Hook formulae from  Segre–MacPherson classes

Mihalcea, Leonardo C.; Naruse, Hiroshi; Su, Changjian

doi:10.5802/alco.428

Hook formulae from Segre–MacPherson classes

Mihalcea, Leonardo C.¹; Naruse, Hiroshi ²; Su, Changjian ³

¹ Department of Mathematics Virginia Tech University Blacksburg, VA 24061 USA
² Graduate School of Education University of Yamanashi Kofu, 400-8510 Japan
³ Yau Mathematical Sciences Center Tsinghua University Beijing China

Algebraic Combinatorics, Volume 8 (2025) no. 3, pp. 655-685.

Abstract

Nakada’s colored hook formula is a vast generalization of many important formulae in combinatorics, such as the classical hook length formula and the Peterson’s formula for the number of reduced expressions of minuscule Weyl group elements. In this paper, we use cohomological properties of Segre–MacPherson classes of Schubert cells and varieties to prove a generalization of a cohomological version of Nakada’s formula, in terms of smoothness properties of Schubert varieties. A key ingredient in the proof is the study of a decorated version of the Bruhat graph. Weights of the paths in this graph give the terms in the generalized Nakada’s formula, and the summation over all paths is equal to the equivariant multiplicity of the Chern–Schwartz–MacPherson class of a Richardson variety. Among the applications we mention an algorithm to calculate structure constants of multiplications of Segre–MacPherson classes of Schubert cells, and a skew version of Nakada–Peterson’s formula.

Received: 2023-02-19
Accepted: 2025-03-11
Published online: 2025-06-26

DOI: 10.5802/alco.428

Classification: 05A19, 14M15, 05E14, 14N15
Keywords: hook formula, Segre–MacPherson classes, Schubert varieties, equivariant multiplicity

Author's affiliations:

Mihalcea, Leonardo C. ¹; Naruse, Hiroshi ²; Su, Changjian ³

¹ Department of Mathematics Virginia Tech University Blacksburg, VA 24061 USA
² Graduate School of Education University of Yamanashi Kofu, 400-8510 Japan
³ Yau Mathematical Sciences Center Tsinghua University Beijing China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{ALCO_2025__8_3_655_0,
     author = {Mihalcea, Leonardo C. and Naruse, Hiroshi and Su, Changjian},
     title = {Hook formulae from  {Segre{\textendash}MacPherson} classes},
     journal = {Algebraic Combinatorics},
     pages = {655--685},
     publisher = {The Combinatorics Consortium},
     volume = {8},
     number = {3},
     year = {2025},
     doi = {10.5802/alco.428},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.428/}
}

TY  - JOUR
AU  - Mihalcea, Leonardo C.
AU  - Naruse, Hiroshi
AU  - Su, Changjian
TI  - Hook formulae from  Segre–MacPherson classes
JO  - Algebraic Combinatorics
PY  - 2025
SP  - 655
EP  - 685
VL  - 8
IS  - 3
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.428/
DO  - 10.5802/alco.428
LA  - en
ID  - ALCO_2025__8_3_655_0
ER  -

%0 Journal Article
%A Mihalcea, Leonardo C.
%A Naruse, Hiroshi
%A Su, Changjian
%T Hook formulae from  Segre–MacPherson classes
%J Algebraic Combinatorics
%D 2025
%P 655-685
%V 8
%N 3
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.428/
%R 10.5802/alco.428
%G en
%F ALCO_2025__8_3_655_0

Mihalcea, Leonardo C.; Naruse, Hiroshi; Su, Changjian. Hook formulae from  Segre–MacPherson classes. Algebraic Combinatorics, Volume 8 (2025) no. 3, pp. 655-685. doi : 10.5802/alco.428. https://alco.centre-mersenne.org/articles/10.5802/alco.428/

References
Cited by

[1] Aluffi, Paolo; Mihalcea, Leonardo C.; Schürmann, Jörg; Su, Changjian Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells, Duke Math. J., Volume 172 (2023) no. 17, pp. 3257-3320 | DOI | MR | Zbl

[2] Aluffi, Paolo; Mihalcea, Leonardo C.; Schürmann, Jörg; Su, Changjian Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman’s problem, Ann. Sci. Éc. Norm. Supér. (4), Volume 57 (2024) no. 1, pp. 87-141 | DOI | MR | Zbl

[3] Billey, Sara; Coskun, Izzet Singularities of generalized Richardson varieties, Comm. Algebra, Volume 40 (2012) no. 4, pp. 1466-1495 | DOI | MR | Zbl

