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.
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References
Akin, K., Buchsbaum, D.A., Weyman, J.: Resolutions of determinantal ideals: the submaximal minors. Adv. in Math. 39(1), 1–30 (1981)
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)
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)
Grayson, D.R., Stillman, M.E.: Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
Lascoux, A.: Syzygies des variétés déterminantales. Adv. Math. 30(3), 202–237 (1978)
Pragacz, P., Weyman, J.: Complexes associated with trace and evaluation. Another approach to Lascoux’s resolution. Adv. Math. 57(2), 163–207 (1985)
Raicu, C.: Regularity and cohomology of determinantal thickenings. Proc. Lond. Math. Soc. 116(2), 248–280 (2018)
Raicu, C., Weyman, J.: Local cohomology with support in generic determinantal ideals. Algebra & Number Theory 8(5), 1231–1257 (2014)
Raicu, C., Weyman, J.: The syzygies of some thickenings of determinantal varieties. Proc. Am. Math. Soc. 145(1), 49–59 (2017)
Sam, S.V.: Derived supersymmetries of determinantal varieties. J. Commut. Algebra. 6(2), 261–286 (2014)
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)
Su, Y., Zhang, R.B.: Character and dimension formulae for general linear superalgebra. Adv. Math. 211(1), 1–33 (2007)
Su, Y., Zhang, R.B.: Generalised Jantzen filtration of Lie superalgebras I. J. Eur. Math. Soc. (JEMS) 14(4), 1103–1133 (2012)
Shigechi, K., Zinn-Justin, P.: Path representation of maximal parabolic Kazhdan-Lusztig polynomials. J. Pure Appl. Algebra 216(11), 2533–2548 (2012)
Weyman, J.: Cohomology of vector bundles and syzygies. Cambridge Tracts in Mathematics, vol. 149. Cambridge University Press, Cambridge (2003)
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.
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Raicu, C., Weyman, J. Syzygies of Determinantal Thickenings and Representations of the General Linear Lie Superalgebra. Acta Math Vietnam 44, 269–284 (2019). https://doi.org/10.1007/s40306-018-0282-z
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DOI: https://doi.org/10.1007/s40306-018-0282-z
Keywords
- Determinantal thickenings
- Syzygies
- BGG correspondence
- General linear Lie superalgebra
- Kac modules
- Dyck paths