Auslander-Reiten Numbers Games

Abstract

Let \(\mathfrak {g}\) be a simple complex Lie algebra of types A n , D n , E n , and Q a quiver obtained by orienting its Dynkin diagram. Let λ be a dominant weight, and E(λ) the corresponding simple highest weight representation. We show that the weight multiplicities of E(λ) may be recovered by playing a numbers game Λ Q (λ), generalizing the well known Mozes game, constructing the orbit of λ under the action of the Weyl group W. The game board is provided by the Auslander-Reiten quiver Γ Q of Q. The game moves are obtained by constructing Nakajima’s monomial crystal M(λ) directly out of Γ Q . As an application, we consider Kashiwara’s parameterizations of the canonical basis. Let w 0 be a reduced expression of the longest element w 0 of W, adapted to a quiver Q of type A n . We show that a set of inequalities defining the string (Kashiwara) cone with respect to w 0, may be obtained by playing subgames of the numbers games Λ Q (ω i ) associated to fundamental representations.