Combinatorial \(R\) matrices for a family of crystals: \(B_n^{(1)}\), \(D_n^{(1)}\), \(A_{2n}^{(2)}\) and \(D_{n+1}^{(2)}\) cases (Q1348680)

\textit{M. Kashiwara} [Duke Math. J. 63, 465-516 (1991; Zbl 0739.17005)] has defined a so-called crystal basis to the quantized enveloping algebra associated to a symmetrizable Kac-Moody Lie algebra. This basis describes the combinatorial behaviour of the algebra when the deformation parameter is specialized to zero. (In the physical model, this parameter corresponds to temperature, and the simplifying behaviour of the algebra when the parameter is specialized to zero corresponds to the physical behaviour close to absolute zero). In this paper, the authors consider the quantized enveloping algebra associated to an affine Lie algebra. This algebra possesses a quantum \(R\)-matrix (which, amongst other things, describes relationships between representations) which, on specialisation of the parameter to zero as above, gives rise to a combinatorial \(R\)-matrix, defined on the tensor product of two affine crystals, \(B\) and \(B'\). It consists of an isomorphism between \(B\otimes B'\) and \(B'\otimes B\) and an energy function \(H:B\otimes B'\rightarrow\mathbb{Z}\). In this paper, the authors give an explicit description of these items in the case where \(B\) and \(B'\) are not isomorphic (the so-called inhomogeneous case), for the cases \(B_n^{(1)}\), \(D_n^{(1)}\), \(A_{2n}^{(2)}\) and \(D_{n+1}^{(2)}\). This continues their previous work [in Physical combinatorics, M. Kashiwara and T. Miwa (eds.), Birkhäuser, Boston, Prog. Math. 191, 105-139 (2000; Zbl 0976.17009)], in which the cases \(C_n^{(1)}\) and \(A_{2n-1}^{(2)}\) were considered. The description is given in terms of a generalised Robinson-Schensted insertion algorithm, employing results of \textit{M. Kashiwara} and \textit{T. Nakashima} [J. Algebra 165, 295-345 (1994; Zbl 0808.17005)] and \textit{T. H. Baker} [in Physical combinatorics, M. Kashiwara and T. Miwa (eds.), Birkhäuser, Boston, Prog. Math. 191, 1-48 (2000; Zbl 0974.05080)] and [in Proc. Int. Workshop on Special Functions, Hong Kong, 1999, 16-30 (2000; Zbl 1189.17013)].