Representation Dimension in Algebras & Groups

• Representation Dimension is an invariant that quantifies the complexity of module categories in algebras and the minimal faithful representation size in groups.
• It employs homological methods such as Auslander generators, resolution techniques, and radical embeddings to compute and bound the invariant.
• Understanding this invariant is crucial for classifying finite representation types, analyzing character theory, and advancing computational approaches in representation theory.

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 $D(A_A) = \Hom_k(A_A, k) \in \mathrm{add}\,M$ (Assem et al., 2010, Chaio et al., 2011, Assem et al., 2015). 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 $k$0 for which $k$1 admits a faithful complex representation—equivalently,

$k$2

(Moretó, 2021, Kaur et al., 2023, Cohen, 2022).

2. Characterizations, Criteria, and Homological Invariants

Auslander's original characterization $k$3 if and only if $k$4 is of finite representation type is foundational: when $k$5 the category $k$6 contains infinitely many indecomposables (Assem et al., 2010, Assem et al., 2015). 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 $k$7, of right (and left) $k$8-resolutions of prescribed length. If every $k$9 admits a resolution of length at most $\mathrm{rep.dim}(A)$0 by summands of $\mathrm{rep.dim}(A)$1, then $\mathrm{rep.dim}(A)$2 (Assem et al., 2017, Ringel, 2011). Generators and cogenerators are essential: if $\mathrm{rep.dim}(A)$3 is not both, the minimal global dimension may not realize $\mathrm{rep.dim}(A)$4.

The principle extends to triangulated categories: if $\mathrm{rep.dim}(A)$5 is self-injective, then the Rouquier dimension of the stable module category $\mathrm{rep.dim}(A)$6 satisfies

$\mathrm{rep.dim}(A)$7

(Zheng et al., 2016, Bergh et al., 2012).

For groups, $\mathrm{rep.dim}(A)$8 can be characterized via character theory: the minimal sum of degrees of irreducible characters whose intersecting kernels are trivial determines the representation dimension (Moretó, 2021, Kaur et al., 2023, Cohen, 2022).

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)$9 (Assem et al., 2010, Ringel, 2011).
• Tame algebras—those of polynomial growth and controlled module type—often satisfy $\mathrm{rep.dim}(A) = \inf\left\{\,\mathrm{gl.dim}\,\End_A(M)\;\middle|\;M \text{ a generator-cogenerator of } \mathrm{mod}\,A\,\right\}$0 under additional hypotheses such as strong simple connectedness or specific homological features (e.g., being a multicoil algebra or of Euclidean type) (Assem et al., 2010, Assem et al., 2015, Schroll, 2017, Assem et al., 2016).
• For selfinjective algebras of wild tilted type, Euclidean type, or those admitting an acyclic generalized standard Auslander–Reiten component, $\mathrm{rep.dim}(A) = \inf\left\{\,\mathrm{gl.dim}\,\End_A(M)\;\middle|\;M \text{ a generator-cogenerator of } \mathrm{mod}\,A\,\right\}$1 holds via explicit Auslander generator constructions (Assem et al., 2015, Assem et al., 2016, Assem et al., 2017).
• The representation dimension of exterior algebras is $\mathrm{rep.dim}(A) = \inf\left\{\,\mathrm{gl.dim}\,\End_A(M)\;\middle|\;M \text{ a generator-cogenerator of } \mathrm{mod}\,A\,\right\}$2 for dimension $\mathrm{rep.dim}(A) = \inf\left\{\,\mathrm{gl.dim}\,\End_A(M)\;\middle|\;M \text{ a generator-cogenerator of } \mathrm{mod}\,A\,\right\}$3 (Zheng et al., 2016).
• For tensor products of $\mathrm{rep.dim}(A) = \inf\left\{\,\mathrm{gl.dim}\,\End_A(M)\;\middle|\;M \text{ a generator-cogenerator of } \mathrm{mod}\,A\,\right\}$4 representation-infinite path algebras of bipartite quivers, the exact value is $\mathrm{rep.dim}(A) = \inf\left\{\,\mathrm{gl.dim}\,\End_A(M)\;\middle|\;M \text{ a generator-cogenerator of } \mathrm{mod}\,A\,\right\}$5 (Ringel, 2011).

