Timeline for Is the root cone is contained in the weight cone?

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Jul 24, 2018 at 13:29	comment	added	Sam Hopkins		(In fact an even stronger statement is true: the Cartan matrix is positive definite.)
Jul 24, 2018 at 13:14	comment	added	Sam Hopkins		If you want more details: the Cartan matrix, which expresses the simple roots in the basis of fundamental weights, is an M-matrix- en.m.wikipedia.org/wiki/M-matrix. Thus the inverse of the Cartan matrix, which expresses the fundamental weights in the basis of simple roots, has all nonnegative entries, implying exactly the inclusion of cones you want. To see that the Cartan matrix is an M-matrix, you can use the fact that the Weyl vector $\rho$ is equal to the half-sum of the positive roots as well as the sum of the fundamental weights.
Jul 23, 2018 at 21:07	comment	added	Gro-Tsen		I'm not sure whether your initial setup is relevant or whether you could ask the question for an abstract (reduced crystallographic) root system, but assuming the latter, the closure of $A$ is simply generated by the fundamental weights (it is the Weyl chamber) and the closure of $B$ is generated by the positive roots, and the inclusion $A\subseteq B$ is correct and states that fundamental weights are combinations with nonnegative coefficients of the simple roots.
Jul 23, 2018 at 17:19	history	asked	D_S	CC BY-SA 4.0