Arithmetic Kac Theory for Ramified Species

Abstract

Let $K=k((t))$, where $k$ is algebraically closed of characteristic zero, and let $Λ$ be an acyclic ramified species over $K$. An indecomposable $Λ$-module can split after extending $K$. The number of its distinct geometric indecomposable summands is the degree of its residue endomorphism field $\operatorname{End}_Λ(M)/\operatorname{rad}\operatorname{End}_Λ(M)$. We use the positive roots and cyclic automorphism of the unfolded quiver to determine the number of isomorphism classes at every nonzero rank and residue degree $e\ge1$: it is zero, an explicitly computable finite number, or $|K|$. For every invariant imaginary root of the unfolding, including at nonprimitive ranks, we construct a smooth invariant family of absolutely indecomposable representations whose rational quotient by the change-of-bases group has dimension $1-q_d(β)$. We characterize ranks admitting absolutely Schur representations by the Schur roots of the unfolded quiver. For the associated species over finite fields, we realize the polynomials counting absolutely indecomposable representations as graded traces of the unfolding automorphism on preprojective BPS cohomology. Galois-orbit formulas and Möbius inversion then recover the counts at each residue degree. Finally, we prove an all-rank symmetrizable analogue of Kac's constant-term conjecture: the constant term of the polynomial counting all indecomposables, rather than only absolutely indecomposable ones, equals the corresponding Kac--Moody root multiplicity. This constant is independent of the symmetrizer and modulation.