On the structure of cyclotomic nilHecke algebras

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