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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
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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  • [ABW81] Kaan Akin, David A. Buchsbaum, and Jerzy Weyman, Resolutions of determinantal ideals: the submaximal minors, Adv. in Math. 39 (1981), no. 1, 1-30. MR 605350,
  • [AW97] Kaan Akin and Jerzy Weyman, Minimal free resolutions of determinantal ideals and irreducible representations of the Lie superalgebra $ {\rm gl}(m\vert n)$, J. Algebra 197 (1997), no. 2, 559-583. MR 1483781,
  • [AW07] Kaan Akin and Jerzy Weyman, Primary ideals associated to the linear strands of Lascoux's resolution and syzygies of the corresponding irreducible representations of the Lie superalgebra $ {\bf gl}(m\vert n)$, J. Algebra 310 (2007), no. 2, 461-490. MR 2308168,
  • [Bru03] Jonathan Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra $ \mathfrak{g}\mathfrak{l}(m\vert n)$, J. Amer. Math. Soc. 16 (2003), no. 1, 185-231. MR 1937204,
  • [dCEP80] C. de Concini, David Eisenbud, and C. Procesi, Young diagrams and determinantal varieties, Invent. Math. 56 (1980), no. 2, 129-165. MR 558865,
  • [Eis05] David Eisenbud, The geometry of syzygies, A second course in commutative algebra and algebraic geometry. Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. MR 2103875
  • [FH91] William Fulton and Joe Harris, Representation theory, A first course; Readings in Mathematics. Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. MR 1153249
  • [HKVdJ92] J. W. B. Hughes, R. C. King, and J. Van der Jeugt, On the composition factors of Kac modules for the Lie superalgebras $ {\rm sl}(m/n)$, J. Math. Phys. 33 (1992), no. 2, 470-491. MR 1145343,
  • [GS] Daniel R. Grayson and Michael E. Stillman, Macaulay 2, a software system for research in algebraic geometry, available at
  • [Kac77] V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8-96. MR 0486011
  • [Las78] Alain Lascoux, Syzygies des variétés déterminantales, Adv. in Math. 30 (1978), no. 3, 202-237 (French). MR 520233,
  • [Mac95] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., with contributions by A. Zelevinsky. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. MR 1354144
  • [NSS16] Rohit Nagpal, Steven V. Sam, and Andrew Snowden, Noetherianity of some degree two twisted commutative algebras, Selecta Math. (N.S.) 22 (2016), no. 2, 913-937. MR 3477338,
  • [RW14] Claudiu Raicu and Jerzy Weyman, Local cohomology with support in generic determinantal ideals, Algebra Number Theory 8 (2014), no. 5, 1231-1257. MR 3263142,
  • [Ser96] Vera Serganova, Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra $ {\mathfrak{g}}{\mathfrak{l}}(m\vert n)$, Selecta Math. (N.S.) 2 (1996), no. 4, 607-651. MR 1443186,
  • [Sno13] Andrew Snowden, Syzygies of Segre embeddings and $ \Delta $-modules, Duke Math. J. 162 (2013), no. 2, 225-277. MR 3018955,
  • [SHK00] Yucai Su, J. W. B. Hughes, and R. C. King, Primitive vectors of Kac-modules of the Lie superalgebras $ {\rm sl}(m/n)$, J. Math. Phys. 41 (2000), no. 7, 5064-5087. MR 1765833,
  • [Su06] Yucai Su, Composition factors of Kac modules for the general linear Lie superalgebras, Math. Z. 252 (2006), no. 4, 731-754. MR 2206623,
  • [Wey03] Jerzy Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, vol. 149, Cambridge University Press, Cambridge, 2003. MR 1988690

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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

Jerzy Weyman
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

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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