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We're looking for a large set of exact sequences of vector bundles on Grassmannians. Here's the set up:

$V$ and $Q$ are complex vector spaces of dimensions $d$ and $r$ respectively $(d\geq r)$, and we're working on the Grassmannian $Gr(V,Q)$. For simplicity let's fix a trivialization of $det(V)$.

Now let $\alpha$ be a partition/Young diagram with at most $(r-1)$ rows and at most $(d-r)$ columns. Let $\beta$ be the Young diagram obtained from $\alpha$ by adding an extra row of length $(d-r)$ at the beginning. What we want is an exact sequence of vector bundles that goes $$\mathbb{S}_\alpha(Q)\otimes det(Q)^{-1} \rightarrow\;\; ... \;\;\rightarrow \mathbb{S}_\beta(Q) $$ ($\mathbb{S}$ denotes a Schur functor). For $r=1$ there's only one choice for $\alpha$, and the Koszul complex is the required sequence. For $d-r=1$ we have the short exact sequences $$\wedge^k Q \otimes det(Q)^{-1}\rightarrow \wedge^{k+1}V\ \rightarrow \wedge^{k+1} Q$$ We can also solve $r=2$ using Eagon-Northcott complexes. These known cases suggest that the exact sequence should have $(d-r+2)$ terms.

Does anyone know a general construction?

Update: we have a precise conjecture for the terms in the sequence. Let $\beta_0$ be the partition obtained from $\alpha$ by deleting the first column. Now define $\beta_i$ recursively as the partition obtained from $\beta_{i-1}$ by adding boxes to the $i$th column until it agrees with the $i$th column of $\beta$. In particular $\beta_{d-r}=\beta$. Then the terms in middle of the exact sequence should be $$...\rightarrow \wedge^{(|\beta| - |\beta_i|)} V \otimes \mathbb{S}_{\beta_i} Q \rightarrow... $$ If we fix a single point on the Grassmannian and split the tautological short exact sequence there then we can show that this works, which is pretty good evidence. Surely this isn't a new discovery?

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I may be missing something but it seems that you may be able to get these sequences by applying the standard Schur complex functors (as in section 2.4 of J.Weyman's book "Cohomology of vector bundles and syzygies) to the tautological short exact sequence on the Grassmanian. – Tony Pantev Aug 23 '11 at 12:17
I think you are looking for Schur complexes, which generalize the Koszul complex. These are described in Jerzy Weyman's book Syzygies and vector bundles and in Boffi-Buchsbaum's book Threading homology... The basic construction starts with a map of vector bundles $E \rightarrow F$ and a partition and produces a complex whose terms have filtrations with associated graded given as tensor products of Schur functors of $E$ and $F$. I don't have references to hand at the moment, but will say something more precise when I can. – Chris Brav Aug 23 '11 at 12:19
Oh, Tony got there first. – Chris Brav Aug 23 '11 at 12:19
Here's an older and less general question:… – Chris Brav Aug 23 '11 at 12:23
Actually I'm not sure that this solves the problem. Even the Eagon-Northcott complexes aren't single Schur complexes, you get them by patching together two Schur complexes (one of which is applied to the dual tautological SES). So the sequences I'm looking for might be built out of Schur complexes, but how? – Ed Segal Aug 23 '11 at 14:47
up vote 2 down vote accepted

Hi Ed. Look at arXiv:1108.2292, Proposition 5.3. I guess this exact sequence is what you need.

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Thanks! So it was almost a new discovery! – Ed Segal Aug 30 '11 at 12:00

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