Auslander-Reiten Numbers Games

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.