A DG-extension of symmetric functions arising from higher representation theory
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.
Keywords
Cite
@article{arxiv.1704.00713,
title = {A DG-extension of symmetric functions arising from higher representation theory},
author = {Andrea Appel and Ilknur Egilmez and Matthew Hogancamp and Aaron D. Lauda},
journal= {arXiv preprint arXiv:1704.00713},
year = {2018}
}
Comments
v2. Minor corrections, reference updated. Material added: connection between extended symmetric functions and the work of Solomon (Section 4). 30 pages