ALGEBRAIC COMBINATORICS

no. 2
Euler obstructions for the Lagrangian Grassmannian
LeVan, Paul1; Raicu, Claudiu1
1 Department of Mathematics University of Notre Dame 255 Hurley Notre Dame IN 46556, USA
Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 299-318.

We prove a case of a positivity conjecture of Mihalcea–Singh, concerned with the local Euler obstructions associated to the Schubert stratification of the Lagrangian Grassmannian LG(n,2n). Combined with work of Aluffi–Mihalcea–Schürmann–Su, this further implies the positivity of the Mather classes for Schubert varieties in LG(n,2n), which Mihalcea–Singh had verified for the other cominuscule spaces of classical Lie type. Building on the work of Boe and Fu, we give a positive recursion for the local Euler obstructions, and use it to show that they provide a positive count of admissible labelings of certain trees, analogous to the ones describing Kazhdan–Lusztig polynomials. Unlike in the case of the Grassmannians in types A and D, for LG(n,2n) the Euler obstructions e y,w may vanish for certain pairs (y,w) with y≤w in the Bruhat order. Our combinatorial description allows us to classify all the pairs (y,w) for which e y,w =0. Restricting to the big opposite cell in LG(n,2n), which is naturally identified with the space of n×n symmetric matrices, we recover the formulas for the local Euler obstructions associated with the matrix rank stratification.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.211
Classification: 14M15, 14M12, 05C05, 32S05, 32S60
Keywords: Local Euler obstructions, Schubert stratification, Lagrangian Grassmannian, tree labelings.
LeVan, Paul 1; Raicu, Claudiu 1

1 Department of Mathematics University of Notre Dame 255 Hurley Notre Dame IN 46556, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2022__5_2_299_0,
     author = {LeVan, Paul and Raicu, Claudiu},
     title = {Euler obstructions for the {Lagrangian} {Grassmannian}},
     journal = {Algebraic Combinatorics},
     pages = {299--318},
     publisher = {The Combinatorics Consortium},
     volume = {5},
     number = {2},
     year = {2022},
     doi = {10.5802/alco.211},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.211/}
}
TY  - JOUR
AU  - LeVan, Paul
AU  - Raicu, Claudiu
TI  - Euler obstructions for the Lagrangian Grassmannian
JO  - Algebraic Combinatorics
PY  - 2022
SP  - 299
EP  - 318
VL  - 5
IS  - 2
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.211/
DO  - 10.5802/alco.211
LA  - en
ID  - ALCO_2022__5_2_299_0
ER  - 
%0 Journal Article
%A LeVan, Paul
%A Raicu, Claudiu
%T Euler obstructions for the Lagrangian Grassmannian
%J Algebraic Combinatorics
%D 2022
%P 299-318
%V 5
%N 2
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.211/
%R 10.5802/alco.211
%G en
%F ALCO_2022__5_2_299_0
LeVan, Paul; Raicu, Claudiu. Euler obstructions for the Lagrangian Grassmannian. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 299-318. doi : 10.5802/alco.211. https://alco.centre-mersenne.org/articles/10.5802/alco.211/

[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 (2017) (https://arxiv.org/abs/1709.08697)

[2] Boe, Brian D. Kazhdan–Lusztig polynomials for Hermitian symmetric spaces, Trans. Amer. Math. Soc., Volume 309 (1988) no. 1, pp. 279-294 | DOI | MR | Zbl

[3] Boe, Brian D.; Fu, Joseph H. G. Characteristic cycles in Hermitian symmetric spaces, Canad. J. Math., Volume 49 (1997) no. 3, pp. 417-467 | DOI | MR | Zbl

[4] Bressler, Paul; Finkelberg, Michael; Lunts, Valery Vanishing cycles on Grassmannians, Duke Math. J., Volume 61 (1990) no. 3, pp. 763-777 | DOI | MR | Zbl

[5] Grayson, Daniel R.; Stillman, Michael E. Macaulay 2, a software system for research in algebraic geometry (Available at http://www.math.uiuc.edu/Macaulay2/)

[6] Lakshmibai, Venkatramani; Raghavan, Komaranapuram N. Standard monomial theory. Invariant theoretic approach., Encyclopaedia of Mathematical Sciences, 137, Springer-Verlag, Berlin, 2008, xiv+265 pages | Zbl

[7] Lascoux, Alain; Schützenberger, Marcel-Paul Polynômes de Kazhdan & Lusztig pour les grassmanniennes, Young tableaux and Schur functors in algebra and geometry (Toruń, 1980) (Astérisque), Volume 87, Soc. Math. France, Paris, 1981, pp. 249-266 | MR | Zbl

[8] Lőrincz, András C; Raicu, Claudiu Local Euler obstructions for determinantal varieties (2021) (https://arxiv.org/abs/2105.00271)

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

[10] Mihalcea, Leonardo C.; Singh, Rahul Mather classes and conormal spaces of Schubert varieties in cominuscule spaces (https://arxiv.org/abs/2006.04842)

[11] Zhang, Xiping Geometric invariants of recursive group orbit stratification (https://arxiv.org/abs/2009.09362)

Cited by Sources: