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

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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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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Correspondence to Claudiu Raicu.

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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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Keywords

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

Mathematics Subject Classification (2010)

  • Primary 13D02
  • 14M12
  • 17B10