Degree cones and monomial bases of Lie algebras and quantum groups

On toric degenerations of flag varieties

“…In general, varying the function w will change the monoid Γ(S, >), yet in [5] we give a family of such functions such that the global monoid stays constant, i.e., we have always Γ(S, >) = Γ P .…”

“…Let ">" be the lexicographic order. One obtains ( [5]) that the global essential monoid Γ 1 := Γ(S, >) is finitely generated and saturated.…”

“…Let Γ ′ := Γ(S ′ , > ′ ) be the associated global essential monoid. The monoid Γ ′ is not saturated [5].…”

“…Is it possible to generalize the filtration aspect to the setting of quantum groups [25]? Some recent work in this direction can be found in [5]. As both have a Gröbner background, it might be interesting to study the relation between the quantum degree cone [5] and the tropical Graßmannian [83], or, in general, tropical flag varieties.…”