Euler obstructions for the Lagrangian Grassmannian
Abstract
We prove a case of a positivity conjecture of MihalceaSingh, concerned with the local Euler obstructions associated to the Schubert stratification of the Lagrangian Grassmannian LG(n,2n). Combined with work of AluffiMihalceaSchürmannSu, this further implies the positivity of the Mather classes for Schubert varieties in LG(n,2n), which MihalceaSingh 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 KazhdanLusztig 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 x n symmetric matrices, we recover the formulas for the local Euler obstructions associated with the matrix rank stratification.
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 arXiv:
 arXiv:2105.08823
 Bibcode:
 2021arXiv210508823L
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics;
 14M15;
 14M12;
 05C05;
 32S05;
 32S60