We construct irreducible graded representations of simply laced Khovanov–Lauda algebras which are concentrated in one degree. The underlying combinatorics of skew shapes and standard tableaux corresponding to arbitrary simply laced types has been developed previously by Peterson, Proctor and Stembridge. In particular, the Peterson–Proctor hook formula gives the dimensions of the homogeneous irreducible modules corresponding to straight shapes.
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Arun Ram, Alexander S. Kleshchev, Homogeneous representations of Khovanov–Lauda Algebras. J. Eur. Math. Soc. 12 (2010), no. 5, pp. 1293–1306DOI 10.4171/JEMS/230