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Algebras and Representation Theory

, Volume 21, Issue 2, pp 277–307 | Cite as

Auslander-Reiten Numbers Games

  • Shmuel Zelikson
Article
  • 48 Downloads

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.

Keywords

Simply laced Lie algebras Numbers games Crystals Auslander-Reiten quivers 

Mathematics Subject Classification (2010)

05E10 16G70 17B10 17B37 

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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Laboratoire de Mathmatiques Nicolas Oresme (LMNO), Unit Mixte de Recherche (UMR) CNRSCaen UniversityCaenFrance
  2. 2.Mathematics DepartmentCaen UniversityCaenFrance

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