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We study graded dimension formulas for finite quiver Hecke algebras$R^{\Lambda_0}(\beta)$ of type $A^{(2)}_{2\ell}$ and $D^{(2)}_{\ell+1}$ usingcombinatorics of Young walls. We introduce the notion of standard tableaux forproper Young walls and show that the standard tableaux form a graded poset withlattice structure. We next investigate Laurent polynomials associated withproper Young walls and their standard tableaux arising from the Fock spacerepresentations consisting of proper Young walls. Then we prove the gradeddimension formulas described in terms of the Laurent polynomials. Whenevaluating at $q=1$, the graded dimension formulas recover the dimensionformulas for $R^{\Lambda_0}(\beta)$ described in terms of standard tableaux ofstrict partitions.

Young walls and graded dimension formulas for finite quiver Hecke algebras of type $A^{(2)}_{2\ell}$ and $D^{(2)}_{\ell+1}$