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)