Combinatorics of canonical bases revisited: String data in type $A$

Volker Genz, Gleb Koshevoy, Bea Schumann

英文摘要：

We give a formula for the crystal structure on the integer points of the string polytopes and the $*$-crystal structure on the integer points of the string cones of type $A$ for arbitrary reduced words. As a byproduct we obtain defining inequalities for Nakashima-Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara $*$-involution on string data for a special choice of reduced word.

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