Minuscule elements of Weyl groups

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On a conjecture of Kottwitz and Rapoport (2008, background): “This statement is proved in [22], Proposition 2.”
Frobenius splitting, point-counting, and degeneration (2009, methodology): “ω 2 =c2c4 −c3, andmω 3 =c3. Notethat theyneednot behomogeneous; homogeneity should only be expected when v is 321-avoiding, as that is the condition (more generally known as “λ-cominuscule”; see e.g. [Ste01]) for the T-action on Xv ◦ to contain the scaling action.7 c c c c 1 2 4 3 1 2 3 4 1 2 3 4 FI”
Homogeneous representations of Khovanov-Lauda algebras (2008, methodology): “[ S2 , Proposition 2.1], and can be characterized in terms of their reduced expressions as follows.”