# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.