Colored five-vertex models and Demazure atoms. (English) Zbl 1458.05257
Summary: Type A Demazure atoms are pieces of Schur functions, or sets of tableaux whose weights sum to such functions. Inspired by colored vertex models of Borodin and Wheeler, we will construct solvable lattice models whose partition functions are Demazure atoms; the proof of this makes use of a Yang-Baxter equation for a colored five-vertex model. As a byproduct, we will construct Demazure atoms on Kashiwara’s \(\mathcal{B}_\infty\) crystal and give new algorithms for computing Lascoux-Schützenberger keys.
MSC:
| 05E10 | Combinatorial aspects of representation theory |
| 16T25 | Yang-Baxter equations |
| 82B23 | Exactly solvable models; Bethe ansatz |
| 05E05 | Symmetric functions and generalizations |
References:
| [1] | Akutsu, Y.; Deguchi, T.; Ohtsuki, T., Invariants of colored links, J. Knot Theory Ramif., 1, 2, 161-184 (1992) · Zbl 0758.57004 |
| [2] | Assaf, S.; Schilling, A., A Demazure crystal construction for Schubert polynomials, Algebraic Combin., 1, 2, 225-247 (2017) · Zbl 1390.14162 |
| [3] | Aval, J.-C., Keys and alternating sign matrices, Sémin. Lothar. Comb., 59, Article B59f pp. (2007/10), 13 · Zbl 1179.05117 |
| [4] | Baxter, R. J., Exactly Solved Models in Statistical Mechanics (1982), Academic Press Inc. [Harcourt Brace Jovanovich Publishers]: Academic Press Inc. [Harcourt Brace Jovanovich Publishers] London · Zbl 0538.60093 |
| [5] | Berenstein, A.; Zelevinsky, A., Canonical bases for the quantum group of type \(A_r\) and piecewise-linear combinatorics, Duke Math. J., 82, 3, 473-502 (1996) · Zbl 0898.17006 |
| [6] | Bernstein, I. N.; Gel’fand, I. M.; Gel’fand, S. I., Schubert cells, and the cohomology of the spaces \(G / P\), Usp. Mat. Nauk, 28, 3(171), 3-26 (1973) · Zbl 0289.57024 |
| [7] | Borodin, A., On a family of symmetric rational functions, Adv. Math., 306, 973-1018 (2017) · Zbl 1355.05250 |
| [8] | Borodin, A.; Bufetov, A.; Wheeler, M., Between the stochastic six vertex model and Hall-Littlewood processes (2016) |
| [9] | Borodin, A.; Wheeler, M., Coloured stochastic vertex models and their spectral theory (2018) |
| [10] | Brubaker, B.; Buciumas, V.; Bump, D., A Yang-Baxter equation for metaplectic ice, Commun. Number Theory Phys., 13, 1, 101-148 (2019) · Zbl 1454.11096 |
| [11] | Brubaker, B.; Buciumas, V.; Bump, D.; Gustafsson, H., Colored vertex models and Iwahori Whittaker functions (2019) |
| [12] | Brubaker, B.; Bump, D.; Friedberg, S., Schur polynomials and the Yang-Baxter equation, Commun. Math. Phys., 308, 2, 281-301 (2011) · Zbl 1232.05234 |
| [13] | Brubaker, B.; Bump, D.; Friedberg, S., Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory, Annals of Mathematics Studies, vol. 175 (2011), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1288.11052 |
| [14] | Brubaker, B.; Frechette, C.; Hardt, A.; Tibor, E.; Weber, K., Frozen pipes: lattice models for Grothendieck polynomials (2020) |
| [15] | Buciumas, V.; Scrimshaw, T., Double Grothendieck polynomials and colored lattice models (2020) |
| [16] | Buciumas, V.; Scrimshaw, T.; Weber, K., Colored five-vertex models and Lascoux polynomials and atoms (2019) · Zbl 1447.05219 |
| [17] | Bump, D., Lie Groups, Graduate Texts in Mathematics, vol. 225 (2013), Springer: Springer New York · Zbl 1279.22001 |
| [18] | Bump, D.; Schilling, A., Crystal Bases, Representations and Combinatorics (2017), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ · Zbl 1440.17001 |
