A DG-extension of symmetric functions arising from higher representation theory

  • Andrea Appel

    University of Edinburgh, UK
  • Ilknur Egilmez

    University of Southern California, Los Angeles, USA
  • Matthew Hogancamp

    University of Southern California, Los Angeles, USA
  • Aaron D. Lauda

    University of Southern California, Los Angeles, USA

Abstract

We investigate analogs of symmetric functions arising from an extension of the nilHecke algebra defined by Naisse and Vaz. These extended symmetric functions form a subalgebra of the polynomial ring tensored with an exterior algebra.We define families of bases for this algebra and show that it admits a family of differentials making it a sub-DG-algebra of the extended nilHecke algebra. The ring of extended symmetric functions equipped with this differential is quasi-isomorphic to the cohomology of a Grassmannian. We also introduce new deformed differentials on the extended nilHecke algebra that when restricted makes extended symmetric functions quasi-isomorphic to -equivariant cohomology of Grassmannians.

Cite this article

Andrea Appel, Ilknur Egilmez, Matthew Hogancamp, Aaron D. Lauda, A DG-extension of symmetric functions arising from higher representation theory. J. Comb. Algebra 2 (2018), no. 2, pp. 169–214

DOI 10.4171/JCA/2-2-3