Weight polytopes and saturation of Demazure characters

Multiplicity in root components via geometric Satake

“…Finally, with this in hand, representation-theoretic techniques could be applied to extend the saturation result of [16] to Demazure characters when G is simple of classical type. By additional computational methods, the same result was also obtained in types F 4 and G 2 , and is conjectured to hold for E 6 , E 7 and E 8 (see [5,Conjecture 11.4]).…”

“…While not clear from the definition, the Demazure weight polytope P w λ has a simple description via its vertices. We take the following result from [5,Theorem 4.7], as the proof given there goes through to the affine Lie algebra setting; in the finite case, this result is implicit in the work of Dabrowski [9, page 119] via a similar proof. Proposition 2.4.…”

“…In previous work [5], the authors extended the consideration of Demazure weight polytopes P w λ beyond type A n . First, using geometric invariant theory (GIT) techniques, inequalities were derived which cut out P w λ in all types.…”

Degrees of the stretched Kostka quasi‐polynomials

“…Recall that we have defined the BZ-polytope BZ 𝜆,𝜇 to be wt −1 (𝜇). The following theorem is well known, see, for example, [4]. As a quick summary, we have a point 𝑇 in the interior of BZ 𝜆 with weight 𝜇.…”

“…F I G U R E 4 An interior point in BZ 𝜆 of type 𝐷, where 𝜆 = (2, 1, 0, 0, 0) with 𝑆 ′ = {3, 4, 5}. The fixed coordinates are underlined.…”