Source Permutation Module in Block Theory

• Source permutation module is a block-local refinement of the Sylow permutation module, defined as Bi ⊗ₖP k using source idempotents to capture defect group-controlled structure.
• It retains the weight content of blocks by ensuring every simple module appears as both top and socle, and by linking Green correspondence and controlled endomorphism algebras.
• Its invariance under splendid Morita equivalence makes it a robust tool for studying block theory, self-injectivity, and validating conjectures like Alperin’s weight conjecture.

Searching arXiv for recent and foundational papers on source permutation modules and closely related permutation-module literature. In modular representation theory, the source permutation module is a block-local refinement of the Sylow permutation module. For a finite group $G$, a field $k$ of prime characteristic $p$, a block $B$ of $kG$ with defect group $P$, and a source idempotent $i\in B^P$, it is defined by

$Bi\otimes_{kP}k.$

The notion was introduced to isolate the part of the block Sylow permutation module that is controlled by the defect group and that remains stable under splendid equivalences. In this form, it serves as the “correct block-theoretic analogue” of the Sylow permutation module $\mathrm{Ind}_S^G(k)=kG\otimes_{kS}k$, while retaining links to weights, Green correspondence, and endomorphism algebras (Kessar et al., 20 Jul 2025).

1. Definition and block-theoretic origin

A permutation module for a finite group is, in the standard sense, a $kG$-module of the form $k$0 for a finite $k$1-set $k$2, equivalently a finite direct sum of modules $k$3 for subgroups $k$4 (Balmer et al., 2020). The classical modular object relevant here is the Sylow permutation module

$k$5

where $k$6 is a Sylow $k$7-subgroup of $k$8 (Kessar et al., 20 Jul 2025).

Because $k$9 decomposes into blocks, the Sylow permutation module decomposes as

$p$0

with $p$1 ranging over the blocks of $p$2. The modules $p$3 are natural block summands, but they are not invariant under splendid Morita equivalence. This failure is the immediate motivation for passing from the block algebra to its source algebra data (Kessar et al., 20 Jul 2025).

If $p$4 has defect group $p$5 and $p$6 is a source idempotent, the source permutation module is defined by

$p$7

The construction is formulated in the language of source algebras and defect pointed groups: $p$8 is a primitive idempotent in the fixed-point algebra $p$9, and the tensor product uses the trivial $B$0-module. A basic structural fact is that the isomorphism class of $B$1 does not depend on the choice of $B$2 or of $B$3, provided $B$4 comes from a defect pointed group of $B$5 (Kessar et al., 20 Jul 2025).

2. Relation to source algebras and the Sylow permutation module

Let

$B$6

be the source algebra of $B$7. The canonical Morita equivalence between $B$8 and $B$9 identifies the endomorphism algebra of the source permutation module with the corresponding endomorphism algebra over the source algebra: $kG$0 This makes $kG$1 a source-algebra invariant, and hence a much better behaved block invariant than the raw block component $kG$2 (Kessar et al., 20 Jul 2025).

The relation with the Sylow permutation module is precise. If $kG$3 is large enough, namely a splitting field for the relevant source algebra, then

$kG$4

is isomorphic to a direct summand of

$kG$5

More sharply, every indecomposable summand of $kG$6 with vertex $kG$7 is already a summand of $kG$8. In that sense, the source permutation module captures exactly the full-vertex part of the block Sylow permutation module (Kessar et al., 20 Jul 2025).

There is also a Brauer-correspondent control statement. If $kG$9 denotes the Brauer correspondent of $P$0 in $P$1, and $P$2 is a source idempotent, then $P$3 is a direct summand of

$P$4

This places the source permutation module within the usual local-global architecture of block theory (Kessar et al., 20 Jul 2025).

3. Weights, Green correspondence, and internal structure

One of the main reasons the source permutation module is useful is that it retains the classical “weight” content of the Sylow permutation module. A principal theorem states that every Green correspondent of a $P$5-weight is a direct summand of $P$6. Consequently, the source permutation module contains at least $P$7 non-isomorphic indecomposable summands, where $P$8 is the number of $P$9-weights (Kessar et al., 20 Jul 2025).

At the same time, the module is controlled on the projective side. The paper proves that $i\in B^P$0 has no nonzero projective direct summand. It also proves that every simple $i\in B^P$1-module appears both as a quotient and as a submodule of $i\in B^P$2. This gives the module simultaneous access to the top and the socle of the block (Kessar et al., 20 Jul 2025).

A common misconception is that the block component $i\in B^P$3 already provides the correct block-local replacement for the Sylow permutation module. The introduction of $i\in B^P$4 shows that this is not the relevant invariant: the raw block summand is natural, but it is not preserved by splendid Morita equivalence, whereas the source permutation module is constructed precisely to rectify that defect (Kessar et al., 20 Jul 2025).

