I would like to understand the syzygies of the determinantal ideal $I\_r$, generated by the $r\times r$ minors of a matrix $(X\_{ij})$ of indeterminantes in the polynomial ring over an algebraically closed field of characteristic zero. The original resource for this object of study is the paper *"Syzygies des variétés déterminantales"* by Alain Lascoux [L]. While I would like to read it at some point, my rusty French is making it a bit cumbersome, and hence I was wondering if there were any translations of this treatment in English, possibly in some textbook. Thanks a lot in advance already.


**[L]** A. Lascoux, *Syzygies des variétés déterminantales*, Adv. Math. 30 (1978), 202–237.
  
     Mathematical Reviews (MathSciNet): [MR520233](http://www.ams.org/mathscinet-getitem?mr=520233)
  
     Digital Object Identifier: [doi:10.1016/0001-8708(78)90037-3](http://dx.doi.org/10.1016/0001-8708(78)90037-3)



Syzygies of determinantal varieties: Looking for English text
Weyman's book is a good reference. If you want other references, you can see the paper by myself joint with Snowden and Weyman: <http://arxiv.org/abs/1209.3509>


One can view the coordinate ring of the determinantal variety as a ring of invariants for a natural group action and in that paper we calculate the syzygies of all modules of covariants (see Section 1.6).