| [19] | Corwin, I.; Petrov, L., Stochastic higher spin vertex models on the line, Commun. Math. Phys., 343, 2, 651-700 (2016) · Zbl 1348.82055 |
| [20] | Demazure, M., Désingularisation des variétés de Schubert généralisées, Ann. Sci. Éc. Norm. Supér. (4), 7, 53-88 (1974), Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I · Zbl 0312.14009 |
| [21] | Deodhar, V. V., Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function, Invent. Math., 39, 2, 187-198 (1977) · Zbl 0333.20041 |
| [22] | Ehresmann, C., Sur la topologie de certains espaces homogènes, Ann. of Math. (2), 35, 2, 396-443 (1934) · JFM 60.1223.05 |
| [23] | Foda, O.; Wheeler, M., Colour-independent partition functions in coloured vertex models, Nucl. Phys. B, 871, 2, 330-361 (2013) · Zbl 1262.82009 |
| [24] | Fomin, S.; Kirillov, A. N., The Yang-Baxter equation, symmetric functions, and Schubert polynomials, (Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics. Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics, Florence, 1993, vol. 153 (1996)), 123-143 · Zbl 0852.05078 |
| [25] | Gorbounov, V.; Korff, C., Quantum integrability and generalised quantum Schubert calculus, Adv. Math., 313, 282-356 (2017) · Zbl 1386.14181 |
| [26] | Gorbounov, V.; Korff, C.; Stroppel, C., Yang-Baxter algebras as convolution algebras: the Grassmannian case (2018) |
| [27] | Hamel, A. M.; King, R. C., Bijective proofs of shifted tableau and alternating sign matrix identities, J. Algebraic Comb., 25, 4, 417-458 (2007) · Zbl 1122.05094 |
| [28] | Hersh, P.; Lenart, C., From the weak Bruhat order to crystal posets, Math. Z., 286, 3-4, 1435-1464 (2017) · Zbl 1371.05315 |
| [29] | Jacon, N.; Lecouvey, C., Keys and Demazure crystals for Kac-Moody algebras (2019) |
| [30] | Jimbo, M.; Miwa, T., Algebraic Analysis of Solvable Lattice Models, CBMS Regional Conference Series in Mathematics, vol. 85 (1995), American Mathematical Society: American Mathematical Society Providence, RI, Published for the Conference Board of the Mathematical Sciences, Washington, DC · Zbl 0828.17018 |
| [31] | Joseph, A., Consequences of the Littelmann path theory for the structure of the Kashiwara \(B(\infty)\) crystal, (Highlights in Lie Algebraic Methods. Highlights in Lie Algebraic Methods, Progr. Math., vol. 295 (2012), Birkhäuser/Springer: Birkhäuser/Springer New York), 25-64 · Zbl 1250.17025 |
| [32] | Kashiwara, M., The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J., 71, 3, 839-858 (1993) · Zbl 0794.17008 |
| [33] | Kashiwara, M., Bases cristallines des groupes quantiques, Cours Spécialisés [Specialized Courses], vol. 9 (2002), Société Mathématique de France: Société Mathématique de France Paris, Edited by Charles Cochet · Zbl 1066.17007 |
| [34] | Kashiwara, M.; Nakashima, T., Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra, 165, 2, 295-345 (1994) · Zbl 0808.17005 |
| [35] | Kirillov, A. N.; Berenstein, A. D., Groups generated by involutions, Gelfand-Tsetlin patterns, and combinatorics of Young tableaux, Algebra Anal., 7, 1, 92-152 (1995) · Zbl 0848.20007 |
| [36] | Knutson, A.; Miller, E., Gröbner geometry of Schubert polynomials, Ann. of Math. (2), 161, 3, 1245-1318 (2005) · Zbl 1089.14007 |
| [37] | Knutson, A.; Zinn-Justin, P., Schubert puzzles and integrability I: invariant trilinear forms (2017) |
| [38] | Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G., Quantum Inverse Scattering Method and Correlation Functions, Cambridge Monographs on Mathematical Physics (1993), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0787.47006 |
