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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.

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