Promotion on oscillating and alternating tableaux and rotation of matchings and permutations
Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 107-141.

Using Henriques’ and Kamnitzer’s cactus groups, Schützenberger’s promotion and evacuation operators on standard Young tableaux can be generalised in a very natural way to operators acting on highest weight words in tensor products of crystals.

For the crystals corresponding to the vector representations of the symplectic groups, we show that Sundaram’s map to perfect matchings intertwines promotion and rotation of the associated chord diagrams, and evacuation and reversal. We also exhibit a map with similar features for the crystals corresponding to the adjoint representations of the general linear groups.

We prove these results by applying van Leeuwen’s generalisation of Fomin’s local rules for jeu de taquin, connected to the action of the cactus groups by Lenart, and variants of Fomin’s growth diagrams for the Robinson–Schensted correspondence.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.87
Classification: 05E18, 05E10
Keywords: Promotion, evacuation, cactus group
Pfannerer, Stephan 1; Rubey, Martin 1; Westbury, Bruce 2

1 Fakultät für Mathematik und Geoinformation TU Wien, Austria
2 Department of Mathematics University of Texas at Dallas, U.S.A.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ALCO_2020__3_1_107_0,
     author = {Pfannerer, Stephan and Rubey, Martin and Westbury, Bruce},
     title = {Promotion on oscillating and alternating tableaux and rotation of matchings and permutations},
     journal = {Algebraic Combinatorics},
     pages = {107--141},
     publisher = {MathOA foundation},
     volume = {3},
     number = {1},
     year = {2020},
     doi = {10.5802/alco.87},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.87/}
}
TY  - JOUR
AU  - Pfannerer, Stephan
AU  - Rubey, Martin
AU  - Westbury, Bruce
TI  - Promotion on oscillating and alternating tableaux and rotation of matchings and permutations
JO  - Algebraic Combinatorics
PY  - 2020
SP  - 107
EP  - 141
VL  - 3
IS  - 1
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.87/
DO  - 10.5802/alco.87
LA  - en
ID  - ALCO_2020__3_1_107_0
ER  - 
%0 Journal Article
%A Pfannerer, Stephan
%A Rubey, Martin
%A Westbury, Bruce
%T Promotion on oscillating and alternating tableaux and rotation of matchings and permutations
%J Algebraic Combinatorics
%D 2020
%P 107-141
%V 3
%N 1
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.87/
%R 10.5802/alco.87
%G en
%F ALCO_2020__3_1_107_0
Pfannerer, Stephan; Rubey, Martin; Westbury, Bruce. Promotion on oscillating and alternating tableaux and rotation of matchings and permutations. Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 107-141. doi : 10.5802/alco.87. https://alco.centre-mersenne.org/articles/10.5802/alco.87/

[1] Brauer, Richard On algebras which are connected with the semisimple continuous groups, Ann. Math. (2), Volume 38 (1937) no. 4, pp. 857-872 | DOI | MR | Zbl

[2] Cautis, Sabin; Kamnitzer, Joel; Morrison, Scott Webs and quantum skew Howe duality, Math. Ann., Volume 360 (2014) no. 1-2, pp. 351-390 | DOI | MR | Zbl

[3] Chmutov, Michael; Glick, Max; Pylyavskyy, Pavlo The Berenstein-Kirillov group and cactus groups (2016) (https://arxiv.org/abs/1609.02046)

[4] Corteel, Sylvie Crossings and alignments of permutations, Adv. Appl. Math., Volume 38 (2007) no. 2, pp. 149-163 | DOI | MR | Zbl

[5] Devadoss, Satyan Tessellations of moduli spaces and the mosaic operad, Homotopy invariant algebraic structures (Baltimore, MD, 1998) (Contemp. Math.), Volume 239, Amer. Math. Soc., Providence, RI, 1999, pp. 91-114 | DOI | MR | Zbl

[6] Fomin, Sergey Schensted algorithms for dual graded graphs, J. Algebr. Comb., Volume 4 (1995) no. 1, pp. 5-45 | DOI | MR | Zbl

[7] Henriques, André; Kamnitzer, Joel Crystals and coboundary categories, Duke Math. J., Volume 132 (2006) no. 2, pp. 191-216 | DOI | MR | Zbl

