# 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
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article