mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the tensor representations of the

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

are those that are obtained by taking finitely many

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...

s of the

fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defini ...

and its dual. The irreducible factors of such a representation are also called tensor representations, and can be obtained by applying

Schur functor In mathematics, especially in the field of representation theory, Schur functors (named after Issai Schur) are certain functors from the category of modules over a fixed commutative ring to itself. They generalize the constructions of exterior po ...

s (associated to

Young tableaux In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and ...

). These coincide with the

rational representation In mathematics, in the representation theory of algebraic groups, a linear representation of an algebraic group is said to be rational if, viewed as a map from the group to the general linear group, it is a rational map In mathematics, in particu ...

s of the general linear group. More generally, a

matrix group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a fait ...

is any subgroup of the general linear group. A tensor representation of a matrix group is any representation that is contained in a tensor representation of the general linear group. For example, the

orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

O(''n'') admits a tensor representation on the space of all trace-free symmetric tensors of order two. For orthogonal groups, the tensor representations are contrasted with the

spin representation In mathematics, the spin representations are particular projective representations of the orthogonal group, orthogonal or special orthogonal groups in arbitrary dimension and metric signature, signature (i.e., including indefinite orthogonal groups ...

s. The

classical group In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or ske ...

s, like the

symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic grou ...

, have the property that all finite-dimensional representations are tensor representations (by Weyl's construction), while other representations (like the

metaplectic representation In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil. A natural extension of the representation leads to a semigroup of contr ...

) exist in infinite dimensions.

References

* {{citation, author1=Roe Goodman, author2=Nolan Wallach, title=Symmetry, representations, and invariants, publisher=Springer, year=2009, chapters 9 and 10. * Bargmann, V., & Todorov, I. T. (1977). Spaces of analytic functions on a complex cone as carriers for the symmetric tensor representations of SO(''n''). Journal of Mathematical Physics, 18(6), 1141–1148. Tensors