Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras

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 $e(\tilde{\nu })R^{\Lambda }(\beta )e(\mu )$ and $e(\tilde{\nu })R^{\Lambda }(\beta )e(\mu )$ of $R^{\Lambda }(\beta )$, where $\mu \in I^{\beta }$ is {\it arbitrary} and $\tilde{\nu }\in I^{\beta }$ 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 $m<n$.

arXiv commentTo appear in Communications in Contemporary Mathematics

References in corpus (12)

• A diagrammatic approach to categorification of quantum groups I
• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• A diagrammatic approach to categorification of quantum groups II
• Graded decomposition numbers for cyclotomic Hecke algebras
• Four dimensional topological quantum field theory, Hopf categories, and the canonical bases
• Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras
• Clock and Category; IS QUANTUM GRAVITY ALGEBRAIC
• Homological properties of finite type Khovanov-Lauda-Rouquier algebras
• Specht modules for quiver Hecke algebras of type $C$
• Cellularity for weighted KLRW algebras of types $B$, $A^{(2)}$, $D^{(2)}$
• On the structure of cyclotomic nilHecke algebras
• Nilpotency in type A cyclotomic quotients

Cited by in corpus (4)

• Cellularity and subdivision of KLR and weighted KLRW algebras
• Content systems and deformations of cyclotomic KLR algebras of type $A$ and $C$
• Cellularity of KLR and weighted KLRW algebras via crystals
• A formula to evaluate type A webs and link polynomials