arXiv subjects

Wakako Nakai

Publications and source records attributed to Wakako Nakai.

4 recordsLinked to original sources

arXiv2008

On Frenkel-Mukhin algorithm for q-character of quantum affine algebras

The q-character is a strong tool to study finite-dimensional representations of quantum affine algebras. However, the explicit formula of the q-character of a given representation has not been known so far. Frenkel and Mukhin proposed the iterative algorithm which generates the q-character of a given irreducible representation starting from its highest weight monomial. The algorithm is known to work for various classes of representations. In this note, however, we give an example in which the algorithm fails to generate the q-character.

math.QA↗

arXiv2007

Paths and tableaux descriptions of Jacobi-Trudi determinant associated with quantum affine algebra of type C_n

We study the Jacobi-Trudi-type determinant which is conjectured to be the q-character of a certain, in many cases irreducible, finite-dimensional representation of the quantum affine algebra of type C_n. Like the D_n case studied by the authors recently, applying the Gessel-Viennot path method with an additional involution and a deformation of paths, we obtain a positive sum expression over a set of tuples of paths, which is naturally translated into the one over a set of tableaux on a skew diagram.

math.QA↗

arXiv2007

Paths and tableaux descriptions of Jacobi-Trudi determinant associated with quantum affine algebra of type D_n

We study the Jacobi-Trudi-type determinant which is conjectured to be the q-character of a certain, in many cases irreducible, finite-dimensional representation of the quantum affine algebra of type D_n. Unlike the A_n and B_n cases, a simple application of the Gessel-Viennot path method does not yield a positive sum expression of the determinant over a set of tuples of paths. However, applying an additional involution and a deformation of paths, we obtain a positive sum expression over a set of tuples of paths, which is naturally translated into the one over a set of tableaux on a skew diagram.

math.QA↗

arXiv2006

Paths, tableaux, and q-characters of quantum affine algebras: the C_n case

For the quantum affine algebra $U_q(\hat{\mathfrak{g}})$ with $\mathfrak{g}$ of classical type, let $χ_{λ/μ,a}$ be the Jacobi-Trudi type determinant for the generating series of the (supposed) $q$-characters of the fundamental representations. We conjecture that $χ_{λ/μ,a}$ is the $q$-character of a certain finite dimensional representation of $U_q(\hat{\mathfrak{g}})$. We study the tableaux description of $χ_{λ/μ,a}$ using the path method due to Gessel-Viennot. It immediately reproduces the tableau rule by Bazhanov-Reshetikhin for $A_n$ and by Kuniba-Ohta-Suzuki for $B_n$. For $C_n$, we derive the explicit tableau rule for skew diagrams $λ/μ$ of three rows and of two columns, and give the implicit tableau rule in terms of paths for general $λ/μ$.

math.QA↗