Stability phenomena for Kac-Moody groups

We show that a canonical procedure of extending generalized Dynkin diagrams gives rise to families of Kac-Moody groups that satisfy homological stability. We also briefly sketch some emergent structure that appears on stabilization. Our results are illustrated for the family {E_n} which is of interest in String theory. The techniques used involve homotopy decompositions of classifying spaces of Kac-Moody groups.

💡 Research Summary

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The paper investigates homological stability phenomena for families of Kac‑Moody groups that are obtained by a canonical procedure of extending generalized Dynkin diagrams. The authors focus on a prototypical family, denoted (E_n), which starts with the exceptional finite‑type groups (E_6, E_7, E_8), continues with the affine type (E_9), and then proceeds to highly‑extended types (E_{9+n}). For each generalized Cartan matrix (A_I) they consider the minimal complex Kac‑Moody group (K(A_I)) and its compact unitary form (K(A_I)). A key tool is the homotopy decomposition of the classifying space (BK(A_I)) as a homotopy colimit over the poset of spherical subsets (S(A_I)); each spherical subset (J) gives rise to a compact subgroup (H_J(A_I)=\langle T, K(A_J)\rangle). Theorem 2.5 (due to Kitchloo and Broto–Kitchloo) asserts that the canonical maps \