Cohomological Mackey Algebra Overview

• Cohomological Mackey algebra is a finite-dimensional k-algebra defined as a quotient of the Mackey algebra by imposing the cohomological relation T_H^K R_H^K = [K:H]·id, encoding key functorial properties.
• It plays a central role in modular representation theory and algebraic topology, underpinning derived equivalences and block theory through explicit Cartan matrix analyses.
• The algebra exhibits varied homological properties, including the Gorenstein condition and finite global dimension, which depend on the structure of Sylow p-subgroups.

The cohomological Mackey algebra is a finite-dimensional associative algebra associated to a finite group $G$ and a commutative ring or field $k$, defined as a quotient of the Mackey algebra by imposing cohomological relations. It plays a central role in encoding the algebraic structure of cohomological Mackey functors—functors that model induction, restriction, and transfer in equivariant algebra and topology—and in providing a foundation for the homological algebra of induction-restriction systems in modular representation theory, equivariant cohomology, and algebraic topology.

1. Structural Definition and Fundamental Properties

The cohomological Mackey algebra, denoted $\operatorname{co}\mu_k(G)$ or similar (notation varies), is a quotient of the Mackey algebra $\mu_k(G)$. The Mackey algebra is generated by symbols $T_H^K$ (transfer/induction), $R_H^K$ (restriction), and $c_g$ (conjugation) for subgroups $H \leq K \leq G$ and $g \in G$, subject to relations encoding the axioms of Mackey functors. In the cohomological Mackey algebra, one adds the "cohomological relation":

$T_H^K R_H^K = [K : H] \cdot \mathrm{id}$

for all inclusions $H \leq K$, reflecting the property that composition of restriction and transfer for cohomological functors scales as the group index.

As a $k$-algebra, $\operatorname{co}\mu_k(G)$ is finite-dimensional and is Morita equivalent to certain Hecke algebras of permutation modules due to Yoshida’s equivalence. Its module category is equivalent to the category of cohomological Mackey functors for $G$ over $k$.

Yoshida’s theorem provides an important isomorphism:

$$\operatorname{co}\mu_k(G) \cong \operatorname{End}_{kG}(k\Omega_G)$$

where $k\Omega_G$ is the direct sum over all transitive $G$-sets (i.e., permutation modules), with $G$ acting diagonally.

2. Block Structure and Cartan Matrix Analysis

A key structural feature is the block decomposition of $\operatorname{co}\mu_k(G)$, paralleling that of the group algebra $kG$. There is a canonical one-to-one correspondence between the blocks of $kG$ and the central primitive idempotents (blocks) of $\operatorname{co}\mu_k(G)$. Explicitly, for central elements $z \in Z(kG)$ written as $z = \sum_{x \in G} \lambda_x x$, the corresponding central element in $\operatorname{co}\mu_k(G)$ is:

$$L(z) = \sum_{H \leq G} \frac{1}{|H|} \sum_{x \in G} \lambda_x t_H c_{x,H}$$

where $t_H$ and $c_{x,H}$ denote the algebra generators reflecting transfer and conjugation.

Cartan matrix structure is crucial for representation theory. For the Cartan matrix $C(\operatorname{co}\mu_k(G))$ (whose entries count composition factors of projective indecomposable modules), the rank admits an explicit formula (Bouc, 2010):

$$\operatorname{rk} C(\operatorname{co}\mu_k(G)) = \sum_{R \in [C_p(G)]} |N_G(R) \backslash C_G(R)_{p'}|$$

where $[C_p(G)]$ is a set of $G$-conjugacy class representatives of cyclic $p$-subgroups and $C_G(R)_{p'}$ denotes the $p'$-elements in the centralizer of $R$.

The Cartan matrix is nonsingular if and only if $G$ is $p$-nilpotent with cyclic Sylow $p$-subgroups. This equivalence is preserved when restricted to blocks: a block algebra of $\operatorname{co}\mu_k(G)$ is nonsingular precisely when the associated block of $kG$ is nilpotent with cyclic defect groups.

3. Homological and Finiteness Properties

The homological characteristics of $\operatorname{co}\mu_k(G)$ are governed by the structure of $G$'s $p$-Sylow subgroups.

• Gorenstein property: $\operatorname{co}\mu_k(G)$ is Gorenstein if and only if, for each prime divisor $p$ of $|G|$, the Sylow $p$-subgroups are cyclic or, in characteristic $2$, either cyclic or dihedral. This ensures that projective objects have finite injective dimension and vice versa (Bouc et al., 2015).
• Finite global dimension: $\operatorname{co}\mu_k(G)$ has finite global dimension if and only if $|G|$ is invertible in $k$ or $k$ has characteristic $2$ and Sylow $2$-subgroups are cyclic of order $2$. Over the integers, finite global dimension occurs simultaneously with all Sylow $p$-subgroups being cyclic (for $p$ odd) or cyclic/dihedral for $p=2$ (Bouc et al., 2015).
• Dominant dimension: For a block $B$ of $kG$ with a nontrivial defect group, the dominant dimension of $\operatorname{co}\mu_k(B)$ is $2$ (Linckelmann, 2017).

The finitistic dimension of a block algebra of $\operatorname{co}\mu_k(G)$ is $1 + s(P)$, where $s(P)$ is the sectional $p$-rank of the defect group $P$ (Linckelmann, 2015).

