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EQUIVARIANT ${\mathcal{D}}$ -MODULES ON ALTERNATING SENARY 3-TENSORS

  • ANDRÁS C. LŐRINCZ (a1) and MICHAEL PERLMAN (a2)

Abstract

We consider the space $X=\bigwedge ^{3}\mathbb{C}^{6}$ of alternating senary 3-tensors, equipped with the natural action of the group $\operatorname{GL}_{6}$ of invertible linear transformations of $\mathbb{C}^{6}$ . We describe explicitly the category of $\operatorname{GL}_{6}$ -equivariant coherent ${\mathcal{D}}_{X}$ -modules as the category of representations of a quiver with relations, which has finite representation type. We give a construction of the six simple equivariant ${\mathcal{D}}_{X}$ -modules and give formulas for the characters of their underlying $\operatorname{GL}_{6}$ -structures. We describe the (iterated) local cohomology groups with supports given by orbit closures, determining, in particular, the Lyubeznik numbers associated to the orbit closures.

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[ASS06]Assem, I., Simson, D. and Skowroński, A., Elements of the Representation Theory of Associative Algebras. Vol. 1, London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge, 2006.
[BCP97]Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system. I. The user language, J. Symbolic Comput. 24(3–4) (1997), 235265; Computational algebra and number theory (London, 1993).
[Bri83]Brion, M., Invariants d’un sous-groupe unipotent maximal d’un groupe semi-simple, Ann. Inst. Fourier (Grenoble) 33(1) (1983), 127.
[GLS98]García López, R. and Sabbah, C., Topological computation of local cohomology multiplicities, Collect. Math. 49(2–3) (1998), 317324; Dedicated to the memory of Fernando Serrano, MR 1677136.
[Har77]Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics 52, Springer, New York-Heidelberg, 1977.
[HTT08]Hotta, R., Takeuchi, K. and Tanisaki, T., D-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics 236, Birkhäuser Boston, Inc., Boston, MA, 2008, Translated from the 1995 Japanese edition by Takeuchi.
[Igu00]Igusa, J.-i., An Introduction to the Theory of Local Zeta Functions, AMS/IP Studies in Advanced Mathematics, 14, American Mathematical Society, Providence, RI, 2000, International Press, Cambridge, MA.
[Kas03]Kashiwara, M., D-Modules and Microlocal Calculus, Iwanami Series in Modern Mathematics; Translations of Mathematical Monographs 217, American Mathematical Society, Providence, RI, 2003.
[Kim82]Kimura, T., The b-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces, Nagoya Math. J. 85 (1982), 180.
[KM87]Knop, F. and Menzel, G., Duale Varietäten von Fahnenvarietäten, Comment. Math. Helv. 62(1) (1987), 3861.
[Lőr19]Lőrincz, A., Decompositions of Bernstein–Sato polynomials and slices, Transform. Groups (2019), doi:10.1007/s00031-019-09526-7.
[LM04]Landsberg, J. M. and Manivel, L., Series of Lie groups, Michigan Math. J. 52(2) (2004), 453479.
[LR18]Lőrincz, A. C. and Raicu, C., Iterated local cohomology groups and Lyubeznik numbers for determinantal rings, 2018, arXiv:1805.08895.
[LRW19]Lőrincz, A. C., Raicu, C. and Weyman, J., Equivariant 𝓓-modules on binary cubic forms, Comm. Algebra 47 (2019), 24572487.
[LSW16]Lyubeznik, G., Singh, A. K. and Walther, U., Local cohomology modules supported at determinantal ideals, J. Eur. Math. Soc. (JEMS) 18(11) (2016), 25452578; MR 3562351.
[LW19]Lőrincz, A. C. and Walther, U., On categories of equivariant D-modules, Adv. Math. 351 (2019), 429478.
[Lyu93]Lyubeznik, G., Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra), Invent. Math. 113(1) (1993), 4155.
[Man01]Manivel, L., Symmetric Functions, Schubert Polynomials and Degeneracy Loci, SMF/AMS Texts and Monographs 6, American Mathematical Society, Providence, RI, 2001, Société Mathématique de France, Paris, Translated from the 1998 French original by John R. Swallow, Cours Spécialisés [Specialized Courses], 3, MR 1852463.
[MV86a]MacPherson, R. and Vilonen, K., Elementary construction of perverse sheaves, Invent. Math. 84(2) (1986), 403435.
[MV86b]MacPherson, R. and Vilonen, K., Elementary construction of perverse sheaves, Invent. Math. 84 (1986), 403435.
[NBWZ16]Núñez Betancourt, L., Witt, E. E. and Zhang, W., A survey on the Lyubeznik numbers, Mexican mathematicians abroad: recent contributions (2016), 137163.
[Ogu73]Ogus, A., Local cohomological dimension of algebraic varieties, Ann. of Math. (2) 98 (1973), 327365; MR 506248.
[Per18]Perlman, M., Equivariant 𝓓-modules on 2 × 2 × 2 hypermatrices, J. Algebra (2018), to appear, arXiv:1809.00352.
[Rai16]Raicu, C., Characters of equivariant 𝓓-modules on spaces of matrices, Compos. Math. 152 (2016), 19351965.
[Rai17]Raicu, C., Characters of equivariant 𝓓-modules on Veronese cones, Trans. Amer. Math. Soc. 369(3) (2017), 20872108.
[RW14]Raicu, C. and Weyman, J., Local cohomology with support in generic determinantal ideals, Algebra Number Theory 8(5) (2014), 12311257.
[RW16]Raicu, C. and Weyman, J., Local cohomology with support in ideals of symmetric minors and Pfaffians, J. Lond. Math. Soc. (2) 94(3) (2016), 709725.
[RWW14]Raicu, C., Weyman, J. and Witt, E., Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians, Adv. Math. 250 (2014), 596610.
[Sai93]Saito, M., On b-function, spectrum and rational singularity, Math. Ann. 295 (1993), 5174.
[SK77]Sato, M. and Kimura, T., A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1155.
[Swi15]Switala, N., Lyubeznik numbers for nonsingular projective varieties, Bull. Lond. Math. Soc. 47(1) (2015), 16.
[Vil94]Vilonen, K., Perverse sheaves and finite-dimensional algebras, Trans. Amer. Math. Soc. 341(2) (1994), 665676.
[Wey03]Weyman, J., Cohomology of Vector Bundles and Syzygies, Cambridge Tracts in Mathematics 149, Cambridge University Press, Cambridge, 2003.
[Wey94]Weyman, J., Calculating discriminants by higher direct images, Trans. Amer. Math. Soc. 343(1) (1994), 367389.
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EQUIVARIANT ${\mathcal{D}}$ -MODULES ON ALTERNATING SENARY 3-TENSORS

  • ANDRÁS C. LŐRINCZ (a1) and MICHAEL PERLMAN (a2)

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