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Representation type of cyclotomic quiver Hecke algebras of type \(A_\ell^{(1)}\). (English) Zbl 07765298

Summary: We first investigate a connected quiver consisting of all dominant maximal weights for an integrable highest weight module in affine type A. This quiver provides an efficient method to obtain all dominant maximal weights. Then, we completely determine the representation type of cyclotomic Khovanov-Lauda-Rouquier algebras of arbitrary level in affine type A, by using the quiver we construct. This result gives a complete classification for the representation type of blocks of cyclotomic Hecke algebras since cyclotomic KLR algebras of type \(A_\ell^{(1)}\) form a one-parameter family and cyclotomic Hecke algebras occur at a special parameter, i.e., \(t = -2\) if \(\ell = 1\) and \(t = (-1)^{\ell + 1}\) if \(\ell \geq 2\).

MSC:

20C08 Hecke algebras and their representations
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
17B65 Infinite-dimensional Lie (super)algebras
16G20 Representations of quivers and partially ordered sets
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