| [39] | Kuperberg, G., Another proof of the alternating-sign matrix conjecture, Int. Math. Res. Not., 3, 139-150 (1996) · Zbl 0859.05027 |
| [40] | Lascoux, A., The 6 vertex model and Schubert polynomials, SIGMA Symmetry Integrability Geom. Methods Appl., 3, Article 029 pp. (2007), 12 · Zbl 1138.05073 |
| [41] | Lascoux, A.; Leclerc, B.; Thibon, J.-Y., Flag varieties and the Yang-Baxter equation, Lett. Math. Phys., 40, 1, 75-90 (1997) · Zbl 0918.17013 |
| [42] | Lascoux, A.; Schützenberger, M.-P., Keys & standard bases, (Invariant Theory and Tableaux. Invariant Theory and Tableaux, Minneapolis, MN, 1988. Invariant Theory and Tableaux. Invariant Theory and Tableaux, Minneapolis, MN, 1988, IMA Vol. Math. Appl., vol. 19 (1990), Springer: Springer New York), 125-144 · Zbl 0815.20013 |
| [43] | Lenart, C., A unified approach to combinatorial formulas for Schubert polynomials, J. Algebraic Comb., 20, 3, 263-299 (2004) · Zbl 1056.05146 |
| [44] | Littelmann, P., Crystal graphs and Young tableaux, J. Algebra, 175, 1, 65-87 (1995) · Zbl 0831.17004 |
| [45] | Littelmann, P., Cones, crystals, and patterns, Transform. Groups, 3, 2, 145-179 (1998) · Zbl 0908.17010 |
| [46] | Mason, S., A decomposition of Schur functions and an analogue of the Robinson-Schensted-Knuth algorithm, Sémin. Lothar. Comb., 60, Article B57e pp. (2006) · Zbl 1193.05160 |
| [47] | Mason, S., An explicit construction of type A Demazure atoms, J. Algebraic Comb., 29, 3, 295-313 (2009) · Zbl 1210.05175 |
| [48] | Proctor, R. A.; Willis, M. J., Semistandard tableaux for Demazure characters (key polynomials) and their atoms, Eur. J. Comb., 43, 172-184 (2015) · Zbl 1301.05363 |
| [49] | Reiner, V.; Shimozono, M., Key polynomials and a flagged Littlewood-Richardson rule, J. Comb. Theory, Ser. A, 70, 1, 107-143 (1995) · Zbl 0819.05058 |
| [50] | Santos, J. M., Symplectic keys and Demazure atoms in type C (2019) |
| [51] | Seaborn, J. W., Combinatorial interpretation of the Kumar-Peterson limit for \(s l_n(\mathbb{C})\) Demazure characters and Gelfand pattern description of \(s l_n(\mathbb{C})\) Demazure characters (2014), ProQuest LLC: ProQuest LLC Ann Arbor, MI, Thesis (PhD), The University of North Carolina at Chapel Hill |
| [52] | Stembridge, J. R., A short derivation of the Möbius function for the Bruhat order, J. Algebraic Comb., 25, 2, 141-148 (2007) · Zbl 1150.20028 |
| [53] | Tokuyama, T., A generating function of strict Gelfand patterns and some formulas on characters of general linear groups, J. Math. Soc. Jpn., 40, 4, 671-685 (1988) · Zbl 0639.20022 |
| [54] | Verma, D.-N., Möbius inversion for the Bruhat ordering on a Weyl group, Ann. Sci. Éc. Norm. Supér. (4), 4, 393-398 (1971) · Zbl 0236.20035 |
| [55] | Wheeler, M.; Zinn-Justin, P., Littlewood-Richardson coefficients for Grothendieck polynomials from integrability, J. Reine Angew. Math., 757, 159-195 (2019) · Zbl 1428.05323 |
| [56] | Willis, M. J., New Descriptions of Demazure Tableaux and Right Keys, with Applications to Convexity (2012), ProQuest LLC: ProQuest LLC Ann Arbor, MI, Thesis (PhD), The University of North Carolina at Chapel Hill |
| [57] | Willis, M. J., A direct way to find the right key of a semistandard Young tableau, Ann. Comb., 17, 2, 393-400 (2013) · Zbl 1270.05100 |
| [58] | Willis, M. J., Relating the right key to the type A filling map and minimal defining chains, Discrete Math., 339, 10, 2410-2416 (2016) · Zbl 1339.05425 |
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