#### Vol. 296, No. 1, 2018

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On the structure of cyclotomic nilHecke algebras

### Jun Hu and Xinfeng Liang

Vol. 296 (2018), No. 1, 105–139
##### Abstract

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

##### 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
##### Authors
 Jun Hu School of Mathematics and Statistics Beijing Institute of Technology Beijing China Xinfeng Liang School of Mathematics and Statistics Beijing Institute of Technology Beijing China