Geodesic
Noncommutative LR coefficients and crystal reflection operators
Richmond, Edward1 ; Tewari, Vasu2
1 Oklahoma State University Department of Mathematics Stillwater OK 74708, USA
2 University of Pennsylvania Department of Mathematics Philadelphia PA 19104, USA
Algebraic Combinatorics, Tome 4 (2021) no. 1, pp. 145-162.

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We relate noncommutative Littlewood–Richardson coefficients of Bessenrodt–Luoto–van Willigenburg to classical Littlewood–Richardson coefficients via crystal reflection operators. A key role is played by the combinatorics of frank words.

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DOI : 10.5802/alco.155
Classification : 05E05, 05A05, 05E10, 20C30
Mots-clés : Crystal operator, Littlewood–Richardson coefficient, noncommutative symmetric function, symmetric function, Schur function.

Richmond, Edward 1 ; Tewari, Vasu 2

1 Oklahoma State University Department of Mathematics Stillwater OK 74708, USA
2 University of Pennsylvania Department of Mathematics Philadelphia PA 19104, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Richmond, Edward; Tewari, Vasu. Noncommutative LR coefficients and crystal reflection operators. Algebraic Combinatorics, Tome 4 (2021) no. 1, pp. 145-162. doi : 10.5802/alco.155. https://geodesic-test.mathdoc.fr/articles/10.5802/alco.155/

[1] Alexandersson, Per Non-symmetric Macdonald polynomials and Demazure–Lusztig operators (2016) (https://arxiv.org/abs/1602.05153) | MR

[2] Allen, Edward E.; Hallam, Joshua; Mason, Sarah Dual immaculate quasisymmetric functions expand positively into Young quasisymmetric Schur functions, J. Combin. Theory Ser. A, Volume 157 (2018), pp. 70-108 | Zbl | MR | DOI

[3] Assaf, Sami; Searles, Dominic Kohnert tableaux and a lifting of quasi-Schur functions, J. Combin. Theory Ser. A, Volume 156 (2018), pp. 85-118 | Zbl | MR | DOI

[4] Berg, Chris; Bergeron, Nantel; Saliola, Franco; Serrano, Luis; Zabrocki, Mike A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions, Canad. J. Math., Volume 66 (2014) no. 3, pp. 525-565 | Zbl | MR | DOI

[5] Bessenrodt, Christine; Luoto, Kurt; van Willigenburg, Stephanie Skew quasisymmetric Schur functions and noncommutative Schur functions, Adv. Math., Volume 226 (2011) no. 5, pp. 4492-4532 | Zbl | MR | DOI

[6] Bessenrodt, Christine; Tewari, Vasu; van Willigenburg, Stephanie Littlewood–Richardson rules for symmetric skew quasisymmetric Schur functions, J. Combin. Theory Ser. A, Volume 137 (2016), pp. 179-206 | Zbl | MR | DOI

[7] Buch, Anders Skovsted The saturation conjecture (after A. Knutson and T. Tao), Enseign. Math. (2), Volume 46 (2000) no. 1-2, pp. 43-60 (With an appendix by William Fulton) | Zbl | MR

[8] Bump, Daniel; Schilling, Anne Crystal bases. Representations and combinatorics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017, xii+279 pages | Zbl | MR | DOI

[9] Fomin, Sergey Schur operators and Knuth correspondences, J. Combin. Theory Ser. A, Volume 72 (1995) no. 2, pp. 277-292 | Zbl | MR | DOI

[10] Fomin, Sergey; Greene, Curtis A Littlewood–Richardson miscellany, European J. Combin., Volume 14 (1993) no. 3, pp. 191-212 | Zbl | MR | DOI

[11] Fulton, William Young tableaux. With applications to representation theory and geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997, x+260 pages | Zbl | MR

[12] Gelʼfand, Israel M.; Krob, Daniel; Lascoux, Alain; Leclerc, Bernard; Retakh, Vladimir S.; Thibon, Jean-Yves Noncommutative symmetric functions, Adv. Math., Volume 112 (1995) no. 2, pp. 218-348 | Zbl | MR | DOI

[13] Gessel, Ira M. Multipartite P-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983) (Contemp. Math.), Volume 34, Amer. Math. Soc., Providence, RI, 1984, pp. 289-317 | Zbl | MR | DOI

[14] Haglund, James; Luoto, Kurt; Mason, Sarah; van Willigenburg, Stephanie Quasisymmetric Schur functions, J. Combin. Theory Ser. A, Volume 118 (2011) no. 2, pp. 463-490 | MR | DOI

[15] Knutson, Allen; Tao, Terence The honeycomb model of GLn(C) tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc., Volume 12 (1999) no. 4, pp. 1055-1090 | Zbl | MR | DOI

