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(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 yw 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
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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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