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 $\lambda \in {\mathfrak{h}}^{\ast }$, and let $L(\lambda )$ be the corresponding simple highest weight $\mathfrak{g}$-module), and write their weight-sets $\mathrm{w}\mathrm{t}V$. This is based on and generalizes the Minkowski decompositions for all $\mathrm{w}\mathrm{t}L(\lambda )$ and hulls ${\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}_{\mathbb{R}}(\mathrm{w}\mathrm{t}V)$, 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 $L(\lambda )$: $\mathrm{w}\mathrm{t}L(\lambda )-$ 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 $(\mathfrak{g},\lambda ,V)$ of our $J_V^c$-freeness: 1) Minkowski decompositions of all $\mathrm{w}\mathrm{t}V$, subsuming those above for simples. 1${}^{\mathrm{\prime }}$) Characterization of these formulas. 1${}^{\mathrm{\prime }\mathrm{\prime }}$) For these, we solve the inverse problem of determining all $V$ with fixing $\mathrm{w}\mathrm{t}V=$ weight-set of a Verma, parabolic Verma and $L(\lambda )$ $\mathrm{\forall }$ $\lambda$. 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 $V$.