Enriched Schubert Problems in Lagrangian Grassmannians Frank Sottile C.J. Bott
		The Lagrangian Grassmannian 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.

There are 44 essential Schubert problems on the Lagrangian Grassmannian of isotropic 4-planes in 8-space, LG(4). (By essential, we mean that they do not reduce to a smaller Lagrangian Grassmannian, and have more than two solutions.) Of these three have been determined to be enriched in that they do not have full symmetric Galois groups. The method used is to compute Frobenius elements in the Galois groups over Q, giving a type of a lower bound the Galois group. When the Galois group is the full symmetric group, this often quickly determines that, and it can be used to infer that the problem is enriched.

More specifically, given an enumerative problem, formulating it over a finite field (e.g. such as F₆₅₅₂₁), and then computing and factoring a univariate eliminant gives the cycle type of the correspomding Frobenius elements. It is known that this selects fairly uniformly, so that by computing enough Frobenius elements will reveal the poccible cycle types of Frobenius elements and eventually the distribution of cycle types in the Galois group.

Of the 44 essential Schubert problems, only one, with 768 solutions (¹⁰=768) has resisted efforts to compute an eliminant.

The 40 that were determined to have the full symmetric group as their Galois group took 79.26 days for that determination, of which 78.39 days were spent on the largest problem, which has 384 solutions, (_·⁸=384)

The three enriched problems all have different Galois groups, and these are discussed in .

·· = 4 in LG(4)
Cycle type	Frequency	Fraction	Empirical
2,2	149672	0.7484	2.9934
1,1,1,1	50328	0.2516	1.0066
This computed 200000 Frobenius elements
This took 2969.40 seconds

··· = 4 in LG(4)
Cycle type	Frequency	Fraction	Empirical
4	201551	0.2519	2.0155
2,2	299875	0.3748	2.9987
1,1,2	199196	0.2490	1.9920
1,1,1,1	99378	0.1242	0.9938
This computed 800000 Frobenius elements
This took 4.78 Hours

··· = 8 in LG(4)
Cycle type	Frequency	Fraction	Empirical
4,4	626184386	0.3131	60.1137
2,6	334051444	0.1670	32.0689
1,1,3,3	333472372	0.1667	32.0133
1,1,2,4	249921255	0.1250	23.9924
2,2,2,2	259801406	0.1299	24.9409
1,1,1,1,2,2	186420449	0.0932	17.8964
1,1,1,1,1,1,1,1	10148688	0.0051	0.9743
This computed 2000000000 Frobenius elements
This took 578.82 Days