Euler obstructions for the Lagrangian Grassmannian

TL;DR

This paper proves a positivity conjecture for local Euler obstructions in the Lagrangian Grassmannian, providing combinatorial and recursive methods to compute these invariants and classifying cases where they vanish.

Contribution

It establishes the positivity of local Euler obstructions for LG(n,2n), introduces a positive recursion, and classifies vanishing cases, extending known results to this specific space.

Findings

Proved positivity of local Euler obstructions for LG(n,2n).

Developed a positive recursive formula for these obstructions.

Classified all pairs with vanishing Euler obstructions.

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 Grassmannian LG(n,2n). Combined with work of Aluffi-Mihalcea-Sch\"urmann-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 x n symmetric matrices, we recover the formulas for the local Euler obstructions associated with the matrix rank stratification.