[4] Bourbaki, Nicolas Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002, xii+300 pages (translated from the 1968 French original by Andrew Pressley) | DOI | MR | Zbl

[5] Brasselet, Jean-Paul; Schwartz, Marie-Hélène Sur les classes de Chern d’un ensemble analytique complexe, The Euler-Poincaré characteristic (French) (Astérisque), Volume 82-83, Soc. Math. France, Paris, 1981, pp. 93-147 | Numdam | MR | Zbl

[6] Brenti, Francesco; Fomin, Sergey; Postnikov, Alexander Mixed Bruhat operators and Yang–Baxter equations for Weyl groups, Internat. Math. Res. Notices (1999) no. 8, pp. 419-441 | DOI | MR | Zbl

[7] Brion, Michel Equivariant Chow groups for torus actions, Transform. Groups, Volume 2 (1997) no. 3, pp. 225-267 | DOI | MR | Zbl

[8] Buch, Anders S.; Chaput, Pierre-Emmanuel; Mihalcea, Leonardo C.; Perrin, Nicolas A Chevalley formula for the equivariant quantum $K$ -theory of cominuscule varieties, Algebr. Geom., Volume 5 (2018) no. 5, pp. 568-595 | DOI | MR | Zbl

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

[10] Carrell, James B. Vector fields, flag varieties and Schubert calculus, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras (1991), pp. 23-57 | MR | Zbl

[11] Carrell, James B. The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991) (Proc. Sympos. Pure Math.), Volume 56, Part 1, Amer. Math. Soc., Providence, RI, 1994, pp. 53-61 | DOI | MR | Zbl

[12] Casbi, Elie Equivariant multiplicities of simply-laced type flag minors, Represent. Theory, Volume 25 (2021), pp. 1049-1092 | DOI | MR | Zbl

[13] Casbi, Elie Newton-Okounkov bodies for categories of modules over quiver Hecke algebras, Ann. Inst. Fourier (Grenoble), Volume 72 (2022) no. 5, pp. 1773-1818 | DOI | Numdam | MR | Zbl

[14] Edidin, Dan; Graham, William Equivariant intersection theory, Invent. Math., Volume 131 (1998) no. 3, pp. 595-634 | DOI | MR | Zbl

[15] Edidin, Dan; Graham, William Localization in equivariant intersection theory and the Bott residue formula, Amer. J. Math., Volume 120 (1998) no. 3, pp. 619-636 | DOI | MR | Zbl

[16] Fulton, W.; Woodward, C. On the quantum product of Schubert classes, J. Algebraic Geom., Volume 13 (2004) no. 4, pp. 641-661 | DOI | MR | Zbl

[17] Fulton, William Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2, Springer-Verlag, Berlin, 1998, xiv+470 pages | DOI | MR | Zbl

[18] Graham, William; Kreiman, Victor Excited Young diagrams, equivariant $K$ -theory, and Schubert varieties, Trans. Amer. Math. Soc., Volume 367 (2015) no. 9, pp. 6597-6645 | DOI | MR | Zbl

[19] Graham, William; Kreiman, Victor Cominuscule points and Schubert varieties, Ann. Inst. Fourier (Grenoble), Volume 71 (2021) no. 6, pp. 2519-2548 | DOI | Numdam | MR | Zbl

[20] Ikeda, Takeshi; Naruse, Hiroshi Excited Young diagrams and equivariant Schubert calculus, Trans. Amer. Math. Soc., Volume 361 (2009) no. 10, pp. 5193-5221 | DOI | MR | Zbl

[21] Kawanaka, Noriaki Games and algorithms with hook structure, Sugaku Expositions, Volume 28 (2015) no. 1, pp. 73-93 (translated from the Japanese) | MR

[22] Kleshchev, Alexander; Ram, Arun Homogeneous representations of Khovanov-Lauda algebras, J. Eur. Math. Soc. (JEMS), Volume 12 (2010) no. 5, pp. 1293-1306 | DOI | MR | Zbl

[23] Knutson, Allen; Tao, Terence Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J., Volume 119 (2003) no. 2, pp. 221-260 | DOI | MR | Zbl

[24] Kumar, Shrawan The nil Hecke ring and singularity of Schubert varieties, Invent. Math., Volume 123 (1996) no. 3, pp. 471-506 | DOI | MR | Zbl

[25] Kumar, Shrawan Kac–Moody groups, their flag varieties and representation theory, Progress in Mathematics, 204, Birkhäuser Boston, Inc., Boston, MA, 2002, xvi+606 pages | DOI | MR | Zbl

