Irreducibility of Gelfand-Serganova strata

Question

To keep the notations simple I'll restrict my attention to the complete flag variety although the question should be equally valid for partial flag varieties. Consider $G=SL_n(\mathbb C)$ with Borel $B\subset G$, Cartan $T\subset B$ and flag variety $F=G/B$. The Plücker embedding $$F\subset\mathbb P(\wedge^1\mathbb C^n)\times\ldots\times\mathbb P(\wedge^{n-1}\mathbb C^n)$$ equips $F$ with Plücker coordinates $\{X_{i_1,\ldots,i_k}\}$.

The Gelfand-Serganova strata in $F$ are the equivalence classes with respect to any of the following three equivalence relations (which are shown to coincide).

1. $x\sim y$ iff the set of Plücker coordinates vanishing in $x$ is the same as that for $y$.
2. $x\sim y$ iff the set of $T$-fixed points contained in the orbit closure $\overline{Tx}$ is the same as that in $\overline{Ty}$.
3. $x\sim y$ iff for any Borel $B'\supset T$ the orbits $B'x$ and $B'y$ coincide, i.e. $x$ and $y$ lie in the same Schubert cell with respect to $B'$. In other words, the strata are the nonempty intersections of $n!$ Schubert cells, one for every Borel containing $T$.

Unfortunately, as shown in the above paper, the closure of a stratum is not necessarily a union of strata. This rules out many nice properties one could hope for in such a setting. However, I still have a feeling that the strata (and their closures) might always be irreducible. Is that so and has this been discussed in the literature?

Answer 1

The strata need not be irreducible.

Quoting page 2 of Knutson, Lam, Speyer (https://arxiv.org/abs/0903.3694v1): "the strata can have essentially any singularity [Mn88]. In particular, the nonempty ones need not be irreducible, or even equidimensional."

They are referencing "Mnëv's universality theorem" here.

Answer 2

Here is an explicit example. There are examples of realization spaces of matroids (which are, up to a torus quotient, Gelfand-Serganova strata in the Grassmannian) which are disconnected. I believe the first example was given by Rybnikov, the so called MacLane matroid: there is a realization space in $Gr(3, 8)$ which consists of two points. Many other examples are discussed in this paper by Corey and Luber.

We can obtain a disconnected Gelfand-Serganova strata in the full flag variety by taking the "full Higgs lift" of the MacLane matroid. This corresponds to taking flags such that the third step represents the MacLane matroid but are otherwise generic. The projection of this Gelfand-Serganova strata to $Gr(3, 8)$ is the Gelfand-Serganova strata of the MacLane matroid, which implies that it is disconnected. Any other (realizable) lift would do as well.