Minimal free resolutions of ideals of minors associated to pairs of matrices

Lőrincz, András

doi:10.1090/proc/15284

Minimal free resolutions of ideals of minors associated to pairs of matrices
HTML articles powered by AMS MathViewer

by András Cristian Lőrincz PDF

Proc. Amer. Math. Soc. 149 (2021), 1857-1873 Request permission

Abstract:

Consider the affine space consisting of pairs of matrices $(A,B)$ of fixed size, and its closed subvariety given by the rank conditions $\operatorname {rank} A \leq a, \operatorname {rank} B \leq b$, and $\operatorname {rank} (A\cdot B) \leq c$, for three non-negative integers $a,b,c$. These varieties are precisely the orbit closures of representations for the equioriented $\mathbb {A}_3$ quiver. In this paper we construct the (equivariant) minimal free resolutions of the defining ideals of such varieties. We show how this problem is equivalent to determining the cohomology groups of the tensor product of two Schur functors of tautological bundles on a 2-step flag variety. We provide several techniques for the determination of these groups, which is of independent interest.

References

Kaan Akin, David A. Buchsbaum, and Jerzy Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), no. 3, 207–278. MR 658729, DOI 10.1016/0001-8708(82)90039-1
S. Abeasis and A. Del Fra, Degenerations for the representations of a quiver of type ${\scr A}_m$, J. Algebra 93 (1985), no. 2, 376–412. MR 786760, DOI 10.1016/0021-8693(85)90166-8
S. Abeasis, A. Del Fra, and H. Kraft, The geometry of representations of $A_{m}$, Math. Ann. 256 (1981), no. 3, 401–418. MR 626958, DOI 10.1007/BF01679706
Daniel Simson and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 3, London Mathematical Society Student Texts, vol. 72, Cambridge University Press, Cambridge, 2007. Representation-infinite tilted algebras. MR 2382332
Michael K. Brown, Hang Huang, Robert P. Laudone, Michael Perlman, Claudiu Raicu, Steven V. Sam, and João Santos, Computing Schur complexes, J. Softw. Algebra Geom. 9 (2019), no. 2, 111–119. MR 4020643, DOI 10.2140/jsag.2019.9.111
A. I. Bondal and M. M. Kapranov, Homogeneous bundles, Helices and vector bundles, London Math. Soc. Lecture Note Ser., vol. 148, Cambridge Univ. Press, Cambridge, 1990, pp. 45–55. MR 1074782, DOI 10.1017/CBO9780511721526.006
Grzegorz Bobiński and Grzegorz Zwara, Normality of orbit closures for Dynkin quivers of type $\Bbb A_n$, Manuscripta Math. 105 (2001), no. 1, 103–109. MR 1885816, DOI 10.1007/PL00005871
Grzegorz Bobiński and Grzegorz Zwara, Schubert varieties and representations of Dynkin quivers, Colloq. Math. 94 (2002), no. 2, 285–309. MR 1967381, DOI 10.4064/cm94-2-10
D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.
Lutz Hille, Homogeneous vector bundles and Koszul algebras, Math. Nachr. 191 (1998), 189–195. MR 1621314, DOI 10.1002/mana.19981910109
R. Kinser and A. C. Lőrincz, Representation varieties of algebras with nodes, arXiv:1810.10997 (2018).
Ryan Kinser and Jenna Rajchgot, Type $A$ quiver loci and Schubert varieties, J. Commut. Algebra 7 (2015), no. 2, 265–301. MR 3370487, DOI 10.1216/JCA-2015-7-2-265
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
V. Lakshmibai and Peter Magyar, Degeneracy schemes, quiver schemes, and Schubert varieties, Internat. Math. Res. Notices 12 (1998), 627–640. MR 1635873, DOI 10.1155/S1073792898000397
A. C. Lőrincz, Singularities of zero sets of semi-invariants for quivers, arXiv:1509.04170 (2015). To appear in Journal of Commutative Algebra.
András Cristian Lőrincz, The $b$-functions of semi-invariants of quivers, J. Algebra 482 (2017), 346–363. MR 3646294, DOI 10.1016/j.jalgebra.2017.03.028
András Cristian Lőrincz, Decompositions of Bernstein-Sato polynomials and slices, Transform. Groups 25 (2020), no. 2, 577–607. MR 4098876, DOI 10.1007/s00031-019-09526-7
András C. Lőrincz and Jerzy Weyman, Free resolutions of orbit closures of Dynkin quivers, Trans. Amer. Math. Soc. 372 (2019), no. 4, 2715–2734. MR 3988590, DOI 10.1090/tran/7697
Giorgio Ottaviani and Elena Rubei, Quivers and the cohomology of homogeneous vector bundles, Duke Math. J. 132 (2006), no. 3, 459–508. MR 2219264, DOI 10.1215/S0012-7094-06-13233-7
Markus Reineke, Quivers, desingularizations and canonical bases, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000) Progr. Math., vol. 210, Birkhäuser Boston, Boston, MA, 2003, pp. 325–344. MR 1985731
Kavita Sutar, Resolutions of defining ideals of orbit closures for quivers of type $A_3$, J. Commut. Algebra 5 (2013), no. 3, 441–475. MR 3161742, DOI 10.1216/JCA-2013-5-3-441
Kavita Sutar, Orbit closures of source-sink Dynkin quivers, Int. Math. Res. Not. IMRN 11 (2015), 3423–3444. MR 3373055, DOI 10.1093/imrn/rnu037
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
Grzegorz Zwara, Singularities of orbit closures in module varieties, Representations of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, pp. 661–725. MR 2931906, DOI 10.4171/101-1/13