[16] Lam, Thomas; Lauve, Aaron; Sottile, Frank Skew Littlewood–Richardson rules from Hopf algebras, Int. Math. Res. Not. IMRN (2011) no. 6, pp. 1205-1219 | Zbl | MR | DOI

[17] Lascoux, Alain; Schützenberger, Marcel-Paul Le monoïde plaxique, Noncommutative structures in algebra and geometric combinatorics (Naples, 1978) (Quad. “Ricerca Sci.”), Volume 109, CNR, Rome, 1981, pp. 129-156 | Zbl | MR

[18] Lascoux, Alain; Schützenberger, Marcel-Paul Keys & standard bases, Invariant theory and tableaux (Minneapolis, MN, 1988) (IMA Vol. Math. Appl.), Volume 19, Springer, New York, 1990, pp. 125-144 | Zbl | MR

[19] Littlewood, Dudley E.; Richardson, Archibald R. Group characters and algebra, Phil. Trans. A, Volume 233 (1934), pp. 99-141 | Zbl

[20] Liu, Ricky Ini An algorithmic Littlewood–Richardson rule, J. Algebraic Combin., Volume 31 (2010) no. 2, pp. 253-266 | Zbl | MR | DOI

[21] Lothaire, M. Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications, 90, Cambridge University Press, Cambridge, 2002, xiv+504 pages | Zbl | MR | DOI

[22] Luoto, Kurt; Mykytiuk, Stefan; van Willigenburg, Stephanie An introduction to quasisymmetric Schur functions. Hopf algebras, quasisymmetric functions, and Young composition tableaux, SpringerBriefs in Mathematics, Springer, New York, 2013, xiv+89 pages | Zbl | MR | DOI

[23] Malvenuto, Claudia; Reutenauer, Christophe Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, Volume 177 (1995) no. 3, pp. 967-982 | Zbl | MR | DOI

[24] Mason, Sarah An explicit construction of type A Demazure atoms, J. Algebraic Combin., Volume 29 (2009) no. 3, pp. 295-313 | Zbl | MR | DOI

[25] Monical, Cara; Pechenik, Oliver; Searles, Dominic K-theoretic polynomials, Sém. Lothar. Combin., Volume 82B (2020), 10, 12 pages | Zbl | MR

[26] Pak, Igor; Vallejo, Ernesto Combinatorics and geometry of Littlewood–Richardson cones, European J. Combin., Volume 26 (2005) no. 6, pp. 995-1008 | Zbl | MR | DOI

[27] Reiner, Victor; Shimozono, Mark Key polynomials and a flagged Littlewood–Richardson rule, J. Combin. Theory Ser. A, Volume 70 (1995) no. 1, pp. 107-143 | Zbl | MR | DOI

[28] Reiner, Victor; Shimozono, Mark Plactification, J. Algebraic Combin., Volume 4 (1995) no. 4, pp. 331-351 | Zbl | MR | DOI

[29] Remmel, Jeffrey B.; Shimozono, Mark A simple proof of the Littlewood–Richardson rule and applications, Discrete Math., Volume 193 (1998) no. 1-3, pp. 257-266 Selected papers in honor of Adriano Garsia (Taormina, 1994) | Zbl | MR | DOI

[30] Sagan, Bruce E. The symmetric group. Representations, combinatorial algorithms, and symmetric functions, Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001, xvi+238 pages | Zbl | MR | DOI

[31] Schützenberger, Marcel-Paul La correspondance de Robinson, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) (Lecture Notes in Math.), Volume 579, Springer-Verlag, Berlin-New York, 1977, pp. 59-113 | Zbl | MR

[32] Searles, Dominic Polynomial bases: positivity and Schur multiplication, Trans. Amer. Math. Soc., Volume 373 (2020) no. 2, pp. 819-847 | Zbl | MR | DOI

[33] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) | Zbl | MR | DOI

[34] Tewari, Vasu Backward jeu de taquin slides for composition tableaux and a noncommutative Pieri rule, Electron. J. Combin., Volume 22 (2015) no. 1, 1.42, 50 pages | Zbl | MR

[35] Tewari, Vasu A Murnaghan–Nakayama rule for noncommutative Schur functions, European J. Combin., Volume 58 (2016), pp. 118-143 | Zbl | MR | DOI

[36] Thomas, Glânffrwd P. On Schensted’s construction and the multiplication of Schur functions, Adv. in Math., Volume 30 (1978) no. 1, pp. 8-32 | Zbl | MR | DOI

[37] Vakil, Ravi A geometric Littlewood-Richardson rule, Ann. of Math. (2), Volume 164 (2006) no. 2, pp. 371-421 (Appendix A written with A. Knutson) | Zbl | MR | DOI

[38] van Leeuwen, Marc A. A. The Littlewood–Richardson rule, and related combinatorics, Interaction of combinatorics and representation theory (MSJ Mem.), Volume 11, Math. Soc. Japan, Tokyo, 2001, pp. 95-145 | Zbl | DOI | MR

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