For finite groups:

• The prime result is $\mathrm{rep.dim}(A) = \inf\left\{\,\mathrm{gl.dim}\,\End_A(M)\;\middle|\;M \text{ a generator-cogenerator of } \mathrm{mod}\,A\,\right\}$6, with equality only for some Camina 2-groups with elementary abelian center, which are fully classified in (Moretó, 2021). Typically, $\mathrm{rep.dim}(A) = \inf\left\{\,\mathrm{gl.dim}\,\End_A(M)\;\middle|\;M \text{ a generator-cogenerator of } \mathrm{mod}\,A\,\right\}$7 is substantially smaller than $\mathrm{rep.dim}(A) = \inf\left\{\,\mathrm{gl.dim}\,\End_A(M)\;\middle|\;M \text{ a generator-cogenerator of } \mathrm{mod}\,A\,\right\}$8.
• For $\mathrm{rep.dim}(A) = \inf\left\{\,\mathrm{gl.dim}\,\End_A(M)\;\middle|\;M \text{ a generator-cogenerator of } \mathrm{mod}\,A\,\right\}$9-groups with cyclic center, $M$0 for extraspecial groups of order $M$1 (Kaur et al., 2023).
• For direct products of suitable groups, $M$2, and in many cases $M$3 (Kaur et al., 2023).

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 (Assem et al., 2010).
• 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 (Chaio et al., 2011, Assem et al., 2016, Chaio et al., 2017).
• Radical embeddings: Inductive construction of radical embeddings into representation-finite algebras enables bounds on $M$4 for special multiserial algebras and their self-injective variants (Schroll, 2017).

For groups, character-theoretic decompositions and Clifford theory underlie both lower and upper bounds, while consideration of central quotients classifies minimal faithful dimension (Moretó, 2021, Cohen, 2022).

5. Applications and Representative Examples

Algebras

• Strongly simply connected algebras of polynomial growth are all torsionless-finite, so $M$5 (Assem et al., 2010).
• Hecke algebras and group algebras: For Hecke algebras of type $M$6 (with weight $M$7), $M$8 for blocks, and similar bounds for symmetric group algebras in characteristic $M$9 (Bergh et al., 2012, Bergh et al., 2010).
• Special multiserial algebras have $A_A \in \mathrm{add}\,M$0; this sharp restriction implies the finitistic dimension conjecture holds for all such algebras (Schroll, 2017).
• Exterior algebras: $A_A \in \mathrm{add}\,M$1 for $A_A \in \mathrm{add}\,M$2 with $A_A \in \mathrm{add}\,M$3 (Zheng et al., 2016).

Groups

• Elementary abelian $A_A \in \mathrm{add}\,M$4-groups $A_A \in \mathrm{add}\,M$5 have $A_A \in \mathrm{add}\,M$6 (Cohen, 2022).
• Direct product behavior: For suitable group classes, $A_A \in \mathrm{add}\,M$7, or multiplicative in cases with coprime centers (Kaur et al., 2023).
• The alternating and symmetric groups of degree $A_A \in \mathrm{add}\,M$8 both have representation dimension $A_A \in \mathrm{add}\,M$9 (Kaur et al., 2023, Cohen, 2022).
• The minimal faithful dimension is tightly classified in degrees 1, 2, and 3 (Cohen, 2022).

6. Broader Implications and Open Problems

The representation dimension links algebraic and geometric invariants:

• Finitistic dimension conjecture: $D(A_A) = \Hom_k(A_A, k) \in \mathrm{add}\,M$0 implies finite finitistic dimension, confirmed for large classes including all special multiserial, cluster-concealed, and tame tilting algebras (Assem et al., 2010, Schroll, 2017, Chaio et al., 2011).
• Socle equivalence: For selfinjective algebras, representation dimension is invariant under socle equivalence; thus, calculations extend to entire socle-equivalent classes (Assem et al., 2017).
• For groups, relations to essential dimension are established: $D(A_A) = \Hom_k(A_A, k) \in \mathrm{add}\,M$1, with equality in wide classes such as $D(A_A) = \Hom_k(A_A, k) \in \mathrm{add}\,M$2-groups (Moretó, 2021).
• Open questions persist for the exact boundaries of $D(A_A) = \Hom_k(A_A, k) \in \mathrm{add}\,M$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) (Assem et al., 2015, Bergh et al., 2010, Ringel, 2011).

7. Computational Aspects and Algorithms

For finite groups, the computation of $D(A_A) = \Hom_k(A_A, k) \in \mathrm{add}\,M$4 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 (Kaur et al., 2023).

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.

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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 (Assem et al., 2010, Assem et al., 2015, Chaio et al., 2011, Zheng et al., 2016, Moretó, 2021, Kaur et al., 2023, Cohen, 2022, Schroll, 2017, Ringel, 2011, Assem et al., 2017).