---
title: Representation Dimension in Algebras & Groups
url: https://www.emergentmind.com/topics/representation-dimension
type: topic
---

# Representation Dimension in Algebras & Groups

The notion of **representation dimension** serves as a fundamental invariant measuring the complexity of module categories over algebras and the minimal dimension of faithful linear actions for groups. It is central both in the homological theory of finite-dimensional associative algebras and the character theory of finite groups, with ramifications in modern representation theory, cohomology, and computational complexity.

## 1. Formal Definitions and Foundational Properties

For a finite-dimensional Artin algebra $A$ over a field $k$, the **representation dimension**, denoted $\mathrm{rep.dim}(A)$, is defined by
\[
\mathrm{rep.dim}(A) = \inf\left\{\,\mathrm{gl.dim}\,\End_A(M)\;\middle|\;M \text{ a generator-cogenerator of } \mathrm{mod}\,A\,\right\}
\]
where $M$ is called a **generator** if $A_A \in \mathrm{add}\,M$ and a **cogenerator** if $D(A_A) = \Hom_k(A_A, k) \in \mathrm{add}\,M$ [1006.2901, 1102.1048, 1510.06443]. This definition, due to Auslander, captures the minimal global dimension attainable by endomorphism rings of modules generating and cogenerating the module category.

For a finite group $G$, the **representation dimension** (also called the minimal faithful dimension), denoted $\mathrm{rdim}(G)$ or $\delta(G)$, is the smallest $n$ for which $G$ admits a faithful complex representation—equivalently,
\[
\mathrm{rdim}(G) = \min\{\,n \mid G \hookrightarrow \mathrm{GL}(n,\mathbb C)\,\} = \min\{\chi(1) \mid \chi \text{ is a faithful character}\}
\]
[2102.01463, 2308.01612, 2206.10752].

## 2. Characterizations, Criteria, and Homological Invariants

Auslander's original characterization $\mathrm{rep.dim}(A) \leq 2$ if and only if $A$ is of finite representation type is foundational: when $\mathrm{rep.dim}(A) \geq 3$ the category $\mathrm{mod}\,A$ contains infinitely many indecomposables [1006.2901, 1510.06443]. The representation dimension thus quantifies "how far" an algebra's module category is from being finite or "tame".

Equivalent formulations hinge on the existence, for every module $X$, of right (and left) $\mathrm{add}\,M$-resolutions of prescribed length. If every $X$ admits a resolution of length at most $d$ by summands of $M$, then $\mathrm{gl.dim}\,\End_A(M)\leq d+2$ [1710.02567, 1107.1861]. Generators and cogenerators are essential: if $M$ is not both, the minimal global dimension may not realize $\mathrm{rep.dim}(A)$.

The principle extends to triangulated categories: if $A$ is self-injective, then the Rouquier dimension of the stable module category $\underline{\mathrm{mod}}\,A$ satisfies
\[
\dim(\underline{\mathrm{mod}}\,A) + 2 \leq \mathrm{rep.dim}(A)
\]
[1603.00953, 1204.6510].

For groups, $\mathrm{rdim}(G)$ can be characterized via character theory: the minimal sum of degrees of irreducible characters whose intersecting kernels are trivial determines the representation dimension [2102.01463, 2308.01612, 2206.10752].

## 3. Sharp Bounds and Structure Theorems

The interaction between structural properties and representation dimension is central.

- For algebras, the **torsionless-finite** property, that indecomposable projectives have only finitely many indecomposable submodules, ensures $\mathrm{rep.dim}(A)\leq 3$ [1006.2901, 1107.1861].

- **Tame** algebras—those of polynomial growth and controlled module type—often satisfy $\mathrm{rep.dim}(A)\le 3$ under additional hypotheses such as strong simple connectedness or specific homological features (e.g., being a multicoil algebra or of Euclidean type) [1006.2901, 1510.06443, 1704.00612, 1607.00305].

- For **selfinjective algebras** of wild tilted type, Euclidean type, or those admitting an acyclic generalized standard Auslander–Reiten component, $\mathrm{rep.dim}(A)=3$ holds via explicit Auslander generator constructions [1510.06443, 1607.00305, 1710.02567].

- The **representation dimension of exterior algebras** is $n+1$ for dimension $n$ [1603.00953].

- For **tensor products** of $n$ representation-infinite path algebras of bipartite quivers, the exact value is $\mathrm{rep.dim}(A_1\otimes_k \dots \otimes_k A_n) = n+2$ [1107.1861].

