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
${\mathcal{\mathscr{H}}}_{\ell ,n}^{\left(0\right)}$.
