# Intertwining operator

A continuous linear operator $ T: E _ {1} \rightarrow E _ {2} $
such that $ T \pi _ {1} ( x) = \pi _ {2} ( x) T $,
where $ \pi _ {1} $
and $ \pi _ {2} $
are mappings of a set $ X $
into two topological vector spaces $ E _ {1} $
and $ E _ {2} $
and $ x \in X $.
This concept is especially fruitful in the case when $ X $
is a group or an algebra and $ \pi _ {1} , \pi _ {2} $
are representations of $ X $.
The set of intertwining operators forms the space $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $,
which is a subspace of the space of all continuous linear mappings from $ E _ {1} $
to $ E _ {2} $.
If $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) = ( 0) $
and $ \mathop{\rm Hom} ( \pi _ {2} , \pi _ {1} ) = ( 0) $,
then $ \pi _ {1} $
and $ \pi _ {2} $
are called disjoint representations. If $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $
contains an operator that defines an isomorphism of $ E _ {1} $
and $ E _ {2} $,
then $ \pi _ {1} $
and $ \pi _ {2} $
are equivalent. If $ E _ {1} , E _ {2} $
are locally convex spaces, if $ E _ {1} ^ {*} $
and $ E _ {2} ^ {*} $
are their adjoints, and if $ \pi _ {1} ^ {*} $
and $ \pi _ {2} ^ {*} $
are the representations contragredient to $ \pi _ {1} $
and $ \pi _ {2} $,
respectively (cf. Contragredient representation), then for any $ T \in \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $,
the operator $ T ^ {*} $
is contained in $ \mathop{\rm Hom} ( \pi _ {2} ^ {*} , \pi _ {1} ^ {*} ) $.
If $ \pi _ {1} $
and $ \pi _ {2} $
are finite-dimensional or unitary representations and $ \pi _ {1} $
is irreducible, then $ \pi _ {2} $
admits a subrepresentation equivalent to $ \pi _ {1} $
if and only if $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) \neq ( 0) $.
See also Intertwining number.

#### References

[1] | A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) |

[2] | A.I. Shtern, "Theory of group representations" , Springer (1982) (Translated from Russian) |

**How to Cite This Entry:**

Intertwining operator.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Intertwining_operator&oldid=47402