[8] Jagenteufel, Judith A Sundaram type bijection for SO(3): vacillating tableaux and pairs of standard Young tableaux and orthogonal Littlewood–Richardson tableaux, Electron. J. Comb., Volume 25 (2018) no. 3, Paper no. P3.50, 44 pages | MR | Zbl

[9] Khovanov, Mikhail; Kuperberg, Greg Web bases for sl(3) are not dual canonical, Pac. J. Math., Volume 188 (1999) no. 1, pp. 129-153 | DOI | MR | Zbl

[10] Kuperberg, Greg Spiders for rank 2 Lie algebras, Commun. Math. Phys., Volume 180 (1996) no. 1, pp. 109-151 | DOI | MR | Zbl

[11] Lenart, Cristian On the combinatorics of crystal graphs, II. The crystal commutor, Proc. Am. Math. Soc., Volume 136 (2008) no. 3, pp. 825-837 | DOI | MR | Zbl

[12] Lusztig, George Canonical bases arising from quantized enveloping algebras. II, Prog. Theor. Phys., Suppl. (1990) no. 102, pp. 175-201 Common trends in mathematics and quantum field theories (Kyoto, 1990) | DOI | MR | Zbl

[13] Patrias, Rebecca Promotion on generalized oscillating tableaux and web rotation, J. Comb. Theory, Ser. A, Volume 161 (2019), pp. 1-28 | DOI | MR | Zbl

[14] Petersen, T. Kyle; Pylyavskyy, Pavlo; Rhoades, Brendon Promotion and cyclic sieving via webs, J. Algebr. Comb., Volume 30 (2009) no. 1, pp. 19-41 | DOI | MR | Zbl

[15] Reiner, Victor; Stanton, Dennis; White, Dennis The cyclic sieving phenomenon, J. Comb. Theory, Ser. A, Volume 108 (2004) no. 1, pp. 17-50 | DOI | MR

[16] Roby, Thomas Applications and extensions of Fomin’s generalization of the Robinson-Schensted correspondence to differential posets, Ph. D. Thesis, Massachusetts Institute of Technology (USA) (1991) https://dspace.mit.edu/handle/1721.1/13517 | MR

[17] Rubey, Martin; Westbury, Bruce W. A combinatorial approach to classical representation theory (2014) (https://arxiv.org/abs/1408.3592)

[18] Rubey, Martin; Westbury, Bruce W. Combinatorics of symplectic invariant tensors, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015), pp. 285-296 | Zbl

[19] Rumer, Georg; Teller, Edward; Weyl, Hermann Eine für die Valenztheorie geeignete Basis der binären Vektorinvarianten, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., Volume 1932 (1932), pp. 499-504 | Zbl

[20] Schützenberger, Marcel-Paul Quelques remarques sur une construction de Schensted, Math. Scand., Volume 12 (1963), pp. 117-128 | DOI | Numdam | MR | Zbl

[21] Stanley, Richard Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 1999, xii+581 pages | MR | Zbl

[22] Stembridge, John Rational tableaux and the tensor algebra of gl n , J. Comb. Theory, Ser. A, Volume 46 (1987) no. 1-2, pp. 79-120 | DOI | MR | Zbl

[23] Sundaram, Sheila On the combinatorics of representations of the symplectic group, Ph. D. Thesis, Massachusetts Institute of Technology (USA) (1986) https://dspace.mit.edu/handle/1721.1/15060 | MR

[24] van Leeuwen, Marc A. A. An analogue of jeu de taquin for Littelmann’s crystal paths, Sém. Lothar. Comb., Volume 41 (1998), Paper no. B41b, 23 pages | MR | Zbl

[25] Westbury, Bruce W. Invariant tensors and the cyclic sieving phenomenon, Electron. J. Comb., Volume 23 (2016) no. 4, Paper no. P4.25, 40 pages | MR | Zbl

[26] White, Noah The monodromy of real Bethe vectors for the Gaudin model, J. Comb. Algebra, Volume 2 (2018) no. 3, pp. 259-300 | DOI | MR | Zbl

Cited by Sources: