Peterson and Proctor obtained a formula which expresses the multivariate generating function for -partitions on a -complete poset as a product in terms of hooks in . In this paper, we give a skew generalization of Peterson–Proctor’s hook formula, i.e. a formula for the generating function of -partitions for a -complete poset and an order filter of , by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant -theory of Kac–Moody partial flag varieties. This generalization provides an alternate proof of Peterson–Proctor’s hook formula. As the equivariant cohomology version, we derive a skew generalization of a combinatorial reformulation of Nakada’s colored hook formula for roots.
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.54
Keywords: $d$-complete posets, hook formulas, $P$-partitions, Schubert calculus, equivariant $K$-theory
Naruse, Hiroshi 1; Okada, Soichi 2
CC-BY 4.0
@article{ALCO_2019__2_4_541_0,
author = {Naruse, Hiroshi and Okada, Soichi},
title = {Skew hook formula for $d$-complete posets via equivariant $K$-theory},
journal = {Algebraic Combinatorics},
pages = {541--571},
publisher = {MathOA foundation},
volume = {2},
number = {4},
year = {2019},
doi = {10.5802/alco.54},
zbl = {1417.05011},
mrnumber = {3997510},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.54/}
}
TY - JOUR AU - Naruse, Hiroshi AU - Okada, Soichi TI - Skew hook formula for $d$-complete posets via equivariant $K$-theory JO - Algebraic Combinatorics PY - 2019 SP - 541 EP - 571 VL - 2 IS - 4 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.54/ DO - 10.5802/alco.54 LA - en ID - ALCO_2019__2_4_541_0 ER -
%0 Journal Article %A Naruse, Hiroshi %A Okada, Soichi %T Skew hook formula for $d$-complete posets via equivariant $K$-theory %J Algebraic Combinatorics %D 2019 %P 541-571 %V 2 %N 4 %I MathOA foundation %U https://alco.centre-mersenne.org/articles/10.5802/alco.54/ %R 10.5802/alco.54 %G en %F ALCO_2019__2_4_541_0
Naruse, Hiroshi; Okada, Soichi. Skew hook formula for $d$-complete posets via equivariant $K$-theory. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 541-571. doi : 10.5802/alco.54. https://alco.centre-mersenne.org/articles/10.5802/alco.54/
[1] Kostant polynomials and the cohomology ring for , Duke Math. J., Volume 96 (1999), pp. 205-224 | DOI | MR | Zbl
[2] Curve neighborhoods of Schubert varieties, J. Differential Geom., Volume 99 (2015) no. 2, pp. 255-283 | DOI | MR | Zbl
[3] The hook graphs of the symmetric group, Can. J. Math., Volume 6 (1954), pp. 316-325 | DOI | MR | Zbl
[4] The Hillman–Grassl correspondence and the enumeration of reverse plane partitions, J. Combin. Theory Ser. A, Volume 30 (1981), pp. 71-89 | DOI | MR | Zbl
[5] Excited Young diagrams, equivariant -theory, and Schubert varieties, Trans. Amer. Math. Soc., Volume 367 (2015), pp. 6597-6645 | DOI | MR | Zbl
[6] Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math., 29, Cambridge Univ. Press, 1992 | Zbl
[7] Excited Young diagrams and equivariant Schubert calculus, Trans. Amer. Math. Soc., Volume 361 (2009), pp. 5193-5221 | DOI | MR | Zbl
[8] -theoretic analogues of factorial Schur - and -functions, Adv. Math., Volume 243 (2013), pp. 22-66 | DOI | MR | Zbl
[9] Schur function identities and hook length posets, Proceedings of the 19th International Conference on Formal Power Series and Algebraic Combinatorics (Tianjin, July 2–6, 2007) (2007) (available at http://igm.univ-mlv.fr/~fpsac/FPSAC07/SITE07/PDF-Proceedings/Posters/55.pdf)
[10] Leaf posets and multivariate hook length property, RIMS Kokyuroku, Volume 1913 (2014), pp. 67-80
[11] The flag manifold of Kac–Moody Lie algebra, Algebraic Analysis, Geometry, and Number Theory: Proceedings of the JAMI Inaugural Conference (Igusa, J., ed.), Johns Hopkins Univ. Press, 1989, pp. 161-190 | Zbl
