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?

up vote 5 down vote accepted

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.

  • Nice, thanks! I think the Mnev reference is in regard to the singularity claim though and the reducibility claim is sort of left unexplained. Maybe one of the authors could direct us towards a counterexample? – imakhlin Dec 6 at 15:33
  • Maybe see Theorem 9.8 here for a precise statement of Mnev's universality theorem which says these strata need not be irreducible?: arxiv.org/pdf/1409.3503.pdf – Sam Hopkins Dec 6 at 15:44
  • Thank you. Seeing how this is related to the irreducibility of GS strata will require some processing on my part. But you're positive it is, right? – imakhlin Dec 6 at 15:55
  • Not 100% positive. – Sam Hopkins Dec 6 at 16:02

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