Representationsgraphs

Representationsgraphs are a conceptual tool used to visualize and analyze relationships among representations of algebraic structures such as groups, algebras, or categories. In this framework, each vertex represents an isomorphism class of a representation, and edges encode relations between representations, such as morphisms, subquotients, or how representations transform under a fixed operation like tensoring, induction, or restriction.

Formal definition can vary by context, but a common approach is: given a base structure A and a chosen collection Rep of representations closed under a chosen operation Op, construct a graph where there is a vertex for each [V] in Rep. For representations [V] and [W], an edge [V] → [W] is added, labeled by the operation type and possibly a multiplicity, whenever W appears in Op(V) with that multiplicity. Edges may be directed and can include weights, colors, or labels to distinguish different operations. The graph may be finite or infinite, depending on Rep and Op, and may be augmented with additional data such as edge colors to distinguish morphisms, subquotients, or decomposition pathways.

Variants of representationsgraphs reflect different perspectives. If edges reflect subquotient relations, the graph resembles a lattice of subobjects; if edges encode tensor product decompositions with a fixed representation, it is a tensor-graph. Undirected variants highlight mutual relationships, while hyperedges can encode multi-way decompositions or functorial images.

Applications include teaching and communicating decomposition patterns in representation theory, exploring how representations behave under functors, and modeling spaces of representations in computer science or data modeling contexts. Representationsgraphs are related to quivers and standard tools in representation theory, serving as a flexible visualization framework for studying the structure and interconnections of representations. Example: if a fixed representation T satisfies S1 ⊗ T ≅ S1 ⊕ S2 for simple S1, the Op = ⊗T representationsgraph would show edges from S1 to S1 and from S1 to S2 with appropriate labels.