# Intertwining number

The dimension $ c ( \pi _ {1} , \pi _ {2} ) $
of the space $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $
of intertwining operators (cf. Intertwining operator) for two mappings $ \pi _ {1} $
and $ \pi _ {2} $
of a set $ X $
into topological vector spaces $ E _ {1} $
and $ E _ {2} $,
respectively. The concept of the intertwining number is especially fruitful in the case when $ X $
is a group or an algebra and $ \pi _ {1} , \pi _ {2} $
are representations of $ X $.
Even for finite-dimensional representations, $ c ( \pi _ {1} , \pi _ {2} ) \neq c ( \pi _ {2} , \pi _ {1} ) $
in general, but for finite-dimensional representations $ \pi _ {1} $,
$ \pi _ {2} $,
$ \pi _ {3} $
the following relations hold:

$$ c ( \pi _ {1} \oplus \pi _ {2} , \pi _ {3} ) = \ c ( \pi _ {1} , \pi _ {3} ) + c ( \pi _ {2} , \pi _ {3} ); $$

$$ c ( \pi _ {1} , \pi _ {2} \oplus \pi _ {3} ) = c ( \pi _ {1} , \pi _ {2} ) + c ( \pi _ {1} , \pi _ {3} ), $$

while if $ X $ is a group, then also

$$ c ( \pi _ {1} \otimes \pi _ {2} , \pi _ {3} ) = \ c ( \pi _ {1} , \pi _ {2} ^ {*} \otimes \pi _ {3} ). $$

If $ \pi _ {1} $ and $ \pi _ {2} $ are irreducible and finite dimensional or unitary, then $ c ( \pi _ {1} , \pi _ {2} ) $ is equal to 1 or 0, depending on whether $ \pi _ {1} $ and $ \pi _ {2} $ are equivalent or not. For continuous finite-dimensional representations of a compact group, the intertwining number can be expressed in terms of the characters of the representations (cf. also Character of a representation of a group).

#### 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 number.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Intertwining_number&oldid=47401