On the structure of cyclotomic nilHecke algebras

TL;DR

This paper thoroughly analyzes the structure of cyclotomic nilHecke algebras, constructing bases, proving isomorphisms, and identifying symmetrizing forms, thereby advancing understanding of their algebraic properties.

Contribution

It constructs a monomial basis confirming Mathas's conjecture, shows the algebra's center is commutative and explicitly describes it, and introduces a new symmetrizing form.

Findings

Constructed a monomial basis for $ ext{HH}_{ ext{ell},n}^{(0)}$ confirming Mathas's conjecture.

Proved the algebra's basic algebra is commutative and isomorphic to its center.

Developed a new homogeneous symmetrizing form matching existing forms.

Abstract

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 $\HH_{\ell,n}^{(0)}$.