2021/05/18 by LeVan, Paul, Raicu, Claudiu
#05C05 #14M12 #14M15 #32S05 #32S60 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2105.08823
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 ey,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 ey,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.