4. Derived and Wild Representation Type

The module and derived category of $\operatorname{co}\mu_k(G)$ can be wild or tame, depending on the group structure and field. Over an algebraically closed field $k$ of characteristic $p$:

• $\operatorname{co}\mu_k(G)$ is derived wild if and only if the Sylow $p$-subgroup of $G$ has order greater than $2$. In particular, for $G$ a nontrivial $p$-group other than $C_2$, the algebra is derived wild (Grevstad et al., 22 Sep 2025).
• The wildness transfers to the stable module category of compact $G$-equivariant $H\underline{k}$-modules in homotopy theory, rendering classification of compact objects infeasible whenever $G$ surjects onto a $p$-group of order $>2$.

Conversely, for $G = C_2$, the cohomological Mackey algebra is gentle and derived tame; yet even then, the full Mackey algebra can be wild (Grevstad et al., 22 Sep 2025).

5. Equivalences, Blocks, and Connections to Broué-type Conjectures

There are deep connections between structural equivalences of $\operatorname{co}\mu_k(G)$ and classical block theory:

• A permeable derived equivalence (induced by complexes of $p$-permutation bimodules that respect the subcategory of $p$-permutation modules) between two block algebras $RGb$ and $RHc$ (with $R$ a complete discrete valuation ring or field) induces a derived equivalence between the associated blocks of cohomological Mackey algebras (Rognerud, 2014).
• For group algebras associated to blocks $b$ and $b'$ that are splendidly derived equivalent (as in Broué’s abelian defect group conjecture), the corresponding blocks of $\operatorname{co}\mu_k(G)$ are also derived equivalent. Thus, invariants such as Cartan matrices are preserved under such derived equivalences.
• Conversely, an equivalence between categories of cohomological Mackey functors for blocks induces a permeable Morita equivalence between the block group algebras (Linckelmann et al., 2016).

These results support using the cohomological Mackey algebra as a test for derived equivalences, and provide new invariants—such as the nonsingularity of Cartan matrices—that must be shared by splendidly equivalent blocks.

6. Explicit Presentation and Functorial Perspective

The functorial viewpoint informs the algebra’s structure:

• Intrinsic description via source algebras: For a block $b$ with source algebra $A$, the category of cohomological Mackey functors for $b$ is equivalent to $\mathrm{mod}(\mathrm{End}_A(N)^{\mathrm{op}})$, where $N = \bigoplus_{Q \leq P} A \otimes_{OQ} O$ sums over subgroups of a defect group $P$ (Linckelmann, 2015). This allows the construction of explicit two-sided tilting complexes realizing derived equivalences.
• Idempotent recollement: The module category for a block, the category of cohomological Mackey functors, and the subcategory vanishing at the trivial group form an idempotent recollement, reflecting a tight structural linkage (Linckelmann, 2015).

7. Applications, Computations, and Broader Context

The cohomological Mackey algebra's module categories serve as a framework for:

• The calculation of extension algebras in modular representation theory, especially for elementary abelian $p$-groups and in computing the self-extension algebra of simple Mackey functors (Bouc et al., 2010).
• Understanding the projective classification of $R[G]$-lattices and permutation modules via cohomological vanishing criteria in cyclic $p$-groups, including a new cohomological proof of the classical equivalence between vanishing of Tate cohomology and being a permutation lattice (Torrecillas et al., 2012).
• Providing invariants for Bredon cohomology and refining closure under group extensions, subgroups, and finite quotients for the cohomological dimension of groups with respect to Mackey functors (Degrijse, 2013, John-Green, 2013).
• Functoriality under restriction, inflation, and induction functors, which is essential in the context of higher categorical generalizations (Mackey 2-motives) and their connection to the blocks of group algebras via categorical decompositions (Balmer et al., 2021, Oda et al., 2022).

8. Summary Table: Rank and Nonsingularity of the Cartan Matrix

Object	Rank Formula	Nonsingularity Condition
$\operatorname{co}\mu_k(G)$	$$\displaystyle \sum_{R \in [C_p(G)]} |N_G(R)\backslash C_G(R)_{p'}|$$	$G$ is $p$-nilpotent with cyclic Sylow $p$-subgroups
Block $\operatorname{co}\mu_k(b)$	$$\displaystyle \sum_{(R, c)\in [C_p(b)]} |N_G(R)\backslash \mathrm{Irr}_k(k C_G(R)c)|$$	$b$ nilpotent, cyclic defect

9. Future Directions and Open Problems

Outstanding problems include:

• Refining invariants and equivalences at the level of derived or stable categories for both algebraic and topological versions of Mackey functors, especially when the algebra is wild.
• Explicit computation of Cartan matrices and their ranks for further classes of finite groups and blocks, especially beyond the nilpotent/cyclic case.
• Further development of the structure theory for cohomological Mackey 2-functors, and their connections to equivariant stable homotopy theory and localization phenomena (Balmer et al., 2021, Oda et al., 2022).

The cohomological Mackey algebra thus lies at the intersection of modular representation theory, algebraic topology, and higher category theory, serving as a unifying structure for functorial induction-restriction systems, block theory, and derived equivalence theory.