scholarly journals Keys and Demazure crystals for Kac–Moody algebras

2020 ◽  
Vol 4 (4) ◽  
pp. 325-358
Author(s):  
Nicolas Jacon ◽  
Cédric Lecouvey
Keyword(s):  
Demazure Crystals

scholarly journals Demazure Crystals, Kirillov-Reshetikhin Crystals, and the Energy Function

10.37236/2184 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Anne Schilling ◽  
Peter Tingley
Keyword(s):  
Lie Algebras ◽  
Energy Function ◽  
Tensor Products ◽  
Macdonald Polynomials ◽  
Close Connection ◽  
Whittaker Functions ◽  
Demazure Character ◽  
Demazure Crystals

It has previously been shown that, at least for non-exceptional Kac-Moody Lie algebras, there is a close connection between Demazure crystals and tensor products of Kirillov-Reshetikhin crystals. In particular, certain Demazure crystals are isomorphic as classical crystals to tensor products of  Kirillov-Reshetikhin crystals via a canonically chosen isomorphism. Here we show that this isomorphism intertwines the natural affine grading on Demazure crystals with a combinatorially defined energy function. As a consequence, we obtain a formula of the Demazure character in terms of the energy function, which has applications to Macdonald polynomials and $q$-deformed Whittaker functions.

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