Contraindicated representation

In mathematics , the contra-related representation or dual representation is an important aid in linear algebra , projective geometry and representation theory .

definition

For a given representation

${\displaystyle \rho \colon G\to \operatorname {GL} (V)}$

one can use the dual representation

${\displaystyle \rho ^{*}\colon G\to \operatorname {GL} (V^{*})}$

in the dual vector space define by ${\displaystyle V^{*}}$

${\displaystyle \left(\rho ^{*}(s)\nu \right)(v)=\nu \left(\rho (s^{-1})v\right)}$

for everyone and ${\displaystyle s\in G,v\in V}$ ${\displaystyle \nu \in V^{*}.}$

With this definition, the natural pairing between and applies ${\displaystyle \langle \nu ,v\rangle :=\nu (v)}$ ${\displaystyle V^{*}}$ ${\displaystyle V:}$

${\displaystyle \langle \rho ^{*}(s)(\nu ),\rho (s)(v)\rangle =\langle \nu ,v\rangle }$ for all ${\displaystyle s\in G,v\in V,\nu \in V^{*}.}$

Representation through matrices

After selecting a base and the canonical dual basis is by a matrix and by the transpose of the inverse matrix described, ie . ${\displaystyle \rho (g)}$ ${\displaystyle A}$ ${\displaystyle \rho ^{*}(g)}$ ${\displaystyle \rho ^{*}(g)=(A^{T})^{-1}}$

Proof : Let be a basis of and the dual basis of . Be ${\displaystyle v_{1},\ldots ,v_{n}}$ ${\displaystyle V}$ ${\displaystyle v_{1}^{*},\ldots ,v_{n}^{*}}$ ${\displaystyle V^{*}}$

${\displaystyle gv_{i}=\sum _{j}a_{ij}(g)v_{j}\in V}$

and

${\displaystyle gv_{j}^{*}=\sum _{i}b_{ij}(g)v_{i}^{*}\in V^{*}}$,

then

${\displaystyle b_{ji}(g)=(gv_{j})^{*}v_{i}=v_{j}^{*}(g^{-1}v_{i})=v_{j}^{*}\left(\sum _{k}a_{ik}(g^{-1})v_{k}\right)=a_{ij}(g^{-1})}$.

Unitary representations

If is a unitary representation , then is the complex conjugate representation . ${\displaystyle \rho }$ ${\displaystyle \rho ^{*}}$ ${\displaystyle {\overline {\rho }}}$

example

Let and be the representation of defined by ${\displaystyle G=\mathbb {Z} /3\mathbb {Z} }$ ${\displaystyle \rho \colon \mathbb {Z} /3\mathbb {Z} \to \operatorname {GL} _{2}(\mathbb {C} )}$ ${\displaystyle \mathbb {Z} /3\mathbb {Z} }$

${\displaystyle \rho ({\overline {0}})={\text{Id}},\,\,\rho ({\overline {1}})=\left({\begin{array}{cc}\cos({\frac {2\pi }{3}})&-\sin({\frac {2\pi }{3}})\\\sin({\frac {2\pi }{3}})&\cos({\frac {2\pi }{3}})\end{array}}\right),\,\,{\text{ und }}\,\,\rho ({\overline {2}})=\left({\begin{array}{cc}\cos({\frac {4\pi }{3}})&-\sin({\frac {4\pi }{3}})\\\sin({\frac {4\pi }{3}})&\cos({\frac {4\pi }{3}})\end{array}}\right).}$

Then the dual representation is given by: ${\displaystyle \rho ^{*}\colon \mathbb {Z} /3\mathbb {Z} \to \operatorname {GL} ((\mathbb {C} ^{2})^{*})}$

${\displaystyle \rho ^{*}({\overline {0}})={\text{Id}},\,\,\rho ^{*}({\overline {1}})=\left({\begin{array}{cc}\cos({\frac {4\pi }{3}})&\sin({\frac {4\pi }{3}})\\-\sin({\frac {4\pi }{3}})&\cos({\frac {4\pi }{3}})\end{array}}\right),\,\,{\text{ und }}\,\,\rho ^{*}({\overline {2}})=\left({\begin{array}{cc}\cos({\frac {2\pi }{3}})&\sin({\frac {2\pi }{3}})\\-\sin({\frac {2\pi }{3}})&\cos({\frac {2\pi }{3}})\end{array}}\right).}$

literature

• Bröcker, Theodor; tom Dieck, Tammo: Representations of compact Lie groups. Graduate Texts in Mathematics, 98. Springer-Verlag, New York, 1985. ISBN 0-387-13678-9
• Fulton, William; Harris, Joe: Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag, 1991. ISBN 978-0-387-97495-8