Anchors in Modular Representation Theory

• Anchors are invariants defined as the defect groups of primitive G-interior algebras that capture the minimal p-local subgroup data needed to recover central idempotents.
• They refine traditional invariants by relating block defect groups, Green vertices, and Navarro vertices, particularly in cases with irreducible reduction and non-abelian defects.
• Anchors exhibit Morita invariance and centricity within fusion systems, offering deeper insight into local module structure and p-group configurations in representation theory.

Searching arXiv for the specified paper and closely related work on characters, defect groups, and vertices. Anchors of irreducible characters are a block-theoretic invariant attached to an ordinary irreducible character $\chi\in \mathrm{Irr}(G)$ of a finite group $G$ at a fixed prime $p$. In the formulation of Kessar–Külshammer–Linckelmann, an anchor is defined as a defect group of the primitive $G$-interior $O$-algebra $OGe_\chi$, where $e_\chi\in Z(KG)$ is the unique primitive central idempotent with $\chi(e_\chi)\neq 0$ (Kessar et al., 2015). Green’s theory implies that these defect groups form a single $G$-conjugacy class, so the anchor is a well-defined conjugacy-class invariant of $\chi$. The notion was considered by L. Barker in the context of finite $G$0-solvable groups, and the systematic treatment in (Kessar et al., 2015) develops its structural properties and its relation to defect groups, Green vertices, and Navarro vertices.

1. Definition through primitive $G$1-interior algebras

Fix a complete discrete valuation ring $G$2 with residue field $G$3 of characteristic $G$4 and field of fractions $G$5 of characteristic $G$6, and assume that $G$7 and $G$8 split all groups under discussion. For $G$9, let

$p$0

be the unique primitive central idempotent with $p$1. The quotient

$p$2

is a primitive $p$3-interior $p$4-algebra, and Green’s theory assigns to it a defect subgroup (Kessar et al., 2015).

An anchor of $p$5 is any defect group $p$6 of the primitive $p$7-interior $p$8-algebra $p$9. Equivalently, by Green’s minimal-subgroup formulation, $G$0 is minimal up to conjugacy among the $G$1-subgroups $G$2 for which there exists

$G$3

where $G$4 is the relative trace from $G$5 to its $G$6-fixed points (Kessar et al., 2015).

This definition places anchors directly at the interface of ordinary character theory, block theory, and interior algebra methods. A plausible implication is that the anchor captures the smallest $G$7-local subgroup data sufficient to recover the central idempotent $G$8 via relative trace, rather than only the ambient block-level local structure.

2. Fundamental structural properties

The basic properties established for anchors are formulated in terms of the block $G$9 containing $O$0 and an $O$1-lattice $O$2 affording $O$3 (Kessar et al., 2015).

If $O$4 is an anchor of $O$5, then $O$6 is conjugate to a subgroup of any defect group $O$7 of the block $O$8. Thus anchors are always contained, up to conjugacy, in block defect groups. They therefore refine the local structure visible at the block level without in general replacing it.

The same theorem shows that any vertex of $O$9 in Green’s sense is contained in $OGe_\chi$0, and that the largest normal $OGe_\chi$1-subgroup $OGe_\chi$2 lies in $OGe_\chi$3 (Kessar et al., 2015). In this sense, the anchor sits above all module vertices arising from lattices affording $OGe_\chi$4, while also necessarily containing the normal $OGe_\chi$5-core.

There is also a bimodule-theoretic formulation. Writing $OGe_\chi$6 for the diagonal copy, the diagonal subgroup $OGe_\chi$7 is contained in some vertex of the $OGe_\chi$8-module $OGe_\chi$9, and $e_\chi\in Z(KG)$0 contains a vertex of $e_\chi\in Z(KG)$1. Moreover, $e_\chi\in Z(KG)$2 itself is a vertex exactly when $e_\chi\in Z(KG)$3 has defect zero (Kessar et al., 2015). This identifies anchors with local data visible not only in lattices affording $e_\chi\in Z(KG)$4 but also in the $e_\chi\in Z(KG)$5-module structure of the corresponding character algebra summand.

These results exclude a common oversimplification: an anchor is not merely a reformulation of the defect group of the block. It is always subordinate to block defect groups, but its relation to them depends on additional character-theoretic hypotheses.

3. Special cases: irreducible reduction, abelian defect, and height zero

Several important cases collapse the distinction between anchors and more familiar local invariants. If the reduction $e_\chi\in Z(KG)$6 is irreducible in $e_\chi\in Z(KG)$7, then the $e_\chi\in Z(KG)$8-lattice $e_\chi\in Z(KG)$9 affording $\chi(e_\chi)\neq 0$0 is unique, its vertex $\chi(e_\chi)\neq 0$1 is unique, and $\chi(e_\chi)\neq 0$2 is a vertex of $\chi(e_\chi)\neq 0$3 (Kessar et al., 2015). In that situation, the anchor is exactly the conjugacy class of the vertex of the unique lattice.

If the block $\chi(e_\chi)\neq 0$4 has abelian defect group $\chi(e_\chi)\neq 0$5, then $\chi(e_\chi)\neq 0$6 itself is an anchor of every $\chi(e_\chi)\neq 0$7 (Kessar et al., 2015). This gives a uniform answer throughout the block and shows that in the abelian-defect case the anchor no longer varies with the irreducible character.

If $\chi(e_\chi)\neq 0$8 has $\chi(e_\chi)\neq 0$9-height $G$0, then its anchors coincide with the defect groups of $G$1 (Kessar et al., 2015). Height-zero characters therefore realize the maximal possible anchor inside the block.

