) · Zbl 0587.20002
[45] Stanley, Richard P., Permutations, (2009)
[46] Stembridge, John R., A weighted enumeration of maximal chains in the Bruhat order, J. Algebr. Comb., 15, 3, 291-301, (2002) · Zbl 1008.20037
[47] Wachs, Michelle L., Flagged Schur functions, Schubert polynomials, and symmetrizing operators, J. Comb. Theory, Ser. A, 40, 2, 276-289, (1985) · Zbl 0579.05001
[48] Weigandt, Anna; Yong, Alexander, The prism tableau model for Schubert polynomials, J. Comb. Theory, Ser. A, 154, 551-582, (2018) · Zbl 1373.05219
[49] Young, Benjamin, A Markov growth process for Macdonald’s distribution on reduced words, (2014)
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