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Acta Mathematica Vietnamica

, Volume 44, Issue 1, pp 269–284 | Cite as

Syzygies of Determinantal Thickenings and Representations of the General Linear Lie Superalgebra

  • Claudiu RaicuEmail author
  • Jerzy Weyman
Article

Abstract

We let \(S=\mathbb C[x_{i,j}]\) denote the ring of polynomial functions on the space of \(m\times n\) matrices and consider the action of the group \(\text {GL}=\text {GL}_{m}\times \text {GL}_{n}\) via row and column operations on the matrix entries. For a \(\text {GL}\)-invariant ideal \(I\subseteq S\), we show that the linear strands of its minimal free resolution translate via the BGG correspondence to modules over the general linear Lie superalgebra \(\mathfrak {gl}(m|n)\). When \(I=I_{\lambda }\) is the ideal generated by the \(\text {GL}\)-orbit of a highest weight vector of weight \(\lambda \), we give a conjectural description of the classes of these \(\mathfrak {gl}(m|n)\)-modules in the Grothendieck group, and prove that our prediction is correct for the first strand of the minimal free resolution.

Keywords

Determinantal thickenings Syzygies BGG correspondence General linear Lie superalgebra Kac modules Dyck paths 

Mathematics Subject Classification (2010)

Primary 13D02 14M12 17B10 

Notes

Acknowledgements

Experiments with the computer algebra software Macaulay2 [4] have provided numerous valuable insights. Raicu acknowledges the support of the Alfred P. Sloan Foundation, and of the National Science Foundation Grant No. 1600765. Weyman acknowledges partial support of the Sidney Professorial Fund and of the National Science Foundation grant No. 1400740.

References

  1. 1.
    Akin, K., Buchsbaum, D.A., Weyman, J.: Resolutions of determinantal ideals: the submaximal minors. Adv. in Math. 39(1), 1–30 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brundan, J.: Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra \(\mathfrak {gl}(m|n)\). J. Am. Math. Soc. 16(1), 185–231 (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Eisenbud, D.: The geometry of syzygies. A second course in commutative algebra and algebraic geometry. Graduate Texts in Mathematics, vol. 229. Springer-Verlag, New York (2005)Google Scholar
  4. 4.
    Grayson, D.R., Stillman, M.E.: Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
  5. 5.
    Lascoux, A.: Syzygies des variétés déterminantales. Adv. Math. 30(3), 202–237 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Pragacz, P., Weyman, J.: Complexes associated with trace and evaluation. Another approach to Lascoux’s resolution. Adv. Math. 57(2), 163–207 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Raicu, C.: Regularity and cohomology of determinantal thickenings. Proc. Lond. Math. Soc. 116(2), 248–280 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Raicu, C., Weyman, J.: Local cohomology with support in generic determinantal ideals. Algebra & Number Theory 8(5), 1231–1257 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Raicu, C., Weyman, J.: The syzygies of some thickenings of determinantal varieties. Proc. Am. Math. Soc. 145(1), 49–59 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Sam, S.V.: Derived supersymmetries of determinantal varieties. J. Commut. Algebra. 6(2), 261–286 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Serganova, V.: Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra \(\mathfrak {gl}(m|n)\). Selecta Math. (N.S.). 2(4), 607–651 (1996)CrossRefzbMATHGoogle Scholar
  12. 12.
    Su, Y., Zhang, R.B.: Character and dimension formulae for general linear superalgebra. Adv. Math. 211(1), 1–33 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Su, Y., Zhang, R.B.: Generalised Jantzen filtration of Lie superalgebras I. J. Eur. Math. Soc. (JEMS) 14(4), 1103–1133 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shigechi, K., Zinn-Justin, P.: Path representation of maximal parabolic Kazhdan-Lusztig polynomials. J. Pure Appl. Algebra 216(11), 2533–2548 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Weyman, J.: Cohomology of vector bundles and syzygies. Cambridge Tracts in Mathematics, vol. 149. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA
  2. 2.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  3. 3.Department of MathematicsUniversity of ConnecticutStorrsUSA

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