For finite groups:
- The prime result is $\mathrm{rdim}(G) \le \sqrt{|G|}$, with equality only for some Camina 2-groups with elementary abelian center, which are fully classified in [2102.01463]. Typically, $\mathrm{rdim}(G)$ is substantially smaller than $|G|$.
- For $p$-groups with cyclic center, $\mathrm{rdim}(G)=p^r$ for extraspecial groups of order $p^{2r+1}$ [2308.01612].
- For direct products of suitable groups, $\delta_{\mathrm{irr}}(G_1\times G_2) = \delta_{\mathrm{irr}}(G_1)\delta_{\mathrm{irr}}(G_2)$, and in many cases $\delta(G_1\times G_2)=\delta(G_1)+\delta(G_2)$ [2308.01612].

## 4. Methods of Proof and Construction

Homological techniques are central in algebra:

- **One-point extensions and wildness**: Detailed analysis of module supports, convex subcategory structure, and wild one-point extensions are used to show that if a family of submodules is infinite, one can construct a wild algebra, contradicting tameness; thus finiteness of submodules is enforced [1006.2901].
- **Explicit Auslander generators**: For cluster-concealed, wild tilted, and cluster-tilted algebras, generators are built from complete slices, regular summands, and covering theory in the repetitive category framework [1102.1048, 1607.00305, 1705.06357].
- **Radical embeddings**: Inductive construction of radical embeddings into representation-finite algebras enables bounds on $\mathrm{rep.dim}$ for special multiserial algebras and their self-injective variants [1704.00612].

For groups, character-theoretic decompositions and Clifford theory underlie both lower and upper bounds, while consideration of central quotients classifies minimal faithful dimension [2102.01463, 2206.10752].

## 5. Applications and Representative Examples

### Algebras

- **Strongly simply connected algebras of polynomial growth** are all torsionless-finite, so $\mathrm{rep.dim}(A)\leq 3$ [1006.2901].
- **Hecke algebras and group algebras**: For Hecke algebras of type $A$ (with weight $w$), $w+1 \le \mathrm{rep.dim}B \le 2w$ for blocks, and similar bounds for symmetric group algebras in characteristic $p$ [1204.6510, 1011.3740].
- **Special multiserial algebras** have $\mathrm{rep.dim}(A)\leq 3$; this sharp restriction implies the finitistic dimension conjecture holds for all such algebras [1704.00612].
- **Exterior algebras**: $\mathrm{rep.dim}(A)=n+1$ for $\Lambda=\wedge V$ with $\dim_k V=n$ [1603.00953].

### Groups

- Elementary abelian $p$-groups $(C_p)^n$ have $\mathrm{rdim}((C_p)^n)=n$ [2206.10752].
- Direct product behavior: For suitable group classes, $\delta(G_1\times G_2) = \delta(G_1)+\delta(G_2)$, or multiplicative in cases with coprime centers [2308.01612].
- The alternating and symmetric groups of degree $n$ both have representation dimension $n-1$ [2308.01612, 2206.10752].
- The minimal faithful dimension is tightly classified in degrees 1, 2, and 3 [2206.10752].

## 6. Broader Implications and Open Problems

The representation dimension links algebraic and geometric invariants:

- **Finitistic dimension conjecture**: $\mathrm{rep.dim}(A)\le3$ implies finite finitistic dimension, confirmed for large classes including all special multiserial, cluster-concealed, and tame tilting algebras [1006.2901, 1704.00612, 1102.1048].
- **Socle equivalence**: For selfinjective algebras, representation dimension is invariant under socle equivalence; thus, calculations extend to entire socle-equivalent classes [1710.02567].
- For groups, relations to **essential dimension** are established: $\mathrm{ed}(G)\le\mathrm{rdim}(G)$, with equality in wide classes such as $p$-groups [2102.01463].
- Open questions persist for the exact boundaries of $\mathrm{rep.dim}=3$ in tame versus wild types, explicit determination for Hecke algebra wild blocks, and precise growth rates as algebraic structures are composed (e.g., via tensor product) [1510.06443, 1011.3740, 1107.1861].

## 7. Computational Aspects and Algorithms

For finite groups, the computation of $\delta(G)$ is algorithmically nontrivial, particularly since the minimal sum of degrees of irreducibles with trivial intersection of kernels involves subset sums over the set of irreducible characters. Efficient GAP routines for small groups and heuristic methods for leveraging character tables are provided [2308.01612].

For algebras, the construction and verification of explicit Auslander generators, radical embeddings, and the analysis of module resolutions depend on combinatorial and homological algebraic algorithms; nonetheless, the underlying complexity grows rapidly with the size of the quiver or the rank of the algebra.

---

In summary, the representation dimension functions as a numeric and homological measure of structural complexity in both algebraic and group-theoretic contexts, affording a unifying perspective on module categories, block theory, and the minimal size of linear manifestations of abstract algebraic entities [1006.2901, 1510.06443, 1102.1048, 1603.00953, 2102.01463, 2308.01612, 2206.10752, 1704.00612, 1107.1861, 1710.02567].

Source: https://www.emergentmind.com/topics/representation-dimension