In mathematics, a representation is a very general relationship that expresses similarities between mathematical objects or structures. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the representing objects yiconform, in some consistent way, to those existing among the corresponding represented objects xi. More specifically, given a set Π of properties and relations, a Π-representation of some structure X is a structure Y that is the image ofX under a homomorphism that preserves Π. The label representation is sometimes also applied to the homomorphism itself.
Although the term representation theory is well established in the algebraic sense discussed above, there are many other uses of the term representation throughout mathematics.
An active area of graph theory is the exploration of isomorphisms between graphs and other structures. A key class of such problems stems from the fact that, like adjacency in undirected graphs, intersection of sets is a symmetric relation. This gives rise to the study of intersection graphs for innumerable families of sets. One foundational result here, due to Paul Erdős and his colleagues, is that every n-vertex graph may be represented in terms of intersection among subsets of a set of size no more than n2/4. Representing a graph by such algebraic structures as its adjacency matrix and Laplacian matrix gives rise to the field of spectral graph theory.
Order theory
to the observation above that every graph is an intersection graph is the fact that every partially ordered set is isomorphic to a collection of sets ordered by the inclusion relation ⊆. Some posets that arise as the inclusion orders for natural classes of objects include the Boolean lattices and the orders of dimension n. Many partial orders arise from collections of geometric objects. Among them are the n-ball orders. The 1-ball orders are the interval-containment orders, and the 2-ball orders are the so-called circle orders—the posets representable in terms of containment among disks in the plane. A particularly nice result in this field is the characterization of the planar graphs, as those graphs whose vertex-edge incidence relations are circle orders. There are also geometric representations that are not based on inclusion. Indeed, one of the best studied classes among these are the interval orders, which represent the partial order in terms of what might be called disjoint precedence of intervals on the real line: each element x of the poset is represented by an interval , such that for any y and z in the poset, y is below zif and only ify2 < z1.
Under certain circumstances, a single function f : X → Y is at once an isomorphism from several mathematical structures on X. Since each of those structures may be thought of, intuitively, as a meaning of the image Y, this phenomenon is called polysemy—a term borrowed from linguistics. Some examples of polysemy include:
intersection polysemy—pairs of graphs G1 and G2 on a common vertex set V that can be simultaneously represented by a single collection of sets Sv, such that any distinct vertices u and w in V are adjacent in G1, if and only if their corresponding sets intersect, and are adjacent in G2 if and only if the complements do.
competition polysemy—motivated by the study of ecologicalfood webs, in which pairs of species may have prey in common or have predators in common. A pair of graphs G1 and G2 on one vertex set is competition polysemic, if and only if there exists a single directed graphD on the same vertex set, such that any distinct vertices u and v are adjacent in G1, if and only if there is a vertex w such that both uw and vw are arcs in D, and are adjacent in G2, if and only if there is a vertex w such that both wu and wv are arcs in D.
interval polysemy—pairs of posets P1 and P2 on a common ground set that can be simultaneously represented by a single collection of real intervals, that is an interval-order representation of P1 and an interval-containment representation of P2.
