Skew hook formula for $d$-complete posets via equivariant $K$-theory

Skew hook formula for

d

-complete posets via equivariant

K

-theory

Naruse, Hiroshi ¹; Okada, Soichi ²

¹ Graduate School of Education, University of Yamanashi 4-4-37, Takeda, Kofu, Yamanashi 400-8510, Japan
² Graduate School of Mathematics, Nagoya University Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan

Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 541-571.

Abstract

Peterson and Proctor 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 $(P ∖ F)$ -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 Nakada’s colored hook formula for roots.

Received: 2018-03-11
Revised: 2018-12-06
Accepted: 2018-12-15
Published online: 2019-08-01

MR Zbl

DOI: 10.5802/alco.54

Classification: 05A15, 06A07, 14N15, 19L47
Keywords: $d$-complete posets, hook formulas, $P$-partitions, Schubert calculus, equivariant $K$-theory

Author's affiliations:

Naruse, Hiroshi ¹; Okada, Soichi ²

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@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/

References
Cited by

[1] Billey, S. Kostant polynomials and the cohomology ring for $G / B$ , Duke Math. J., Volume 96 (1999), pp. 205-224 | DOI | MR | Zbl

[2] Buch, A. S.; Mihalcea, L C. Curve neighborhoods of Schubert varieties, J. Differential Geom., Volume 99 (2015) no. 2, pp. 255-283 | DOI | MR | Zbl

[3] Frame, J. S.; Robinson, G. de B.; Thrall, R. W. The hook graphs of the symmetric group, Can. J. Math., Volume 6 (1954), pp. 316-325 | DOI | MR | Zbl

[4] Gansner, E. R. 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] Graham, W.; Kreiman, V. Excited Young diagrams, equivariant $K$ -theory, and Schubert varieties, Trans. Amer. Math. Soc., Volume 367 (2015), pp. 6597-6645 | DOI | MR | Zbl

[6] Humphreys, J. E. Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math., 29, Cambridge Univ. Press, 1992 | Zbl

[7] Ikeda, T.; Naruse, H. Excited Young diagrams and equivariant Schubert calculus, Trans. Amer. Math. Soc., Volume 361 (2009), pp. 5193-5221 | DOI | MR | Zbl

[8] Ikeda, T.; Naruse, H. $K$ -theoretic analogues of factorial Schur $P$ - and $Q$ -functions, Adv. Math., Volume 243 (2013), pp. 22-66 | DOI | MR | Zbl

[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) (available at http://igm.univ-mlv.fr/~fpsac/FPSAC07/SITE07/PDF-Proceedings/Posters/55.pdf)

[10] Ishikawa, M.; Tagawa, H. Leaf posets and multivariate hook length property, RIMS Kokyuroku, Volume 1913 (2014), pp. 67-80

[11] Kashiwara, M. 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] Kim, J.; Yoo, M. Hook length property of $d$ -complete posets via $q$ -integrals, J. Combin. Theory, Ser. A, Volume 162 (2019), pp. 167-221 | DOI | MR | Zbl

[13] Kirillov, A. N.; Naruse, H. 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] Knuth, D. E. The Art of Computer Programming, Volume 3: Sorting and Searching, 3rd Edition, Addison-Wesley, 1973

[15] Kreiman, V. Schubert classes in the equivariant $K$ -theory and equivariant cohomology of the Grassmannian (https://arxiv.org/abs/math/0512204)

[16] Kreiman, V. Schubert classes in the equivariant $K$ -theory and equivariant cohomology of the Lagrangian Grassmannian (https://arxiv.org/abs/math/0602245)

[17] Kumar, S. Kac–Moody Groups, their Flag Varieties and Representation Theory, Prog. Math., 204, Birkhäuser, 2002 | MR | Zbl

[18] Lam, T.; Schilling, A.; Shimozono, M. $K$ -theory Schubert calculus of the affine Grassmannian, Comp. Math., Volume 146 (2010) no. 4, pp. 811-852 | DOI | MR | Zbl

[19] Lenart, C.; Postnikov, A. Affine Weyl groups in $K$ -theory and representation theory, Int. Math. Res. Not. IMRN, Volume 2007 (2007), Paper no. rnm038 | MR | Zbl

[20] Lenart, C.; Shimozono, M. Equivariant $K$ -Chevalley rules for Kac–Moody flag manifolds, Amer. J. Math., Volume 136 (2014) no. 5, pp. 1175-1213 | DOI | MR | Zbl

[21] Mihalcea, L. 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] Morales, A.; Pak, I.; Panova, G. Hook formulas for skew shapes I. $q$ -analogues and bijections, J. Combin. Theory Ser. A, Volume 154 (2018), pp. 350-405 | DOI | MR | Zbl

[23] Nakada, K. $q$ -Hook formula for a generalized Young diagram (in preparation)

[24] Nakada, K. Colored hook formula for a generalized Young diagram, Osaka J. Math., Volume 45 (2008) no. 4, pp. 1085-1120 | MR | Zbl

[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) (DMTCS Proceedings), Volume AK (2009), pp. 685-696 | MR | Zbl

[26] Naruse, H. 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] Okada, S. $(q, t)$ -Deformations of multivariate hook product formulae, J. Algebraic Combin., Volume 32 (2010) no. 3, pp. 399-416 | DOI | MR | Zbl

[28] Pechenik, O.; Yong, A. Equivariant $K$ -theory of Grassmannians, Forum Math. $Π$ , Volume 5 (2017), Paper no. e3 | MR | Zbl

[29] Proctor, R. A. Dynkin diagram classification of $λ$ -minuscule Bruhat lattices and $d$ -complete posets, J. Algebraic Combin., Volume 9 (1999) no. 1, pp. 61-94 | DOI | MR | Zbl

[30] Proctor, R. A. Minuscule elements of Weyl groups, the number game, and $d$ -complete posets, J. Algebra, Volume 213 (1999) no. 1, pp. 272-303 | DOI | MR | Zbl

[31] Proctor, R. A. $d$ -Complete posets generalize Young diagrams for the hook product formula: Partial presentation of proof, RIMS Kokyuroku, Volume 1913 (2014), pp. 120-140

[32] Proctor, R. A.; Scoppetta, L. M. $d$ -Complete posets: Local structural axioms, properties, and equivalent definitions, Order (2018) (https://doi.org/10.1007/s11083-018-9473-4) | DOI | Zbl

[33] Schur, I. 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] Stanley, R. P. Theory and application of plane partitions, Part 2, Studies in Applied Math., Volume 50 (1971) no. 3, pp. 259-279 | DOI

[35] Stanley, R. P. Ordered Structures and Partitions, Mem. Amer. Math. Soc., 119, Amer. Math. Soc., 1972, iii + 104 pages pages | MR | Zbl

[36] Stanley, R. P. Enumerative Combinatorics, Volume I, Cambridge Stud. Adv. Math., 49, Cambridge Univ. Press, 1997

[37] Stembridge, J. R. On the fully commutative elements of Coxeter groups, J. Algebraic Combin., Volume 5 (1996) no. 4, pp. 353-385 | DOI | MR | Zbl

[38] Stembridge, J. R. Minuscule elements of Weyl groups, J. Algebra, Volume 235 (2001) no. 2, pp. 722-743 | DOI | MR | Zbl

[39] Thrall, R. W. A combinatorial problem, Michigan Math. J., Volume 1 (1952) no. 1, pp. 81-88 | Zbl

Cited by Sources: