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The syzygies of some thickenings of determinantal varieties


Authors: Claudiu Raicu and Jerzy Weyman
Journal: Proc. Amer. Math. Soc. 145 (2017), 49-59
MSC (2010): Primary 13D02, 14M12, 17B10
DOI: https://doi.org/10.1090/proc/13197
Published electronically: July 12, 2016
MathSciNet review: 3565359
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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\vert 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.


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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
Email: craicu@nd.edu

Jerzy Weyman
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: jerzy.weyman@uconn.edu

DOI: https://doi.org/10.1090/proc/13197
Keywords: Syzygies, determinantal varieties, permanents, general linear superalgebra
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
Article copyright: © Copyright 2016 American Mathematical Society

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