# Representation (mathematics)

In mathematics, representation is a very general relationship that expresses similarities between objects. 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 yi conform in some consistent way to those existing among the corresponding represented objects xi. Somewhat more formally, for a set Π of properties and relations, a Π-representation of some structure X is a structure Y that is the image of X under a homomorphism that preserves Π. The label representation is sometimes also applied to the homomorphism itself.

## Representation theory

Perhaps the most well-developed example of this general notion is the subfield of abstract algebra called representation theory, which studies the representing of elements of algebraic structures by linear transformations of vector spaces.

## Other examples

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.

### Graph theory

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 (or, more precisely, non-disjointness) is a symmetric relation. This gives rise to the study of intersection graphs for innumerable families of sets.[1] One foundational result here, due to Paul Erdős and 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.[2]

Representing a graph by such algebraic structures as its adjacency matrix and Laplacian matrix gives rise to the field of spectral graph theory.[3]

### Order theory

Dual 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 containment (or inclusion) relation ⊆. Among the posets that arise as the containment orders for natural classes of objects are the Boolean lattices and the orders of dimension n.[4]

Many partial orders arise from (and thus can be represented by) 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.[5]

There are also geometric representations that are not based on containment. Indeed, one of the best studied classes among these are the interval orders,[6] 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 [x1, x2] such that for any y and z in the poset, y is below z if and only if y2 < z1.

### Polysemy

Under certain circumstances, a single function f:XY 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—one of the things that Y is trying to tell us—this phenomenon is called polysemy, a term borrowed from linguistics. Examples 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 ( SuSw ≠ Ø ), and
are adjacent in G2 if and only if the complements do ( SuCSwC ≠ Ø ).[7]
• competition polysemy—motivated by the study of ecological food 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 graph D 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.[8]
• 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.[9]