ALGEBRAIC COMBINATORICS

no. 4

Dual filtered graphs
Algebraic Combinatorics, Volume 1 (2018) no. 4, pp. 441-500.

We define a K-theoretic analogue of Fomin’s dual graded graphs, which we call dual filtered graphs. The key formula in the definition is DU-UD=D+I. Our major examples are K-theoretic analogues of Young’s lattice, of shifted Young’s lattice, and of the Young–Fibonacci lattice. We suggest notions of tableaux, insertion algorithms, and growth rules whenever such objects are not already present in the literature. (See the table below.) We also provide a large number of other examples. Most of our examples arise via two constructions, which we call the Pieri construction and the Möbius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra, as described in [1, 19, 16]. The Möbius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms.

Received: 2017-08-18
Revised: 2018-05-04
Accepted: 2018-05-09
Published online: 2018-09-10
DOI: https://doi.org/10.5802/alco.21
Classification: 05E99,  05E05
Keywords: dual graded graphs, insertion algorithms, K-theory, symmetric functions 
@article{ALCO_2018__1_4_441_0,
     author = {Patrias, Rebecca and Pylyavskyy, Pavlo},
     title = {Dual filtered graphs},
     journal = {Algebraic Combinatorics},
     pages = {441--500},
     publisher = {MathOA foundation},
     volume = {1},
     number = {4},
     year = {2018},
     doi = {10.5802/alco.21},
     mrnumber = {3875073},
     zbl = {1397.05202},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2018__1_4_441_0/}
}
Patrias, Rebecca; Pylyavskyy, Pavlo. Dual filtered graphs. Algebraic Combinatorics, Volume 1 (2018) no. 4, pp. 441-500. doi : 10.5802/alco.21. https://alco.centre-mersenne.org/item/ALCO_2018__1_4_441_0/

[1] Bergeron, Nantel; Lam, Thomas; Li, Huilan Combinatorial Hopf algebras and towers of algebras–dimension, quantization and functorality, Algebr. Represent. Theory, Volume 15 (2012) no. 4, pp. 675-696 | Article | MR 2944437 | Zbl 1281.16036

[2] Björk, Jan-Erik Rings of differential operators, North-Holland mathematical Library, Volume 21, North-Holland, 1979, xvii+374 pages | MR 549189

[3] Björner, Anders The Möbius function of subword order, Invariant theory and tableaux (Minneapolis, USA, 1988) (The IMA Volumes in Mathematics and its Applications) Volume 19, Springer, 1990, pp. 118-124 | Zbl 0706.06007

[4] Björner, Anders; Stanley, Richard P. An analogue of Young’s lattice for compositions (2005) (https://arxiv.org/abs/math/0508043 )

[5] Buch, Anders Skovsted A Littlewood–Richardson rule for the K-theory of Grassmannians, Acta Math., Volume 189 (2002) no. 1, pp. 37-78 | Article | MR 1946917 | Zbl 1090.14015

[6] Buch, Anders Skovsted; Kresch, Andrew; Shimozono, Mark; Tamvakis, Harry; Yong, Alexander Stable Grothendieck polynomials and K-theoretic factor sequences, Math. Ann., Volume 340 (2008) no. 2, pp. 359-382 | Article | MR 2368984 | Zbl 1157.14036

[7] Buch, Anders Skovsted; Samuel, Matthew J K-theory of minuscule varieties, J. Reine Angew. Math., Volume 719 (2016), pp. 133-171 | MR 3552494 | Zbl 06636676

[8] Clifford, Edward; Thomas, Hugh; Yong, Alexander K-theoretic Schubert calculus for OG(n,2n+1) and jeu de taquin for shifted increasing tableaux, J. Reine Angew. Math., Volume 690 (2014), pp. 51-63 | MR 3200334 | Zbl 1348.14127

[9] Fomin, Sergei Vladimirovich Generalized Robinson–Schensted–Knuth correspondence, J. Sov. Math., Volume 41 (1988) no. 2, pp. 979-991 | Article | MR 869582 | Zbl 0698.05003

[10] Fomin, Sergey Duality of graded graphs, J. Algebr. Comb., Volume 3 (1994) no. 4, pp. 357-404 | Article | MR 1293822 | Zbl 0810.05005

[11] Fomin, Sergey Schensted algorithms for dual graded graphs, J. Algebr. Comb., Volume 4 (1995) no. 1, pp. 5-45 | Article | MR 1314558 | Zbl 0817.05077

