A Classification of the Isomorphism Types of Indecomposable and Simple Modules that Refines the Green Theory in Finite Group Modular Representation Theory

In Finite Group Modular Representation Theory, the basic objects are the indecomposable and simple modules. This paper offers a new classification of these objects that refines the Green Theory Classification of indecomposable and simple modules. The sets of isomorphism tyes of these modules is decomposed into disjoint, non-empty subsets such that any two elements in any subset share Green Theory Invariants. We also prove a new formula for the number of isomorphism types of absolutely simple modules of finite groups in prime characteristic.

💡 Research Summary

The paper addresses a fundamental problem in the modular representation theory of finite groups: how to classify indecomposable and simple modules more finely than the classical Green theory allows. Working over a field Φ of prime characteristic p and its algebraic closure \bar F, the author builds on the Green correspondence (which groups modules by common vertices and sources) and introduces a new stratification based on the finite subfields Δ of Φ.

Key constructions:

1. For each subfield K ⊆ Φ the sets ITI(KG) and ITS(KG) denote representatives of isomorphism classes of indecomposable and simple right KG‑modules, respectively.
2. The author defines a relation ↑ on the set I(Ω(G)) = {(K,V) | K subfield, V ∈ ITI(KG)} by (K,V) ↑ (L,U) iff K ⊆ L and U is a direct‑summand of V⊗_K L. This relation is reflexive and transitive and captures the behaviour of modules under field extensions.
3. Restricting to finite subfields Δ yields the subsets FI(Ω(G)) and FS(Ω(G)) (the latter consisting of simple modules).

For a fixed simple F‑module W (where F is the unique subfield of order p), the author defines
 E(W) = {(K,V) ∈ FI(Ω(G)) | (F,W) ↑ (K,V)}.
He proves that E(W) is never empty, that any (K,V) ∈ E(W) gives rise to all its Galois conjugates (K,Vσ) (σ∈Aut(K)), and that all members of E(W) share the same vertex Q and a Q‑source U. Consequently the Green correspondence Gr_H^G (with H = N_G(Q)) induces a bijection between E(W) and E(Gr_H^G(W)).

These observations lead to two “projection” maps:

* Γ: ITI(\bar F G) → ITI(F G) is defined by sending a \bar F‑indecomposable Y to the unique F‑indecomposable W for which there exists (K,V) ∈ E(W) with V⊗_K \bar F ≅ Y. The paper shows Γ is well‑defined, that each W has a non‑empty pre‑image Γ⁻¹(W), and that the family {Γ⁻¹(W) | W∈ITI(F G)} partitions ITI(\bar F G).

* Σ: ITS(\bar F G) → ITS(F G) is defined analogously for simple modules, and the same partition property holds.

Thus the classical Green classes are refined: instead of a single class determined by a vertex‑source pair, one obtains a disjoint union of finer subclasses indexed by finite subfields and the corresponding indecomposable (or simple) KG‑modules.

Finally, exploiting this refined partition, the author derives a new explicit formula for the number of absolutely simple \bar F‑modules (i.e., simple modules that remain simple over the algebraic closure). The formula involves a sum over the finite subfields K ∈ Δ and counts, for each K, the number of simple KG‑modules whose vertex‑source data lift to \bar F. This result improves earlier estimates and provides an exact count in terms of readily computable invariants.

In summary, the paper makes three major contributions:

1. A refined classification of indecomposable and simple modules that respects Green invariants while distinguishing modules that become isomorphic only after extending the base field.
2. Construction of the maps Γ and Σ, establishing a precise correspondence between modules over the algebraic closure and those over the prime field, together with a proof that these correspondences partition the whole set of modules.
3. A new counting formula for absolutely simple modules, offering a concrete numerical invariant that was previously unavailable.

These advances deepen our understanding of modular representation theory, provide new tools for block theory and the study of source algebras, and open avenues for further investigation into how field extensions interact with module structure.