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?


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.

  • $\begingroup$ 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? $\endgroup$
    – imakhlin
    Dec 6 '18 at 15:33
  • $\begingroup$ 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 $\endgroup$ Dec 6 '18 at 15:44
  • $\begingroup$ 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? $\endgroup$
    – imakhlin
    Dec 6 '18 at 15:55
  • $\begingroup$ Not 100% positive. $\endgroup$ Dec 6 '18 at 16:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.