Enriched Schubert Problems in Lagrangian Grassmannians

Frank Sottile
C.J. Bott
The Lagrangian Grassmannian LG(n) is the set of n-dimensional isotropic linear subspaces in 2n-dimension space equipped with a nondegenerate alternating form (a symplectic vector space). This has dimension n(n-1)/2. It has Schubert subvarieties and a rich Schubert calculus of enumerative geometry. Its Schubert problems turn out to have very interesting Galois groups, much more varied than those found in the ordinary (type A) Grassmannian and flag manifolds.
Schubert problems on the Lagrangian Grassmannian are given by two data
● An isotropic flag F∙, which is a sequence of isotropic linear subsaces F1⊂F2⊂ ⋯ ⊂Fn with Fn Lagrangian.
● A strict partition: λ, which is a sequence n ≥ λ1 >λ2 > ⋯ >λk .
Include some explanation here
LG(3) has four essential Schubert problems, none of which are enriched.
LG(4) has 44 essential Schubert problems. (By essential, we mean that they do not reduce to a smaller Lagrangian Grassmannian, and have more than two solutions.) One, with 768 solutions (10=768) resisted efforts to compute an eliminant. Of the remaining 43, exactly three are enriched. We used 294.59 GHz-Days determining that 40 had fully symmetric Galois Group, and 5.87 GHz-Years studying the remaining three, finding three different Galois groups. The enriched problems had 4 and 8 solutions. These are discussed in .
A more complete description of the computation on this Grassmannian is found here.     Raw data for LG4.
LG(5) has significantly more Schubert problems. We studied the 472 essential problems with at most 320 solutions. Of these, 36 were enriched. We used 2.53 GHz-years determining that the 436 had full-symmetric Galois group and the computation of many Frobenius elements tool 11.66 GHz-years. The enriched problems had 4, 8, or 16 solutions, and we found six different Galois groups. These are discussed on this page.     Raw data for LG5.
Raw data for LG5
Created Fri Apr 4 04:42:25 AM CDT 2025 by Frank Sottile