[12] Hook length property of -complete posets via -integrals, J. Combin. Theory, Ser. A, Volume 162 (2019), pp. 167-221 | DOI | MR | Zbl
[13] Construction of double Grothendieck polynomials of classical types using IdCoxeter algebras, Tokyo J. Math., Volume 39 (2017) no. 3, pp. 695-728 | DOI | MR | Zbl
[14] The Art of Computer Programming, Volume 3: Sorting and Searching, 3rd Edition, Addison-Wesley, 1973
[15] Schubert classes in the equivariant -theory and equivariant cohomology of the Grassmannian (https://arxiv.org/abs/math/0512204)
[16] Schubert classes in the equivariant -theory and equivariant cohomology of the Lagrangian Grassmannian (https://arxiv.org/abs/math/0602245)
[17] Kac–Moody Groups, their Flag Varieties and Representation Theory, Prog. Math., 204, Birkhäuser, 2002 | MR | Zbl
[18] -theory Schubert calculus of the affine Grassmannian, Comp. Math., Volume 146 (2010) no. 4, pp. 811-852 | DOI | MR | Zbl
[19] Affine Weyl groups in -theory and representation theory, Int. Math. Res. Not. IMRN, Volume 2007 (2007), Paper no. rnm038 | MR | Zbl
[20] Equivariant -Chevalley rules for Kac–Moody flag manifolds, Amer. J. Math., Volume 136 (2014) no. 5, pp. 1175-1213 | DOI | MR | Zbl
[21] On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms, Duke Math. J., Volume 140 (2007) no. 2, pp. 321-350 | DOI | MR | Zbl
[22] Hook formulas for skew shapes I. -analogues and bijections, J. Combin. Theory Ser. A, Volume 154 (2018), pp. 350-405 | DOI | MR | Zbl
[23] -Hook formula for a generalized Young diagram (in preparation)
[24] Colored hook formula for a generalized Young diagram, Osaka J. Math., Volume 45 (2008) no. 4, pp. 1085-1120 | MR | Zbl
[25] -Hook formula of Gansner type for a generalized Young diagram, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (DMTCS Proceedings), Volume AK (2009), pp. 685-696 | MR | Zbl
[26] Schubert calculus and hook formula (Talk slides at 73rd Sém. Lothar. Combin., Strobl, Austria, 2014, available at https://www.mat.univie.ac.at/~slc/wpapers/s73vortrag/naruse.pdf)
[27] -Deformations of multivariate hook product formulae, J. Algebraic Combin., Volume 32 (2010) no. 3, pp. 399-416 | DOI | MR | Zbl
[28] Equivariant -theory of Grassmannians, Forum Math. , Volume 5 (2017), Paper no. e3 | MR | Zbl
[29] Dynkin diagram classification of -minuscule Bruhat lattices and -complete posets, J. Algebraic Combin., Volume 9 (1999) no. 1, pp. 61-94 | DOI | MR | Zbl
[30] Minuscule elements of Weyl groups, the number game, and -complete posets, J. Algebra, Volume 213 (1999) no. 1, pp. 272-303 | DOI | MR | Zbl
[31] -Complete posets generalize Young diagrams for the hook product formula: Partial presentation of proof, RIMS Kokyuroku, Volume 1913 (2014), pp. 120-140
[32] -Complete posets: Local structural axioms, properties, and equivalent definitions, Order (2018) (https://doi.org/10.1007/s11083-018-9473-4) | DOI | Zbl
[33] Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math., Volume 139 (1911), pp. 155-250 | Zbl
[34] Theory and application of plane partitions, Part 2, Studies in Applied Math., Volume 50 (1971) no. 3, pp. 259-279 | DOI
[35] Ordered Structures and Partitions, Mem. Amer. Math. Soc., 119, Amer. Math. Soc., 1972, iii + 104 pages pages | MR | Zbl
[36] Enumerative Combinatorics, Volume I, Cambridge Stud. Adv. Math., 49, Cambridge Univ. Press, 1997
[37] On the fully commutative elements of Coxeter groups, J. Algebraic Combin., Volume 5 (1996) no. 4, pp. 353-385 | DOI | MR | Zbl
[38] Minuscule elements of Weyl groups, J. Algebra, Volume 235 (2001) no. 2, pp. 722-743 | DOI | MR | Zbl
[39] A combinatorial problem, Michigan Math. J., Volume 1 (1952) no. 1, pp. 81-88 | Zbl
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