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Skew hook formula for \(d\)-complete posets via equivariant \(K\)-theory. (English) Zbl 1417.05011
Summary: Peterson and Proctor (see [R. A. Proctor “\(d\)-complete posets generalize Young diagrams for the hook product formula: partial presentation of proof”, RIMS Kôkyûroku 1913, 120–140 (2014)]) obtained a formula which expresses the multivariate generating function for \(P\)-partitions on a \(d\)-complete poset \(P\) as a product in terms of hooks in \(P\). In this paper, we give a skew generalization of Peterson-Proctor’s hook formula, i.e. a formula for the generating function of \((P \setminus F)\)-partitions for a \(d\)-complete poset \(P\) and an order filter \(F\) of \(P\), by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant \(K\)-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 K. Nakada’s colored hook formula for roots [Osaka J. Math. 45, No. 4, 1085–1120 (2008; Zbl 1204.05099)].

MSC:
05A15 Exact enumeration problems, generating functions
06A07 Combinatorics of partially ordered sets
14N15 Classical problems, Schubert calculus
19L47 Equivariant \(K\)-theory
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[1] Billey, S., Kostant polynomials and the cohomology ring for \(G/B\), Duke Math. J., 96, 205-224, (1999) · Zbl 0980.22018
[2] Buch, A. S.; Mihalcea, L. C., Curve neighborhoods of Schubert varieties, J. Differential Geom., 99, 2, 255-283, (2015) · Zbl 06423472
[3] Frame, J. S.; Robinson, G. De B.; Thrall, R. W., The hook graphs of the symmetric group, Can. J. Math., 6, 316-325, (1954) · Zbl 0055.25404
[4] Gansner, E. R., The Hillman-Grassl correspondence and the enumeration of reverse plane partitions, J. Combin. Theory Ser. A, 30, 71-89, (1981) · Zbl 0474.05008
[5] Graham, W.; Kreiman, V., Excited Young diagrams, equivariant \(K\)-theory, and Schubert varieties, Trans. Amer. Math. Soc., 367, 6597-6645, (2015) · Zbl 1317.05187
[6] Humphreys, J. E., Reflection Groups and Coxeter Groups, 29, (1992), Cambridge Univ. Press · Zbl 0768.20016
[7] Ikeda, T.; Naruse, H., Excited Young diagrams and equivariant Schubert calculus, Trans. Amer. Math. Soc., 361, 5193-5221, (2009) · Zbl 1229.05287
[8] Ikeda, T.; Naruse, H., \(K\)-theoretic analogues of factorial Schur \(P\)- and \(Q\)-functions, Adv. Math., 243, 22-66, (2013) · Zbl 1278.05240
[9] Ishikawa, M.; Tagawa, H., 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)
[10] Ishikawa, M.; Tagawa, H., Leaf posets and multivariate hook length property, RIMS Kokyuroku, 1913, 67-80, (2014)
[11] Kashiwara, M.; Igusa, J., Algebraic Analysis, Geometry, and Number Theory: Proceedings of the JAMI Inaugural Conference, The flag manifold of Kac–Moody Lie algebra, 161-190, (1989), Johns Hopkins Univ. Press · Zbl 0764.17019
[12] Kim, J.; Yoo, M., Hook length property of \(d\)-complete posets via \(q\)-integrals, J. Combin. Theory, Ser. A, 162, 167-221, (2019) · Zbl 1401.05035
[13] Kirillov, A. N.; Naruse, H., Construction of double Grothendieck polynomials of classical types using IdCoxeter algebras, Tokyo J. Math., 39, 3, 695-728, (2017) · Zbl 1364.05081
[14] Knuth, D. E., The Art of Computer Programming, Volume 3: Sorting and Searching, 3rd Edition, (1973), Addison-Wesley · Zbl 0302.68010
[15] Kreiman, V., Schubert classes in the equivariant \(K\)-theory and equivariant cohomology of the Grassmannian
