Integral Basis Theorem of cyclotomic Khovanov-Lauda-Rouquier algebras of Type A

In this paper we prove that the cyclotomic Khovanov-Lauda-Rouquier algebras in type A, ${\mathcal{R}}_n^{\mathrm{\Lambda }}$, are $\mathbb{Z}$-free. We then extend the graded cellular basis of ${\mathcal{R}}_n^{\mathrm{\Lambda }}$ constructed by Hu and Mathas to ${\mathcal{R}}_n$ and use this basis to give a classification of all irreducible ${\mathcal{R}}_n$-modules.