On the structure of cyclotomic nilHecke algebras

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