Euler obstructions for the Lagrangian Grassmannian

Abstract

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 Grassman- nian LG ( n, 2 n ). Combined with work of Aluffi–Mihalcea–Schürmann–Su, this further implies the positivity of the Mather classes for Schubert varieties in LG ( n, 2 n ), 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, 2 n ) the Euler obstructions e y,w may vanish for certain pairs ( y,w ) with y (cid:54) 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, 2 n ), 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.