This paper gives explicit web diagrams and dual webs for quadratic and cubic cluster variables in C[Gr(4,8)].
Graphical Calculus on Representations of Quantum Lie Algebras
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abstract
We develop graphical calculation methods. Jones-Wenzl projectors for U_q(sl(2,C)) are very powerful tools to find not only invariants of links but also invariants of 3-manifolds. We find single clasp expansions of generalized Jones-Wenzl projectors for simple Lie algebras of rank 2. Trihedron coefficients of the representation theory for U_q(sl(2,C)) has significant meaning and it is called 3j symbols. Using single clasp expansions for U_q(sl(3,C)), we find some trihedron coefficients of the representation theory of U_q(sl(3,C)). We study representation theory for U_q(sl(4,C)). We conjecture a complete set of relations for U_q(sl(4,C)).
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Web Diagrams of Cluster Variables for Grassmannian Gr(4,8)
This paper gives explicit web diagrams and dual webs for quadratic and cubic cluster variables in C[Gr(4,8)].