In mathematics, a **representation** is a very general relationship that expresses similarities (or equivalences) between mathematical objects or structures.^{[1]} 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 *y _{i}* conform, in some consistent way, to those existing among the corresponding represented objects

*x*. More specifically, given a set

_{i}*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 (such as group homomorphism in group theory).

^{[2]}

^{[3]}

## Contents

## Representation theory[edit]

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.^{[3]}

## Other examples[edit]

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[edit]

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.^{[4]} 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 *n*^{2}/4.^{[5]}

Representing a graph by such algebraic structures as its adjacency matrix and Laplacian matrix gives rise to the field of spectral graph theory.^{[6]}

### Order theory[edit]

Dual to the observation above that every graph is an intersection graph is the fact that every partially ordered set (a.k.a., poset) is isomorphic to a collection of sets ordered by the containment (or inclusion) relation ⊆. Some posets that arise as the containment orders for natural classes of objects include the Boolean lattices and the orders of dimension *n*.^{[7]}

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.^{[8]}

There are also geometric representations that are not based on containment. Indeed, one of the best studied classes among these are the interval orders,^{[9]} 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 [*x*_{1}, *x*_{2}], such that for any *y* and *z* in the poset, *y* is below *z* if and only if *y*_{2} < *z*_{1}.

### Polysemy[edit]

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* (one of the things that *Y* is trying to tell us), this phenomenon is called **polysemy**—a term borrowed from linguistics. Some examples of polysemy include:

**intersection polysemy**—pairs of graphs*G*_{1}and*G*_{2}on a common vertex set*V*that can be simultaneously represented by a single collection of sets*S*, such that any distinct vertices_{v}*u*and*w*in*V*are adjacent in*G*_{1}, if and only if their corresponding sets intersect (*S*∩_{u}*S*≠ Ø ), and are adjacent in_{w}*G*_{2}if and only if the complements do (*S*_{u}^{C}∩*S*_{w}^{C}≠ Ø ).^{[10]}

**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*G*_{1}and*G*_{2}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*G*_{1,}if and only if there is a vertex*w*such that both*uw*and*vw*are arcs in*D*, and are adjacent in*G*_{2,}if and only if there is a vertex*w*such that both*wu*and*wv*are arcs in*D*.^{[11]}

**interval polysemy**—pairs of posets*P*_{1}and*P*_{2}on a common ground set that can be simultaneously represented by a single collection of real intervals, that is an interval-order representation of*P*_{1}and an interval-containment representation of*P*_{2}.^{[12]}

## See also[edit]

## References[edit]

**^**"The Definitive Glossary of Higher Mathematical Jargon — Mathematical Representation".*Math Vault*. 2019-08-01. Retrieved 2019-12-07.**^**Weisstein, Eric W. "Group Representation".*mathworld.wolfram.com*. Retrieved 2019-12-07.- ^
^{a}^{b}Teleman, Constantin. "Representation Theory" (PDF).*math.berkeley.edu*. Retrieved 2019-12-07. **^***McKee, Terry A.; McMorris, F. R. (1999),*Topics in Intersection Graph Theory*, SIAM Monographs on Discrete Mathematics and Applications, Philadelphia: Society for Industrial and Applied Mathematics, doi:10.1137/1.9780898719802, ISBN 978-0-89871-430-2, MR 1672910**^**Erdős, Paul; Goodman, A. W.; Pósa, Louis (1966), "The representation of a graph by set intersections",*Canadian Journal of Mathematics*,**18**(1): 106–112, CiteSeerX 10.1.1.210.6950, doi:10.4153/cjm-1966-014-3, MR 0186575**^***Biggs, Norman (1994),*Algebraic Graph Theory*, Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-45897-9, MR 1271140**^***Trotter, William T. (1992),*Combinatorics and Partially Ordered Sets: Dimension Theory*, Johns Hopkins Series in the Mathematical Sciences, Baltimore: The Johns Hopkins University Press, ISBN 978-0-8018-4425-6, MR 1169299**^***Scheinerman, Edward (1991), "A note on planar graphs and circle orders",*SIAM Journal on Discrete Mathematics*,**4**(3): 448–451, doi:10.1137/0404040, MR 1105950**^***Fishburn, Peter C. (1985),*Interval Orders and Interval Graphs: A Study of Partially Ordered Sets*, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, ISBN 978-0-471-81284-5, MR 0776781**^***Tanenbaum, Paul J. (1999), "Simultaneous intersection representation of pairs of graphs",*Journal of Graph Theory*,**32**(2): 171–190, doi:10.1002/(SICI)1097-0118(199910)32:2<171::AID-JGT7>3.0.CO;2-N, MR 1709659**^***Fischermann, Miranca; Knoben, Werner; Kremer, Dirk; Rautenbachh, Dieter (2004), "Competition polysemy",*Discrete Mathematics*,**282**(1–3): 251–255, doi:10.1016/j.disc.2003.11.014, MR 2059526**^***Tanenbaum, Paul J. (1996), "Simultaneous representation of interval and interval-containment orders",*Order*,**13**(4): 339–350, CiteSeerX 10.1.1.53.8988, doi:10.1007/BF00405593, MR 1452517