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We give an explicit construction of irreducible modules over Khovanov–Lauda–Rouquier algebras  R  and their cyclotomic quotients  R   λ   for finite classical types using a crystal basis theoretic approach. More precisely, for each element  v  of the crystal(resp.  B ( λ )), we first construct certain modules  Δ ( a ; k ) labeled by the adapted string  a  of  v . We then prove that the head of the induced moduleis irreducible and that every irreducible  R -module (resp.  R   λ  -module) can be realized as the irreducible head of one of the induced modules. Moreover, we show that our construction is compatible with the crystal structure on(resp.  B ( λ )).

Construction of Irreducible Representations over Khovanov–Lauda–Rouquier Algebras of Finite Classical Type