Strong shift equivalence and the generalized spectral conjecture for nonnegative matrices

TL;DR

This paper proves the equivalence of weak and strong forms of the Generalized Spectral Conjecture for nonnegative matrices, linking spectral conditions to shift equivalence via algebraic K-theory.

Contribution

It establishes the equivalence of the weak and strong forms of the GSC, connecting spectral properties with algebraic K-theory for matrices over subrings of reals.

Findings

Weak and strong GSC forms are equivalent.

Algebraic K-theory group NK_1(S) captures shift equivalence refinements.

GSC remains open for S = real numbers.

Abstract

We show that the weak and strong forms of the Generalized Spectral Conjecture (GSC) of Boyle and Handelman are equivalent. The GSC asserts that well understood necessary spectral conditions on a square matrix A over a subring S of the reals are sufficient for that matrix to be shift equivalent over S (in the weak form) or strong shift equivalent over S (in the strong form) to a primitive matrix over S. The foundation of this work is the recent result that the group NK_1(S) of algebraic K-theory exactly captures the refinement of shift equivalence over S by strong shift equivalence over S. The GSC remains open in general even in the case that S equals the real numbers.