On dominance and minuscule Weyl group elements

Abstract

Abstract. Fix a Dynkin diagram and let λ be a coweight. When does there exist an element w of the corresponding Weyl group such that w is λ-minuscule and w(λ) is dominant? We answer this question for general Coxeter groups. We express and prove these results using a variant of Mozes's game of numbers.

Chip firing on Dynkin diagrams and McKay quivers

“…In particular, they thank Sam Payne for a helpful discussion there leading to the formulation of Proposition 5.20, and thank Shaked Koplewitz for allowing them to sketch his proof of Proposition 6. 19. The third author also thanks T. Schedler for helpful comments and gratefully acknowledges partial support by NSF grant DMS-1001933.…”

“…root system Φ ∨ . Looping sequences were characterized by Eriksson [15, §3.2], and studied further by Gashi and Schedler [19], and by Gashi, Schedler and Speyer [20].…”

Normality and Quadraticity for Special Ample Line Bundles on Toric Varieties Arising From Root Systems

“…2.7]) stating that in the usual partial order of σ-dominant weights, a weight ν covers another one ν ′ if and only if the difference ν − ν ′ is a root that is positive with respect to σ. In §3 we recall the numbers game with a cutoff (from [Gas08a]; see also [GS09]), which gives a useful language to prove Theorem 1.12. The proof of the theorem is then given in § §4 and 5.…”

“…In order to prove Theorem 1.4, we use the language of the numbers game with a cutoff, from [Gas08b] (see also [GS09]). In this section we recall what we will need.…”

“…In [GS09], a simple criterion was given to determine when the numbers game with a cutoff is winning. We will restrict to the Dynkin case, with C the standard Cartan matrix.…”