## The syzygies of some thickenings of determinantal varieties

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- by Claudiu Raicu and Jerzy Weyman
- Proc. Amer. Math. Soc.
**145**(2017), 49-59 - DOI: https://doi.org/10.1090/proc/13197
- Published electronically: July 12, 2016
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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 $\mathfrak {gl}(m|n)$, 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

- 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**, DOI 10.1016/0001-8708(81)90055-4 - Kaan Akin and Jerzy Weyman,
*Minimal free resolutions of determinantal ideals and irreducible representations of the Lie superalgebra $\textrm {gl}(m|n)$*, J. Algebra**197**(1997), no. 2, 559–583. MR**1483781**, DOI 10.1006/jabr.1997.7101 - 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 $\textbf {gl}(m|n)$*, J. Algebra**310**(2007), no. 2, 461–490. MR**2308168**, DOI 10.1016/j.jalgebra.2003.11.015 - Jonathan Brundan,
*Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra $\mathfrak {g}\mathfrak {l}(m|n)$*, J. Amer. Math. Soc.**16**(2003), no. 1, 185–231. MR**1937204**, DOI 10.1090/S0894-0347-02-00408-3 - C. de Concini, David Eisenbud, and C. Procesi,
*Young diagrams and determinantal varieties*, Invent. Math.**56**(1980), no. 2, 129–165. MR**558865**, DOI 10.1007/BF01392548 - David Eisenbud,
*The geometry of syzygies*, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. A second course in commutative algebra and algebraic geometry. MR**2103875** - William Fulton and Joe Harris,
*Representation theory*, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR**1153249**, DOI 10.1007/978-1-4612-0979-9 - J. W. B. Hughes, R. C. King, and J. Van der Jeugt,
*On the composition factors of Kac modules for the Lie superalgebras $\textrm {sl}(m/n)$*, J. Math. Phys.**33**(1992), no. 2, 470–491. MR**1145343**, DOI 10.1063/1.529782 - Daniel R. Grayson and Michael E. Stillman,
*Macaulay 2, a software system for research in algebraic geometry*, available at http://www.math.uiuc.edu/Macaulay2/. - V. G. Kac,
*Lie superalgebras*, Advances in Math.**26**(1977), no. 1, 8–96. MR**486011**, DOI 10.1016/0001-8708(77)90017-2 - Alain Lascoux,
*Syzygies des variétés déterminantales*, Adv. in Math.**30**(1978), no. 3, 202–237 (French). MR**520233**, DOI 10.1016/0001-8708(78)90037-3 - I. G. Macdonald,
*Symmetric functions and Hall polynomials*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR**1354144** - 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**, DOI 10.1007/s00029-015-0205-y - Claudiu Raicu and Jerzy Weyman,
*Local cohomology with support in generic determinantal ideals*, Algebra Number Theory**8**(2014), no. 5, 1231–1257. MR**3263142**, DOI 10.2140/ant.2014.8.1231 - Vera Serganova,
*Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra ${\mathfrak {g}}{\mathfrak {l}}(m|n)$*, Selecta Math. (N.S.)**2**(1996), no. 4, 607–651. MR**1443186**, DOI 10.1007/PL00001385 - Andrew Snowden,
*Syzygies of Segre embeddings and $\Delta$-modules*, Duke Math. J.**162**(2013), no. 2, 225–277. MR**3018955**, DOI 10.1215/00127094-1962767 - Yucai Su, J. W. B. Hughes, and R. C. King,
*Primitive vectors of Kac-modules of the Lie superalgebras $\textrm {sl}(m/n)$*, J. Math. Phys.**41**(2000), no. 7, 5064–5087. MR**1765833**, DOI 10.1063/1.533392 - Yucai Su,
*Composition factors of Kac modules for the general linear Lie superalgebras*, Math. Z.**252**(2006), no. 4, 731–754. MR**2206623**, DOI 10.1007/s00209-005-0874-x - Jerzy Weyman,
*Cohomology of vector bundles and syzygies*, Cambridge Tracts in Mathematics, vol. 149, Cambridge University Press, Cambridge, 2003. MR**1988690**, DOI 10.1017/CBO9780511546556

## Bibliographic 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