Minkowski difference weight formulas

Fix any complex Kac-Moody Lie algebra \(\mathfrak{g}\), and Cartan subalgebra \(\mathfrak{h}\subset \mathfrak{g}\). We study arbitrary highest weight \(\mathfrak{g}\)-modules \(V\) (with any highest weight \(\lambda\in \mathfrak{h}^*\), and let \(L(\lambda)\) be the corresponding simple highest weight \(\mathfrak{g}\)-module), and write their weight-sets \(\mathrm{wt} V\). This is based on and generalizes the Minkowski decompositions for all \(\mathrm{wt} L(\lambda)\) and hulls \(\mathrm{conv}_{\mathbb{R}}(\mathrm{wt} V)\), of Khare [J. Algebra. 2016 & Trans. Amer. Math. Soc. 2017] and Dhillon-Khare [Adv. Math. 2017 & J. Algebra. 2022]. Those works need a freeness property of the Dynkin graph nodes of integrability \(J_{\lambda}\) of \(L(\lambda)\): \(\mathrm{wt} L(\lambda)\ -\) any sum of simple roots over \(J_{\lambda}^c\) are all weights of \(L(\lambda)\). We generalize it for all \(V\), by introducing nodes \(J_V\) that record all the lost 1-dim. weights in \(V\). We show three applications (seemingly novel) for all \(\big(\mathfrak{g}, \lambda, V\big)\) of our \(J_V^c\)-freeness: 1) Minkowski decompositions of all \(\mathrm{wt} V\), subsuming those above for simples. 1\('\)) Characterization of these formulas. 1\(''\)) For these, we solve the inverse problem of determining all \(V\) with fixing \(\mathrm{wt} V \ =\) weight-set of a Verma, parabolic Verma and \(L(\lambda)\) \(\forall\) \(\lambda\). 2) At module level (by raising operators' actions), construction of weight vectors along \(J_V^c\)-directions. 3) Lower bounds on the multiplicities of such weights, in all \(V\).