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I am looking for an (incomplete) list of partial flag varieties for which all Schubert cells admit small resolutions of singularities. This is interesting, for many reasons. My motivation is, that a description of a small resolution will give the corresponding IC sheaves very explicitly and hence explicit formulas for the KL-polynomials.

For example I know that Zelevinsky showed that this is the case for all type A Grassmannians.

What about other $G/P$ for say $P$ Hermitian symmetric? I think in this case there are at least explicit formulas for KL polynomials.

What about other partial flag varieties?

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@Jan: This is outside my range of knowledge, but I wonder what searches you have tried? For example, beyond Zelevinsky's 1983 paper, MathSciNet turns up quickly five others which may be relevant to your question: B.F.Jones (2010), N. Perrin (2007), J. van Hamel (2003), P.Sankaran and P. Parameswaran (1994, 1995). At least some of these must be on arXiv. There has certainly been some related study though it might not fit your question exactly. –  Jim Humphreys Sep 3 '12 at 14:21
The book "Singular Loci of Schubert Varieties" by Billey and Lakshmibai has some results about small resolutions (cf in particular section 9.1); you might want to take a look there. –  Chuck Hague Sep 3 '12 at 15:34
It is a result of Billey and Warrington that "321 hexagon avoiding" permutations in the symmetric group admit small resolutions. See: Billey and Warrington, Kazhdan-Lusztig Polynomials for 321-hexagon-avoiding permutations, J. of Algebraic Combinatorics, v. 13 (2001), no. 2, 111--136. –  Geordie Williamson Mar 6 at 15:06
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1 Answer

up vote 7 down vote accepted

For the Hermitian symmetric $G/P$, Nicolas Perrin explicitly classified all the Schubert varieties admitting a small resolution. (More explicitly, he classifies all the minimal models and quotes a theorem that says a small resolution is a smooth minimal model.) If I remember correctly, except for some very small rank exceptions, type A, and the obvious cases (e.g. projective space), all $G/P$ have a Schubert without a small resolution. Fortunately, there are few enough minimal models for the Hermitian symmetric cases that even in the non-smooth case, their intersection cohomology have a reasonable description, leading to the (previously known) explicit formulas for KL-polynomials. (These formulas are summarized in a paper by Boe that proves the final cases and summarizes and refers to the cases done earlier.)

In type A, there is a Schubert without a small resolution for any 2-step flag variety where the two steps are not adjacent. (The example mentioned in the Zelevinsky paper (which he notes as being due to MacPherson) generalizes to all such cases.) This also means there is a Schubert without a small resolution for any 3-or-more-step flag variety.

This leaves as the only case (barring low rank exceptions) adjacent 2-step flag varieties. I have not seen this resolved in the literature. I suspect there is a Schubert variety in those cases without a small resolution provided the steps are at least two away from both ends. In fact I have an explicit expected counterexample, but I have never learned the tools necessary to prove it actually is one.

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That's interesting! In case you edit your answer again, I think MacPherson has a capital "P". –  tweetie-bird Sep 4 '12 at 2:12
Oops -- thanks. –  Alexander Woo Sep 5 '12 at 19:51
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