On the tensor semigroup of affine Kac-Moody lie algebras

The support of the tensor product decomposition integrable irreducible highest weight representations a symmetrizable Kac-Moody Lie algebra g \mathfrak {g} defines semigroup triples weights. Namely, given alttext="lamda"> λ encoding="application/x-tex">\lambda in set alttext="upper P Subscript plus"> P + encoding="application/x-tex">P_+ dominant integral weights, V left-parenthesis lamda right-parenthesis"> V ( stretchy="false">) encoding="application/x-tex">V(\lambda ) denotes representation with . We are interested tensor semigroup mathvariant="normal">Γ 3 stretchy="false">| ⊂<!-- ⊂ <mml:mo>⊗<!-- ⊗ stretchy="false">} encoding="application/x-tex">\begin{equation*} \Gamma _{\mathbb {N}}(\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+}^3\,|\, V(\mu )\subset V(\lambda _1)\otimes _2)\}, \end{equation*} and cone encoding="application/x-tex">\Gamma (\mathfrak {g}) it generates: Q there-exists greater-than-or-equal-to period"> mathvariant="double-struck">Q mathvariant="normal">∃<!-- ∃ <mml:mi>N ≥<!-- ≥ width="1em" <mml:mo>. P_{+,{\mathbb {Q}}}^3\,|\,\exists N\geq \quad V(N\mu V(N\lambda _2)\}. Here, Q"> encoding="application/x-tex">P_{+,{\mathbb {Q}}} rational convex cone generated by In special case when is finite-dimensional semisimple algebra, known to be finitely hence polyhedral. Moreover, described Belkale Kumar [Invent. Math. 166 (2006), pp. 185–228] an explicit finite list inequalities. general, neither polyhedral, nor closed. this article we describe closure countable family linear inequalities for any untwisted affine which most important class infinite-dimensional algebra. This solves Brown-Kumar’s conjecture (see Brown [Math. Ann. 360 (2014), 901–936]). difference between measured saturation factors. positive integer alttext="d"> d encoding="application/x-tex">d called factor, if encoding="application/x-tex">V(N\lambda _2) contains encoding="application/x-tex">V(N\mu some N"> encoding="application/x-tex">N then d encoding="application/x-tex">V(d\lambda V(d\lambda encoding="application/x-tex">V(d\mu , assuming that alttext="mu minus 2"> −<!-- − encoding="application/x-tex">\mu -\lambda _1-\lambda _2 belongs root lattice. For equals s l n"> = mathvariant="fraktur">s mathvariant="fraktur">l n {g}={\mathfrak {sl}}_n famous Knutson-Tao theorem asserts alttext="d 1"> encoding="application/x-tex">d=1 saturation factor Knutson Tao [J. Amer. Soc. 12 (1999), 1055–1090]). More generally, simple factors known. case, not necessarily existence such unclear priori. obtain example, type A overTilde A stretchy="false">~<!-- ~ </mml:mover> encoding="application/x-tex">\tilde A_n prove encoding="application/x-tex">d\geq 2 factor, generalizing A_1 shown 901–936].