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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)
Zbl 1085.17011
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