A definition of the crystal commutor using Kashiwara’s involution

Abstract

Henriques and Kamnitzer defined and studied a commutor for the category of crystals of a finite dimensional complex reductive Lie algebra. We show that the action of this commutor on highest weight elements can be expressed very simply using Kashiwara's involution on the Verma crystal.

Coboundary Categories and Local Rules

“…In particular they construct an analogue of the Lusztig involution in §4. 10. This then gives a coboundary structure on the category of continuous crystals by the construction in [7].…”

“…There is a second construction of the coboundary structure in [10] using the Kashiwara involution. This construction is given on highest weight elements by the formula…”

On the combinatorics of crystal graphs, II. The crystal commutor

“…More recently, Kamnitzer and Tingley [9] proved that the action of the above commutor on the highest weight elements (which determines it) is given by Kashiwara's involution on the Verma crystal [11]. Henriques and Kamnitzer proved that the category g-Crystals with this commutor is a coboundary category (cf.…”

“…(3) We can combine our main result with both realizations of the commutor of Henriques and Kamnitzer (in [6] and [9]) in order to obtain explicit constructions related to: (i) Lusztig's involution on highest weight elements in a tensor product of two irreducible crystals; (ii) Kashiwara's involution. Indeed, by Schützenberger's evacuation, we have η B π (b π ) = (ε 3 , ε 2 , ε 3 ), and η B π (p) = (ε 2 , ε 1 ).…”

“…[4]). More recently, Kamnitzer and Tingley [9] proved that the action of σ A,B on the highest weight elements (which determines it) is given by Kashiwara's involution on the Verma crystal; as remarked in [9], Kashiwara's involution can be realized in terms of Mirković-Vilonen polytopes [8]. Both of the above constructions of σ A,B depend on some maps of crystals whose explicit construction is nontrivial.…”