## The syzygies of some thickenings of determinantal varieties

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- by Claudiu Raicu and Jerzy Weyman PDF
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**145**(2017), 49-59 Request permission

## Abstract:

The vector space of $m\times n$ complex matrices ($m\geq n$) admits a natural action of the group $\operatorname {GL}=\operatorname {GL}_m\times \operatorname {GL}_n$ via row and column operations. For positive integers $a,b$, we consider the ideal $I_{a\times b}$ defined as the smallest $\operatorname {GL}$-equivariant ideal containing the $b$-th powers of the $a\times a$ minors of the generic $m\times n$ matrix. We compute the syzygies of the ideals $I_{a\times b}$ for all $a,b$, together with their $\operatorname {GL}$-equivariant structure, generalizing earlier results of Lascoux for the ideals of minors ($b=1$), and of Akin–Buchsbaum–Weyman for the powers of the ideals of maximal minors ($a=n$). Our methods rely on a nice connection between commutative algebra and the representation theory of the superalgebra $\mathfrak {gl}(m|n)$, as well as on our previous calculation of $\operatorname {Ext}$ modules done in the context of describing local cohomology with determinantal support. Our results constitute an important ingredient in the proof by Nagpal–Sam–Snowden of the first non-trivial Noetherianity results for twisted commutative algebras which are not generated in degree one.## References

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## Additional Information

**Claudiu Raicu**- Affiliation: Department of Mathematics, 255 Hurley Hall, University of Notre Dame, Notre Dame, Indiana 46556 – and – Institute of Mathematics “Simion Stoilow” of the Romanian Academy
- MR Author ID: 909516
- Email: craicu@nd.edu
**Jerzy Weyman**- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
- MR Author ID: 182230
- ORCID: 0000-0003-1923-0060
- Email: jerzy.weyman@uconn.edu
- Received by editor(s): January 27, 2016
- Received by editor(s) in revised form: March 15, 2016
- Published electronically: July 12, 2016
- Communicated by: Irena Peeva
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 49-59 - MSC (2010): Primary 13D02, 14M12, 17B10
- DOI: https://doi.org/10.1090/proc/13197
- MathSciNet review: 3565359