[16] Kreiman, V., Schubert classes in the equivariant \(K\)-theory and equivariant cohomology of the Lagrangian Grassmannian
[17] Kumar, S., Kac-Moody Groups, their Flag Varieties and Representation Theory, 204, (2002), Birkhäuser · Zbl 1026.17030
[18] Lam, T.; Schilling, A.; Shimozono, M., \(K\)-theory Schubert calculus of the affine Grassmannian, Comp. Math., 146, 4, 811-852, (2010) · Zbl 1256.14056
[19] Lenart, C.; Postnikov, A., Affine Weyl groups in \(K\)-theory and representation theory, Int. Math. Res. Not. IMRN, 2007, (2007) · Zbl 1137.14037
[20] Lenart, C.; Shimozono, M., Equivariant \(K\)-Chevalley rules for Kac-Moody flag manifolds, Amer. J. Math., 136, 5, 1175-1213, (2014) · Zbl 1328.19012
[21] Mihalcea, L., On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms, Duke Math. J., 140, 2, 321-350, (2007) · Zbl 1135.14042
[22] Morales, A.; Pak, I.; Panova, G., Hook formulas for skew shapes I. \(q\)-analogues and bijections, J. Combin. Theory Ser. A, 154, 350-405, (2018) · Zbl 1373.05026
[23] Nakada, K., \(q\)-Hook formula for a generalized Young diagram · Zbl 1392.05117
[24] Nakada, K., Colored hook formula for a generalized Young diagram, Osaka J. Math., 45, 4, 1085-1120, (2008) · Zbl 1204.05099
[25] Nakada, K., \(q\)-Hook formula of Gansner type for a generalized Young diagram, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), AK, 685-696, (2009), Discrete Math. Theor. Comput. Sci.: Discrete Math. Theor. Comput. Sci., Hagenberg, Austria · Zbl 1392.05117
[26] Naruse, H., Schubert calculus and hook formula
[27] Okada, S., \((q,t)\)-Deformations of multivariate hook product formulae, J. Algebraic Combin., 32, 3, 399-416, (2010) · Zbl 1228.05048
[28] Pechenik, O.; Yong, A., Equivariant \(K\)-theory of Grassmannians, Forum Math. \[, 5, (201\] · Zbl 1369.14060
[29] Proctor, R. A., Dynkin diagram classification of \(\lambda \)-minuscule Bruhat lattices and \(d\)-complete posets, J. Algebraic Combin., 9, 1, 61-94, (1999) · Zbl 0920.06003
[30] Proctor, R. A., Minuscule elements of Weyl groups, the number game, and \(d\)-complete posets, J. Algebra, 213, 1, 272-303, (1999) · Zbl 0969.05068
[31] Proctor, R. A., \(d\)-Complete posets generalize Young diagrams for the hook product formula: Partial presentation of proof, RIMS Kokyuroku, 1913, 120-140, (2014)
[32] Proctor, R. A.; Scoppetta, L. M., \(d\)-Complete posets: Local structural axioms, properties, and equivalent definitions, Order, (2018)
[33] Schur, I., Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math., 139, 155-250, (1911) · JFM 42.0154.02
[34] Stanley, R. P., Theory and application of plane partitions, Part 2, Studies in Applied Math., 50, 3, 259-279, (1971) · Zbl 0225.05012
[35] Stanley, R. P., Ordered Structures and Partitions, 119, iii + 104ages p. pp., (1972), Amer. Math. Soc. · Zbl 0246.05007
[36] Stanley, R. P., Enumerative Combinatorics, Volume I, 49, (1997), Cambridge Univ. Press
[37] Stembridge, J. R., On the fully commutative elements of Coxeter groups, J. Algebraic Combin., 5, 4, 353-385, (1996) · Zbl 0864.20025
[38] Stembridge, J. R., Minuscule elements of Weyl groups, J. Algebra, 235, 2, 722-743, (2001) · Zbl 0973.17034
[39] Thrall, R. W., A combinatorial problem, Michigan Math. J., 1, 1, 81-88, (1952) · Zbl 0049.01001
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