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

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dc.creatorOkado, Masato
dc.creatorSchilling, Anne
dc.creatorShimozono, Mark
dc.date2001-05-02
dc.date.accessioned2026-07-07T06:22:43Z
dc.date.available2026-07-07T06:22:43Z
dc.descriptionWe 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.
dc.description63 pages, style file axodraw.sty required
dc.identifierhttps://arxiv.org/abs/math/0105017
dc.identifierhttp://arxiv.org/abs/math/0105017
dc.identifierRepresent. Theory 7 (2003) 101-163
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/95991
dc.subjectQuantum Algebra
dc.subject81R50; 81R10; 17B37; 05A30; 82B23
dc.titleVirtual crystals and fermionic formulas of type $D_{n+1}^{(2)}$, $A_{2n}^{(2)}$, and $C_n^{(1)}$
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