Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras

TL;DR

This paper derives a closed formula for the graded dimension of cyclotomic quiver Hecke algebras, providing conditions for nonzero idempotents, level reduction decompositions, explicit monomial bases, and examples illustrating their structural properties.

Contribution

It introduces a general graded dimension formula, necessary and sufficient conditions for idempotent non-vanishing, and explicit monomial bases for cyclotomic quiver Hecke algebras, advancing understanding of their structure.

Findings

Derived a closed formula for graded dimensions.

Established conditions for nonzero idempotents.

Constructed explicit monomial bases.

Abstract

In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra $R^\Lambda(\beta)$ associated to an {\it arbitrary} symmetrizable Cartan matrix $A=(a_{ij})_{i,j}\in I$, where $\Lambda\in P^+$ and $\beta\in Q_n^+$. As applications, we obtain some {\it necessary and sufficient conditions} for the KLR idempotent $e(\nu)$ (for any $\nu\in I^\beta$) to be nonzero in the cyclotomic quiver Hecke algebra $R^\Lambda(\beta)$. We prove several level reduction results which decomposes $\dim R^\Lambda(\beta)$ into a sum of some products of $\dim R^{\Lambda^i}(\beta_i)$ with $\Lambda=\sum_i\Lambda^i$ and $\beta=\sum_{i}\beta_i$, where $\Lambda^i\in P^+, \beta^i\in Q^+$ for each $i$. We construct some explicit monomial bases for the subspaces $e(\widetilde{\nu})R^\Lambda(\beta)e(\mu)$ and $e(\widetilde{\nu})R^\Lambda(\beta)e(\mu)$ of $R^\Lambda(\beta)$, where $\mu\in I^\beta$ is {\it arbitrary} and $\widetilde{\nu}\in I^\beta$ is a certain specific $n$-tuple (see Section 4).Finally, we use our graded dimension formulae to provide some examples which show that $R^\Lambda(n)$ is in general not graded free over its natural embedded subalgebra $R^\Lambda(m)$ with $m<n$.