Virtual crystals and fermionic formulas of type 𝐷_{𝑛+1}⁽²⁾, 𝐴_{2𝑛}⁽²⁾, and 𝐶_{𝑛}⁽¹⁾

Abstract

We introduce “virtual” crystals of the affine types g = D n + 1 ( 2 ) \mathfrak {g}=D_{n+1}^{(2)} , A 2 n ( 2 ) A_{2n}^{(2)} and C n ( 1 ) C_n^{(1)} by naturally extending embeddings of crystals of types B n B_n and C n C_n into crystals of type A 2 n − 1 A_{2n-1} . Conjecturally, these virtual crystals are the crystal bases of finite dimensional U q ′ ( g ) U_q’(\mathfrak {g}) -modules associated with multiples of fundamental weights. We provide evidence and in some cases proofs of this conjecture. Recently, fermionic formulas for the one-dimensional configuration sums associated with tensor products of the finite dimensional U q ′ ( g ) U_q’(\mathfrak {g}) -modules were conjectured by Hatayama et al. We provide proofs of these conjectures in specific cases by exploiting duality properties of crystals and rigged configuration techniques. For type A 2 n ( 2 ) A_{2n}^{(2)} we also conjecture a new fermionic formula coming from a different labeling of the Dynkin diagram.