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

TL;DR

This paper introduces virtual crystals for certain affine types, provides evidence for their connection to finite-dimensional modules, and proves related fermionic formulas, advancing understanding of crystal bases and combinatorial formulas in quantum algebra.

Contribution

It defines virtual crystals for affine types $D_{n+1}^{(2)}$, $A_{2n}^{(2)}$, and $C_n^{(1)}$, and proves fermionic formulas conjectured for their tensor products.

Findings

Virtual crystals extend embeddings of types $B_n$ and $C_n$ into type $A_{2n-1}$ crystals.

Proved fermionic formulas for specific cases using duality and rigged configurations.

Conjectured a new fermionic formula for type $A_{2n}^{(2)}$ with a different Dynkin diagram labeling.

Abstract

We introduce ``virtual'' crystals of the affine types $g=D_{n+1}^{(2)}$, $A_{2n}^{(2)}$ and $C_n^{(1)}$ by naturally extending embeddings of crystals of types $B_n$ and $C_n$ into crystals of type $A_{2n-1}$. Conjecturally, these virtual crystals are the crystal bases of finite dimensional $U_q'(g)$-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 $U_q'(g)$-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 $A_{2n}^{(2)}$ we also conjecture a new fermionic formula coming from a different labeling of the Dynkin diagram.