Virtual crystals and fermionic formulas of type \(D_{n+1}^{(2)}\), \(A_{2n}^{(2)}\), and \(C_n^{(1)}\). (English) Zbl 1026.81024
Fermionic formulae (\(q\)-polynomials or infinite \(q\)-series being sums of products of \(q\)-binomial coefficients) originate in the Bethe ansatz study of exactly solvable lattice models [V. G. Drinfel’d, Dokl. Akad. Nauk SSSR 283, 1060–1064 (1985; Zbl 0588.17015)]. This work belongs to a series of articles by the authors in which fermionic formulae and crystal bases associated with quantum affine algebras are studied [M. Okado, A. Schilling and M. Shimozone, Contemp. Math. 291, 29–53 (2001; Zbl 1020.17011); A. N. Kirillov, A. Schilling and M. Shimozono, Sel. Math., New Ser. 8, 67–135 (2002; Zbl 0986.05013)]. Specifically this paper is devoted to the virtual crystals and fermionic formulae for the types \(D_{n+1}^{(2)}\), \(A_{2n}^{(2)}\) and \(C_{n}^{(1)}\). After reviewing the general crystal theory (R-matrices, crystal graphs and duals, configuration sums, etc), the combinatorial structure of \(A_{n}^{(1)}\)-type crystals is analyzed. The Littlewood-Richardson (LR) tableaux are used to parametrize the (irreducible) components of tensor products of these crystals. For the types \(D_{n+1}^{(2)}\), \(A_{2n}^{(2)}\), \(C_{n}^{(1)}\) and \(A_{2n}^{(2)†}\) virtual crystals \(V^{r,s}\) are introduced and characterized for the value \(s=1\). Fermionic formulae for these types are proved using the characterization of rigged configurations in the image of virtual crystals under the bijection with the LR-tableaux. The type \(A_{2n}^{(2)†}\) is analyzed separately, and new fermionic formulae are obtained. The paper solves some conjectures proposed earlier in the literature, and constitutes an interesting advance in the theory of virtual crystals.
Reviewer: Rutwig Campoamor-Stursberg (Mulhouse)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 05A30 | \(q\)-calculus and related topics |
| 05E10 | Combinatorial aspects of representation theory |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
| 82B23 | Exactly solvable models; Bethe ansatz |
Keywords:
virtual crystal; fermionic formulae; affine algebras; Bethe ansatz; exactly solvable lattice modelsReferences:
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