An arXiv preprint has classified the irreducible building blocks—called composition factors—that can appear in certain finite-group modules when an element of order p, for a prime p, acts with exactly k non-trivial Jordan blocks.
Jordan blocks are standard pieces used to record how a linear transformation acts. The paper calls the situation k-active when exactly k of those blocks are non-trivial, and studies which composition factors can support that action.
The main theorem applies to a faithful module V for a finite group X with a strongly p-embedded subgroup, p-rank m of at least 2, and k at most m. It classifies the non-trivial composition factors W that can occur under those assumptions.
The list changes by group family
In the linear rank-one branch, the allowed factors include the basic modules V_i in the paper’s stated index range. A separate two-factor, Galois-twisted construction has dimension 4m when p is odd, subject to an additional restriction on its index j.
The remaining linear cases add irreducible constructions subject to evenness or divisibility conditions. Their listed dimensions are 2m, 8m/3, 4m and 9m/2.
Unitary examples show why the Jordan form matters. On the natural factor, if x lies in the centre Z(S), the dimension is 6n and the action has 2n blocks of size 1 plus 2n blocks of size 2. If x lies outside Z(S), the dimension remains 6n but the action has 2n blocks of size 3.
Other unitary possibilities include a non-trivial irreducible factor of M tensor its dual: its dimension is 7n when p=3 and 8n when p is at least 5. The symmetric square of M appears with dimension 12n when p is at least 5, and the paper gives an explicit Jordan form for each case.
For the Ree family, the allowed factor is the natural module of dimension 7n. The active element must lie in the specified subgroup difference Ω1(S)\Z(S), excluding Z(S).
The exceptional branches are equally specific: an M11 code or cocode module of dimension 5; 2·PSL3(4) and 4·PSL3(4) cases with dimensions 6 and 8; and an alternating-group branch with the paper’s stated possibilities for a subgroup L.
A sharper result at smaller dimensions
A separate low-dimensional theorem sets a clear cutoff. Under its group assumptions, a faithful irreducible module of dimension at most 4m(p−1) must fall into the paper’s quasisimple-core case or one of the explicitly listed exceptional configurations.
With the extra conditions that the module is generated by its commutators with X, has no vectors fixed by all of X, and k<m, the possibilities narrow to an odd-p, even-m PSL2 family. It has a 2m-dimensional irreducible summand, on which x acts with two Jordan blocks of size 3.
A proof-driven classification
The work reduces the main theorem to rank-one quasisimple groups, uses Steinberg tensor-product decompositions and Jordan-block calculations, and turns to Magma for selected small cases. The associated code is stated to be attached as an ancillary file.
The resulting Jordan decompositions for W restricted to the group generated by x are given for every case in the main classification, with the data collected in Table 2 of Appendix B.
A tightly bounded result
The conclusions remain conditional on the stated strongly p-embedded, rank and module hypotheses. The exceptional branches also carry their own specified subgroup conditions.
The result is a version-1 arXiv preprint dated 20 Aug 2026, presenting its classification within that defined setting.
Paper data and sources
Original title: Modules with few Jordan blocks for rank $1$ groups of Lie type and related groups
Authors: Valentina Grazian, Justin Lynd, Chris Parker et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text