# Syzygies of determinantal varieties: Looking for English text

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
Digital Object Identifier: doi:10.1016/0001-8708(78)90037-3

• Is the Bruns-Vetter book ( home.uni-osnabrueck.de/wbruns/brunsw/detrings.pdf ) of any relevance? Apr 10, 2013 at 22:22
• Bruns-Vetter only treats special cases of the syzygies of determinantal ideals. At the end of Chapter 2 they mention Lascoux's work and ask whether a minimal free resolution exists in general over the integers. The answer is no: Hashimoto (in "Determinantal ideals without minimal free resolutions") showed that in characteristic 3, the Betti numbers change for the $2 \times 2$ minors of a $5 \times 5$ matrix (this example can also be seen in Macaulay2 now). Apr 11, 2013 at 13:12

## 3 Answers

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).

A less sophisticated search was more successful. It's all there in Chapter 6 of the book Cohomology of Vector Bundles and Syzygies by Weyman.

Since I can't accept my own answer for another two days, feel free to post any other suggestions, though. More reading material can't hurt.

• You have the right reference. Lascoux's paper isn't a complete solution; he has to assume some constants he constructs are nonzero, but does not prove that the constants actually are nonzero. (This is acknowledged in the paper; it was the best that could be done at the time.) The paper that finishes off this problem in characteristic zero is by Pragacz and Weyman in the late 80s; I don't have the reference offhand but you'll find it in Weyman's book. Apr 10, 2013 at 20:45

For a bit different angle on this problem look at the review article "Diagrams of Hermitian type, highest weight modules, and syzygies of determinantal varieties" by Hunziker & al.