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We first investigate a connected quiver consisting of all dominant maximalweights for an integrable highest weight module in affine type A. This quiverprovides an efficient method to obtain all dominant maximal weights. Then, wecompletely determine the representation type of cyclotomicKhovanov-Lauda-Rouquier algebras of arbitrary level in affine type A, by usingthe quiver we construct. This result gives a complete classification for therepresentation type of blocks of cyclotomic Hecke algebras since cyclotomic KLRalgebras of type $A^{(1)}_\ell$ form a one-parameter family and cyclotomicHecke algebras occur at a special parameter, i.e., $t=-2$ if $\ell=1$ and$t=(-1)^{\ell+1}$ if $\ell\geq2$.

Representation type of cyclotomic quiver Hecke algebras of type $A_\ell^{(1)}$