In
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 \otime ...
s of the
fundamental representation and its dual. The irreducible factors of such a representation are also called tensor representations, and can be obtained by applying
Schur functors (associated to
Young tableaux). 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 which is compatible with its structure as an algebraic variety. Thus the study of algeb ...
s of the general linear group.
More generally, a
matrix group 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 of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
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 representations.
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 s ...
s, like the
symplectic group, 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