[12] Hamaker, Zachary; Keilthy, Adam; Patrias, Rebecca; Webster, Lillian; Zhang, Yinuo; Zhou, Shuqi Shifted Hecke insertion and the K-theory of OG(n,2n+1), J. Comb. Theory, Ser. A, Volume 151 (2017), pp. 207-240 | Article | MR 3663495 | Zbl 1366.05118

[13] Knuth, Donald Permutations, matrices, and generalized Young tableaux, Pac. J. Math., Volume 34 (1970) no. 3, pp. 709-727 | Article | MR 272654 | Zbl 0199.31901

[14] Lam, Thomas Quantized dual graded graphs, Electron. J. Comb., Volume 17 (2010) no. 1, 11 pages | MR 2661391 | Zbl 1230.05163

[15] Lam, Thomas; Pylyavskyy, Pavlo Combinatorial Hopf algebras and K-homology of Grassmanians, Int. Math. Res. Not., Volume 2007 (2007) no. 24, 48 pages | Zbl 1134.16017

[16] Lam, Thomas; Shimozono, Mark (unpublished)

[17] Lam, Thomas; Shimozono, Mark Dual graded graphs for Kac–Moody algebras, Algebra Number Theory, Volume 1 (2007) no. 4, pp. 451-488 | Article | MR 2368957 | Zbl 1200.05249

[18] Macdonald, Ian Grant Symmetric functions and Hall polynomials, Oxford Science Publications, Clarendon Press, 1998, x+475 pages | Zbl 0899.05068

[19] Nzeutchap, Janvier Dual graded graphs and Fomin’s r-correspondences associated to the Hopf algebras of planar binary trees, quasi-symmetric functions and noncommutative symmetric functions (2006) (in Formal Power Series and Algebraic Combinatorics (San Diego, 2006), available at http://garsia.math.yorku.ca/fpsac06/papers/53.pdf)

[20] Ore, Oystein Theory of non-commutative polynomials, Ann. Math., Volume 34 (1933), pp. 480-508 | Article | MR 1503119 | Zbl 0007.15101

[21] Patrias, Rebecca; Pylyavskyy, Pavlo Combinatorics of K-theory via a K-theoretic Poirier–Reutenauer bialgebra, Discrete Mathematics, Volume 339 (2016) no. 3, pp. 1095-1115 | Article | MR 3433916 | Zbl 1328.05193

[22] Poirier, Stéphane; Reutenauer, Christophe Algèbres de Hopf de tableaux, Ann. Sci. Math. Qué., Volume 19 (1995) no. 1, pp. 79-90 | Zbl 0835.16035

[23] Robinson, Gilbert de B. On the representations of the symmetric group, Am. J. Math., Volume 60 (1938), pp. 745-760 | Article | Zbl 0019.25102

[24] Sagan, Bruce E. Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley, J. Comb. Theory, Ser. A, Volume 45 (1987) no. 1, pp. 62-103 | Article | MR 883894 | Zbl 0661.05010

[25] Schensted, Craige Longest increasing and decreasing subsequences, Classic Papers in Combinatorics (Modern Birkhäuser Classics), Birkhäuser, 2009, pp. 299-311 | Article

[26] Stanley, Richard P. Differential posets, J. Am. Math. Soc., Volume 1 (1988) no. 4, pp. 919-961 | Article | MR 941434 | Zbl 0658.05006

[27] Stanley, Richard P. Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, Volume 62, Cambridge University Press, 1999, xii+581 pages | MR 1676282 | Zbl 0928.05001

[28] Stanley, Richard P. Enumerative Combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, Volume 49, Cambridge University Press, 2012, xiii+626 pages | Zbl 1247.05003

[29] Thomas, Hugh; Yong, Alexander A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus, Algebra Number Theory, Volume 3 (2009) no. 2, pp. 121-148 | Article | MR 2491941 | Zbl 1229.05285

[30] Thomas, Hugh; Yong, Alexander The direct sum map on Grassmannians and jeu de taquin for increasing tableaux, Int. Math. Res. Not., Volume 2011 (2011) no. 12, pp. 2766-2793 | MR 2806593 | Zbl 1231.05280

[31] Thomas, Hugh; Yong, Alexander Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm, Adv. Appl. Math., Volume 46 (2011) no. 1-4, pp. 610-642 | Article | MR 2794040 | Zbl 1227.05262

[32] Worley, Dale Raymond A theory of shifted Young tableaux (1984) (Ph. D. Thesis) | MR 2941073

[33] Young, Alfred Qualitative substitutional analysis (third paper), Proc. Lond. Math. Soc., Volume 28 (1927), pp. 255-292 | MR 1575854 | Zbl 54.0150.01