[26] Lenart, Cristian; Naito, Satoshi; Sagaki, Daisuke; Schilling, Anne; Shimozono, Mark A uniform model for Kirillov–Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph, Int. Math. Res. Not. IMRN, Volume 2015 (2015) no. 7, pp. 1848-1901 | DOI | MR | Zbl

[27] MacPherson, Robert D. Chern classes for singular algebraic varieties, Ann. of Math. (2), Volume 100 (1974), pp. 423-432 | DOI | MR | Zbl

[28] Mihalcea, Leonardo Equivariant quantum Schubert calculus, Adv. Math., Volume 203 (2006) no. 1, pp. 1-33 | DOI | MR | Zbl

[29] Mihalcea, Leonardo C.; Naruse, Hiroshi; Su, Changjian Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K-theory, Int. Math. Res. Not. IMRN (2022) no. 16, pp. 12096-12147 | DOI | MR | Zbl

[30] Mihalcea, Leonardo C.; Naruse, Hiroshi; Su, Changjian Hook formula for Coxeter groups via the twisted group ring, Proc. Japan Acad. Ser. A Math. Sci., Volume 100 (2024) no. 6, pp. 31-36 | DOI | MR | Zbl

[31] Mihalcea, Leonardo Constantin 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

[32] Molev, Alexander I.; Sagan, Bruce E. A Littlewood-Richardson rule for factorial Schur functions, Trans. Amer. Math. Soc., Volume 351 (1999) no. 11, pp. 4429-4443 | DOI | MR | Zbl

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

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

[35] Nakada, Kento A hook formula for the standard tableaux of a generalized shape, Combinatorial representation theory and related topics (RIMS Kôkyûroku Bessatsu, B8), Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, pp. 55-62 | MR | Zbl

[36] Naruse, Hiroshi Schubert calculus and hook formula, slides at the 73rd Sém. Lothar. Combin., Strobl, Austria, 2014 https://www.emis.de/...

[37] Naruse, Hiroshi; Okada, Soichi Skew hook formula for $d$ -complete posets via equivariant $K$ -theory, Algebr. Comb., Volume 2 (2019) no. 4, pp. 541-571 | DOI | Numdam | MR | Zbl

[38] Ohmoto, Toru Equivariant Chern classes of singular algebraic varieties with group actions, Math. Proc. Cambridge Philos. Soc., Volume 140 (2006) no. 1, pp. 115-134 | DOI | MR

[39] Panova, Greta; Petrov, Leonid Hook-length Formulas for Skew Shapes via Contour Integrals and Vertex Models, 2024 | arXiv

[40] Proctor, Robert A. Bruhat lattices, plane partition generating functions, and minuscule representations, European J. Combin., Volume 5 (1984) no. 4, pp. 331-350 | DOI | MR | Zbl

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

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

[43] Schürmann, Jörg Chern classes and transversality for singular spaces, Singularities in geometry, topology, foliations and dynamics (Trends Math.), Birkhäuser/Springer, Cham, 2017, pp. 207-231 | DOI | MR | Zbl

[44] Schürmann, Jörg; Simpson, Connor; Wang, Botong A new generic vanishing theorem on homogeneous varieties and the positivity conjecture for triple intersections of Schubert cells, Compos. Math., Volume 161 (2025) no. 1, pp. 1-12 | DOI | MR | Zbl

[45] Schwartz, Marie-Hélène Classes caractéristiques définies par une stratification d’une variété analytique complexe. I, C. R. Acad. Sci. Paris, Volume 260 (1965), pp. 3262-3264 | MR | Zbl

[46] Schwartz, Marie-Hélène Classes caractéristiques définies par une stratification d’une variété analytique complexe. II, C. R. Acad. Sci. Paris, Volume 260 (1965), pp. 3535-3537 | MR | Zbl

[47] Stanley, Richard P. On the number of reduced decompositions of elements of Coxeter groups, European J. Combin., Volume 5 (1984) no. 4, pp. 359-372 | DOI | MR | Zbl

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

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

[50] Su, Changjian Equivariant quantum cohomology of cotangent bundle of $G / P$ , Adv. Math., Volume 289 (2016), pp. 362-383 | DOI | MR | Zbl

[51] Su, Changjian Structure constants for Chern classes of Schubert cells, Math. Z., Volume 298 (2021) no. 1-2, pp. 193-213 | DOI | MR | Zbl

Cited by Sources: