Convexity properties of the canonical S-graphs

Abstract

Let n be a positive integer and set N = {1, 2,...,n}. Let {ck}k∈N be non-negative integers. A convex set (ck')k⊂ Qn, given by a family of linear relations in the {ck}k∈N and depending on their natural order, is defined. The extremal points of this convex set is shown to be the S-set constructed in A. Joseph, A preparation theorem for the Kashiwara B(∞) crystal, Selecta Mathematica 23 (2017), 1309–1353. A main application of this result is towards a precise description of the Kashiwara B(∞) crystal given in A. Joseph, Trails S-graphs and identities in Demazure modules, arXiv:1702.00243.