# ALGEBRAIC COMBINATORICS

Euler obstructions for the Lagrangian Grassmannian
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 $LG\left(n,2n\right)$. Combined with work of Aluffi–Mihalcea–Schürmann–Su, this further implies the positivity of the Mather classes for Schubert varieties in $LG\left(n,2n\right)$, 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 $LG\left(n,2n\right)$ the Euler obstructions ${e}_{y,w}$ may vanish for certain pairs $\left(y,w\right)$ with $y\le w$ in the Bruhat order. Our combinatorial description allows us to classify all the pairs $\left(y,w\right)$ for which ${e}_{y,w}=0$. Restricting to the big opposite cell in $LG\left(n,2n\right)$, 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.

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
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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/

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