mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the tensor representations of the

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

are those that are obtained by taking finitely many

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

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 defi ...

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 pow ...

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 ...

). These coincide with the

rational representation In mathematics, in the representation theory of algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebrai ...

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 fai ...

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 or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equi ...

s. The

classical group In mathematics, the classical groups are defined as the special linear groups over the reals \mathbb, the complex numbers \mathbb and the quaternions \mathbb together with special automorphism groups of Bilinear form#Symmetric, skew-symmetric an ...

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 gr ...

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