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Paths and tableaux descriptions of Jacobi-Trudi determinant associated with quantum affine algebra of type \(C_n\). (English) Zbl 1142.17009
Let \(\chi_{\lambda/\mu,a}\) be the Jacobi–Trudi–type determinant associated with quantum affine algebra of type \(C_n\). Tableaux description means an expression of \(\chi_{\lambda/\mu,a}\) by a positive sum over a certain set of tableaux on \(\lambda/\mu\), where \(\lambda/\mu\) is a skew diagram and \(a\) is a complex parameter. For types \(C_n\) the authors first obtained in [J. Phys. A, Math. Phys. 39, 2083–2115 (2006; Zbl 1085.17011)] that a tableaux description of \(\chi_{\lambda/\mu,a}\) for the skew diagram \(\lambda/\mu\) of at most two columns or of at most three rows. The same problem for \(D_n\) was also considered in the other paper.
The paper under review uses the Gessel-Viennot path method with an additional involution and a deformation of paths to obtain a tableaux description of \(\chi_{\lambda/\mu,a}\) by a positive sum over a set of tuples of paths for a general skew diagram \(\lambda/\mu\), which is naturally translated into the one over a set of tableaux on a skew diagram. The method may be applied to study the twisted quantum affine algebras of classical type and the crystal bases of the representations of quantum affine algebra.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
05E15 Combinatorial aspects of groups and algebras (MSC2010)
Zbl 1085.17011
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