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 $\HH_{ \ell,n}^{ (0)}$.
Categories: math.RT