This paper shows that a wide class of “generalized” symmetries can split a many‑body quantum system into exponentially many dynamically disconnected pieces. That splitting is known as Hilbert space fragmentation. Each piece, called a Krylov sector, is the small part of the full state space that a given initial product state can explore under the system’s dynamics. Fragmentation has often been taken as evidence that a system fails to thermalize, or is nonergodic. The authors argue that this conclusion can be too quick: generalized symmetries alone can produce many sectors without necessarily breaking thermalization in every sector.

The authors give a simple, general mechanism. If a conserved operator U can be translated many times so that the translated copies act on disjoint regions, and if U is diagonal in the product basis used to build the many‑body states with at least two distinct eigenvalues, then the different local eigenvalues combine like independent choices across space. That combinatorics produces exponentially many symmetry sectors. Since Krylov sectors sit inside symmetry sectors whenever the symmetry is diagonal in the same product basis, the number of Krylov sectors grows exponentially as well. They formalize this with a proposition and a counting argument that links the geometry of the symmetry operators to the number of sectors.

The paper shows this mechanism applies to several types of generalized symmetry. Gauge symmetries, higher‑form or subsystem symmetries, and certain non‑invertible symmetries can meet the conditions needed for fragmentation. As a concrete example, the authors discuss a three‑dimensional U(1) quantum link model on a cubic lattice. A local gauge generator in that model can take seven values and can be translated about L3/2 times, giving at least 7·L3/2 symmetry sectors in one product basis. A subsystem “1‑form” operator that measures planar magnetization has L2+1 eigenvalues on each plane and can be translated L times, yielding roughly (L2+1)^L symmetry sectors when counted across planes. The paper also shows how non‑invertible operators called partial isometries can further fragment individual symmetry sectors.