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?



Exact sequences of bundles on Grassmannians
Look at Fonarev's [*On minimal Lefschetz decompositions for Grassmannians*](https://arxiv.org/abs/1108.2292v1), specifically Proposition 5.3 ([link to proposition in the PDF](https://arxiv.org/pdf/1108.2292v1.pdf#page=14)). I guess this exact sequence is what you need.


