Colored lattice models for Demazure characters in types ABC (Recent developments in Combinatorial Representation Theory)

Abstract

Characters of irreducible representations of 𝖌𝖑ₙ, the Schur functions are wellknown to be described as partition functions of solvable lattice models. One method to prove this uses the Gelfand-Tsetlin interpretation of semistandard tableaux, which are naturally in bijection with states of the lattice model. This approach has been extended to type C in 2012 by Ivanov using Proctor patterns. A recent trend in solvable lattice models has been to use "colored" lattice models that are based on the R-matrices of finite dimensional quantum affine general linear group 𝘜𝘲(𝖌𝖑ₙ) representations instead of 𝘜𝘲(𝖌𝖑₂). In this note, we obtain Demazure characters, which are representations of the corresponding Borel subalgebra and can be thought of as "partial" versions of the irreducible simple Lie algebra representations, and their Demazure atoms analogs by using a colored lattice model built from that of Bump-Brubaker-Buciumas-Gustafsson for type A Demazure atoms. A key component of these lattice models is a natural relationship with the wiring diagram of the defining element of the corresponding Weyl group. We then will discuss the colored version of Ivanov's lattice model to obtain Demazure characters in type B and C and the related tableaux.