4. Invariance under splendid equivalence and the self-injectivity problem

The central structural theorem is its invariance under splendid equivalence. If $i\in B^P$5 and $i\in B^P$6 are blocks with defect groups $i\in B^P$7 and $i\in B^P$8, and a $i\in B^P$9-$Bi\otimes_{kP}k.$0-bimodule $Bi\otimes_{kP}k.$1 induces a splendid stable equivalence of Morita type, then

$Bi\otimes_{kP}k.$2

is the non-projective part of

$Bi\otimes_{kP}k.$3

with $Bi\otimes_{kP}k.$4 and $Bi\otimes_{kP}k.$5 source idempotents for $Bi\otimes_{kP}k.$6 and $Bi\otimes_{kP}k.$7. If the equivalence is in fact a splendid Morita equivalence, then the source permutation modules correspond directly (Kessar et al., 20 Jul 2025).

This invariance is the basis for the paper’s treatment of endomorphism algebras. A key theorem states that if the endomorphism algebra of the full Sylow permutation module is self-injective, then so is

$Bi\otimes_{kP}k.$8

The converse is not asserted in this form, but the source permutation module is the object for which self-injectivity interacts well with local structure and splendid equivalences (Kessar et al., 20 Jul 2025).

The same framework yields consequences for Alperin’s weight conjecture. Using the facts that Green correspondents of weights occur as summands and that every simple $Bi\otimes_{kP}k.$9-module occurs in the top and socle, the paper derives

$\mathrm{Ind}_S^G(k)=kG\otimes_{kS}k$0

Under additional splendid stable equivalence hypotheses, self-injectivity of $\mathrm{Ind}_S^G(k)=kG\otimes_{kS}k$1 forces

$\mathrm{Ind}_S^G(k)=kG\otimes_{kS}k$2

that is, Alperin’s weight conjecture for $\mathrm{Ind}_S^G(k)=kG\otimes_{kS}k$3 (Kessar et al., 20 Jul 2025).

5. Computed families and explicit cases

The source permutation module can be calculated explicitly in several important defect types. The resulting endomorphism algebras are notably better behaved than those of the full Sylow permutation module.

Defect type / case	Endomorphism algebra of $\mathrm{Ind}_S^G(k)=kG\otimes_{kS}k$4	Consequence
Cyclic defect group	Direct product of self-injective Nakayama algebras	Self-injective
Klein four defect group	$\mathrm{Ind}_S^G(k)=kG\otimes_{kS}k$5, or $\mathrm{Ind}_S^G(k)=kG\otimes_{kS}k$6, or $\mathrm{Ind}_S^G(k)=kG\otimes_{kS}k$7	Self-injective
Certain symmetric-group blocks	Better behaved than $\mathrm{Ind}_S^G(k)=kG\otimes_{kS}k$8 in the cited cases	Motivates block refinement

For blocks with cyclic defect group $\mathrm{Ind}_S^G(k)=kG\otimes_{kS}k$9, the paper proves that

$kG$0

is a direct product of self-injective Nakayama algebras. This yields self-injectivity uniformly in that family (Kessar et al., 20 Jul 2025).

For blocks with defect group $kG$1 in characteristic $kG$2, the endomorphism algebra is shown to be one of

$kG$3

where $kG$4 is a $kG$5-dimensional basic Nakayama algebra with two simple modules and projective indecomposables of length $kG$6. In every case it is self-injective. The paper also stresses that replacing the source permutation module by a Heller translate can destroy self-injectivity, which underscores that the exact module $kG$7 is the relevant object (Kessar et al., 20 Jul 2025).

For symmetric groups, the source permutation module is studied partly in response to a question of Diaconis–Giannelli–Guralnick–Law–Navarro–Sambale–Spink. When $kG$8 and the principal block has cyclic defect, the source permutation module is the non-projective part of the Sylow permutation module for $kG$9, but not in general. For $k$00, the full endomorphism algebra of $k$01 is not self-injective; an explicit $k$02 example exhibits this failure. The source permutation module is introduced precisely as the better block invariant in such situations (Kessar et al., 20 Jul 2025).

6. Position within the theory of permutation modules

The source permutation module belongs to the broader theory of permutation modules, but it is more specific than the usual linearization $k$03 of a finite $k$04-set. In the general modular setting, permutation modules generate large parts of representation theory: every modular representation becomes, after adding a suitable complement, the endpoint of a finite exact complex of permutation modules (Balmer et al., 2020). In tensor-triangular geometry, finitely generated permutation modules form the category $k$05, and their direct summands are the $k$06-permutation modules or trivial source modules (Balmer et al., 2022).

This context helps clarify the role of $k$07. It is not merely another permutation module attached to a $k$08-set, but a block-theoretic object extracted from a source algebra and a defect pointed group. A plausible implication is that its significance comes from combining two layers of structure that are usually treated separately: the combinatorics of permutation modules and the local structure of blocks. The 2025 construction makes that combination explicit by showing that the block-level replacement for $k$09 should be formed at the source-algebra level rather than at the level of the block summand $k$10 (Kessar et al., 20 Jul 2025).

In that sense, the source permutation module is best viewed as a local-global bridge. It is local through the defect group $k$11 and source idempotent $k$12; global through its recovery of weights, its interaction with induction from Brauer correspondents, and its control of block invariants such as self-injectivity and weight counts.