On the structure of cyclotomic nilHecke algebras

Jun Hu and Xinfeng Liang

Vol. 296 (2018), No. 1, 105–139

DOI: 10.2140/pjm.2018.296.105

Abstract

In this paper we study the structure of the cyclotomic nilHecke algebras ${\mathcal{\mathscr{H}}}_{\ell ,n}^{\left(0\right)}$, where $\ell ,n\in \mathbb{N}$. We construct a monomial basis for ${\mathcal{\mathscr{H}}}_{\ell ,n}^{\left(0\right)}$ which verifies a conjecture of Mathas. We show that the graded basic algebra of ${\mathcal{\mathscr{H}}}_{\ell ,n}^{\left(0\right)}$ is commutative and hence isomorphic to the center $Z$ of ${\mathcal{\mathscr{H}}}_{\ell ,n}^{\left(0\right)}$. We further prove that ${\mathcal{\mathscr{H}}}_{\ell ,n}^{\left(0\right)}$ is isomorphic to the full matrix algebra over $Z$ and construct an explicit basis for the center $Z$. We also construct a complete set of pairwise orthogonal primitive idempotents of ${\mathcal{\mathscr{H}}}_{\ell ,n}^{\left(0\right)}$. Finally, we present a new homogeneous symmetrizing form Tr on ${\mathcal{\mathscr{H}}}_{\ell ,n}^{\left(0\right)}$ by explicitly specifying its values on a given homogeneous basis of ${\mathcal{\mathscr{H}}}_{\ell ,n}^{\left(0\right)}$ and show that it coincides with Shan–Varagnolo–Vasserot’s symmetrizing form Tr${}^{SVV}$ on ${\mathcal{\mathscr{H}}}_{\ell ,n}^{\left(0\right)}$.

Keywords

cyclotomic nilHecke algebras, graded cellular bases, trace forms

Mathematical Subject Classification 2010

Primary: 16G99, 20C08

Milestones

Received: 8 September 2017

Revised: 12 December 2017

Accepted: 20 December 2017

Published: 1 May 2018