Derived supersymmetries of determinantal varieties. (English) Zbl 1351.17009

Sam, Steven V.

J. Commut. Algebra 6, No. 2, 261-286 (2014).

Summary: We show that the linear strands of the Tor of determinantal varieties in spaces of symmetric and skew-symmetric matrices are irreducible representations for the periplectic (strange) Lie superalgebra. The structure of these linear strands is explicitly known, so this gives an explicit realization of some representations of the periplectic Lie superalgebra. This complements results of P. Pragacz and J. Weyman [J. Algebra 128, 1–44 (1990; Zbl 0688.13003)], who showed an analogous statement for the generic determinantal varieties and the general linear Lie superalgebra. We also give a simpler proof of their result. Via Koszul duality, this is an odd analogue of the fact that the coordinate rings of these determinantal varieties are irreducible representations for a certain classical Lie algebra.

MSC:

17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
13D02	Syzygies, resolutions, complexes and commutative rings
14M12	Determinantal varieties

Keywords:

periplectic (strange) Lie superalgebra; irreducible representations; determinantal varieties

Citations:

Zbl 0688.13003

References:

[1]	Kaan Akin, David A. Buchsbaum and Jerzy Weyman, Schur functors and Schur complexes , Adv. Math. 44 (1982), 207-278. · Zbl 0497.15020 · doi:10.1016/0001-8708(82)90039-1
[2]	Kaan Akin and Jerzy Weyman, Minimal free resolutions of determinantal ideals and irreducible representations of the Lie superalgebra \({\mathrm gl}(m| n)\) , J. Algebra 197 (1997), 559-583. · Zbl 0898.17003 · doi:10.1006/jabr.1997.7101
[3]	—-, The irreducible tensor representations of \({\mathrm gl}(m| 1)\) and their generic homology , J. Algebra 230 (2000), 5-23. · Zbl 1041.17005 · doi:10.1006/jabr.1999.7986
[4]	—-, Primary ideals associated to the linear strands of Lascoux’s resolution and syzygies of the corresponding irreducible representations of the Lie superalgebra \({\mathbf gl}(m| n)\) , J. Algebra 310 (2007), 461-490. · Zbl 1171.17002 · doi:10.1016/j.jalgebra.2003.11.015
[5]	A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras , Adv. Math. 64 (1987), 118-175. · Zbl 0617.17002 · doi:10.1016/0001-8708(87)90007-7
[6]	Winfried Bruns and Udo Vetter, Determinantal rings , Lect. Notes Math. 1327 , Springer-Verlag, Berlin, 1988. · Zbl 0673.13006 · doi:10.1007/BFb0080378
[7]	Thomas J. Enright and Markus Hunziker, Resolutions and Hilbert series of determinantal varieties and unitary highest weight modules , J. Algebra 273 (2004), 608-639. · Zbl 1088.17007 · doi:10.1016/S0021-8693(03)00159-5
[8]	Thomas J. Enright and Jeb F. Willenbring, Hilbert series, Howe duality and branching for classical groups , Ann. Math. 159 (2004), 337-375. · Zbl 1087.22011 · doi:10.4007/annals.2004.159.337
[9]	William Fulton and Joe Harris, Representation theory. A first course , Grad. Texts Math. 129 , Springer-Verlag, New York, 1991. · Zbl 0744.22001
[10]	Roger Howe, Remarks on classical invariant theory , Trans. Amer. Math. Soc. 313 (1989), 539-570. · Zbl 0674.15021 · doi:10.2307/2001418
[11]	T. Józefiak, P. Pragacz and J. Weyman, Resolutions of determinantal varieties and tensor complexes associated with symmetric and antisymmetric matrices , Astérisque 87 - 88 , Soc. Math., Paris, France, 1981. · Zbl 0488.14012
[12]	V. G. Kac, Lie superalgebras , Adv. Math. 26 (1977), 8-96. · Zbl 0366.17012 · doi:10.1016/0001-8708(77)90017-2
[13]	—-, A sketch of Lie superalgebra theory , Comm. Math. Phys. 53 (1977), 31-64. · Zbl 0359.17009 · doi:10.1007/BF01609166
[14]	V. Lakshmibai and K. N. Raghavan, Standard monomial theory. Invariant theoretic approach , Inv. Th. Alg. Trans. Groups 8 , Springer-Verlag, Berlin, 2008. · Zbl 1137.14036
[15]	Alain Lascoux, Syzygies des variétés déterminantales , Adv. Math. 30 (1978), 202-237. · Zbl 0394.14022 · doi:10.1016/0001-8708(78)90037-3
[16]	T. Levasseur and J.T. Stafford, Rings of differential operators on classical rings of invariants , Mem. Amer. Math. Soc. 81 , 1989. · Zbl 0691.16019 · doi:10.1090/memo/0412
[17]	Piotr Pragacz, Symmetric polynomials and divided differences in formulas of intersection theory, Parameter spaces , Banach Cent. Publ. 36 , Polish Academy of Sciences, Warsaw, 1996. · Zbl 0851.05094
[18]	Piotr Pragacz and Jerzy Weyman, Complexes associated with trace and evaluation. Another approach to Lascoux’s resolution , Adv. Math. 57 (1985), 163-207. · Zbl 0591.14035 · doi:10.1016/0001-8708(85)90052-0
[19]	Steven V Sam, Schubert complexes and degeneracy loci , J. Alg. 337 (2011), 103-125. · Zbl 1242.13017 · doi:10.1016/j.jalgebra.2011.04.032
[20]	T. Tanisaki, Highest weight modules associated to parabolic subgroups with commutative unipotent radicals , NATO Adv. Sci. Inst. Math. Phys. Sci. 517 , Kluwer Academic Publishers, Dordrecht, 1998. · Zbl 1012.17004 · doi:10.1007/978-94-011-5308-9_5
[21]	Jerzy Weyman, Cohomology of vector bundles and syzygies , Cambr. Tracts Math. 149 , Cambridge University Press, Cambridge, 2003. · Zbl 1075.13007