These cases can be summarized as follows.

Hypothesis on $G$2 or $G$3	Consequence for anchors
$G$4 irreducible in $G$5	Unique lattice and unique vertex $G$6; $G$7 is a vertex of $G$8
$G$9 has abelian defect group $\chi$0	$\chi$1 is an anchor of every $\chi$2
$\chi$3 has $\chi$4-height $\chi$5	Anchors coincide with defect groups of $\chi$6

Together these results show that anchors interpolate between character-specific and block-wide local structure. This suggests that the invariant is most informative precisely when the character is not height zero and when the defect structure is not already forced by abelianity.

4. Relation to vertices and other $\chi$7-group invariants

The conceptual significance of anchors lies in their relation to three established $\chi$8-local invariants: defect groups of blocks, Green vertices of modules, and Navarro vertices of characters.

For defect groups, the general containment is one-sided: anchors always lie inside defect groups of the block, and equality holds for height-zero characters (Kessar et al., 2015). Thus the block defect group provides an ambient upper bound, while the anchor supplies finer character-level localization.

For Green vertices, the relation is especially tight when $\chi$9 is irreducible. In that case $G$00 is the endomorphism algebra of any lattice $G$01 affording $G$02, and the anchor is exactly the conjugacy class of any vertex of $G$03. In general, anchors “sit above” all vertices of lattices affording $G$04 (Kessar et al., 2015). This identifies anchors as an upper envelope for the module vertices associated with the same character.

For Navarro vertices in the $G$05-solvable setting, if $G$06 has Navarro vertex $G$07, then under the hypothesis that $G$08 is irreducible and $G$09 is $G$10-solvable, the subgroup $G$11 always contains an anchor. If moreover $G$12 is odd or the $G$13-special part of $G$14 is trivial, then $G$15 itself is an anchor (Kessar et al., 2015). For groups of odd order every Navarro $G$16 is contained in some anchor, but there are examples in even order where no anchor contains $G$17.

A concise comparison is useful.

Invariant	Relation to anchor
Defect group of the block	Anchor is conjugate to a subgroup; equality for height-zero characters
Green vertex of a lattice affording $G$18	Any such vertex is contained in the anchor
Navarro subgroup $G$19	In $G$20-solvable settings, $G$21 contains an anchor under irreducibility of $G$22; in further cases $G$23 itself is an anchor

These comparisons clarify that anchors should not be identified with any single pre-existing invariant. Rather, they organize several local invariants around the character $G$24, sometimes coinciding with them and sometimes strictly refining or bounding them.

5. Morita invariance and local structure

Anchors satisfy a Morita-invariance property under a specific class of equivalences. If two blocks $G$25 and $G$26 of groups $G$27 and $G$28 are Morita equivalent via a bimodule with endopermutation source, then under the induced bijection

$G$29

the anchors correspond via any fixed identification of the common defect group (Kessar et al., 2015).

This result indicates that anchors are not merely artifacts of a particular group realization; they are preserved by a strong form of equivalence compatible with local block structure. A plausible implication is that the pair “block plus irreducible character” carries anchor data in a way robust under representation-theoretic transport, at least within the endopermutation-source Morita framework.

The same body of work also states that anchors are proved centric in the fusion system of their block, relating them to local structure and source algebras (Kessar et al., 2015). This places anchors inside the standard network of fusion-theoretic notions used in modern block theory.

The combination of Morita-invariance and centricity explains why anchors are more than an ad hoc subgroup assignment. They are designed to interact with both equivalence theory and the internal local structure of blocks.

6. Examples, computations, and open directions

The paper develops explicit examples illustrating both computation and divergence from neighboring invariants. For the dihedral group $G$30 at $G$31, the unique irreducible character of degree $G$32 has anchor equal to a Klein four subgroup, and its block has defect $G$33. Concretely, $G$34 is realized as a $G$35-dimensional quotient of $G$36, and its Brauer quotients can be computed directly (Kessar et al., 2015).

For symmetric groups, if $G$37 and $G$38 is labelled by a partition $G$39 whose reduction is irreducible, then Specht-module lattices show that the unique lattice is a permutation module; consequently the anchor equals the vertex of a $G$40-permutation module (Kessar et al., 2015).

For $G$41 at $G$42, there is a $G$43-special character of degree $G$44 whose reduction has vertex of order $G$45, while the unique $G$46-lattice has vertex of order $G$47; hence the defect of $G$48 is strictly smaller than the anchor $G$49 (Kessar et al., 2015). This example shows that anchors may exceed the modular vertex information visible after reduction.

Further wreath-product and $G$50-solvable constructions show that Navarro vertices and anchors need not coincide (Kessar et al., 2015). This directly addresses a likely misconception that all natural $G$51-local subgroups attached to a character should agree once one passes to $G$52-solvable groups.

The work also identifies several applications and open questions. Anchors provide new upper bounds on vertices of lattices affording a given $G$53, refine Barker’s work on Robinson’s conjecture for $G$54-solvable groups, and connect to local structure through fusion systems and source algebras (Kessar et al., 2015). Open directions include full comparison with Navarro vertices beyond the odd-order case, explicit classification of anchors in sporadic or exceptional groups, and potential applications to counting conjectures in block theory (Kessar et al., 2015).

In that sense, anchors occupy a precise but still developing position in modular representation theory: they are rigorously defined through interior algebra defect groups, constrained by block defect groups and vertices, stable under suitable Morita equivalences, and sufficiently delicate to separate phenomena that coarser invariants do not distinguish.