DUAL REPRESENTATION

In mathematics, if ''G'' is a group and ρ is a linear representation of it on the vector space ''V'', then the 'dual representation'
:
ho}
is defined over the dual vector space as follows^[1]:
:
ho}(g) is the transpose of ρ(''g''⁻¹)
for all ''g'' in ''G''. Then
ho} is also a representation, as may be checked explicitly. The dual representation is also known as the ''contragredient representation''.
If $mathfrak\{g\}$ is a Lie algebra and ρ is a representation of it over the vector space ''V'', then the dual representation
ho} is defined over the dual vector space as follows^[2]:
:
ho}(u) is the transpose of −ρ(u) for all u in $mathfrak\{g\}$.
:
ho} is also a representation, as you may check explicitly.
For a unitary representation, the conjugate representation and the dual representation coincide, up to equivalence of representations.

Contents

Generalization

See also

References

Generalization

A general ring module does not admit a dual representation. Modules of Hopf algebras do, however.

See also

★ Complex conjugate representation

★ Kirillov Character Formula

References

1. Lecture 1 of William Fulton & Joe Harris, 'Representation Theory. A First Course', Graduate Texts in Mathematics, 129, Springer-Verlag, 1991. ISBN: 0-387-97527-6; 0-387-97495-4
2. Lecture 8 of William Fulton & Joe Harris, 'Representation Theory. A First Course', Graduate Texts in Mathematics, 129, Springer-Verlag, 1991. ISBN: 0-387-97527-6; 0-387-97495-4