The dimension of the space of intertwining operators (cf. Intertwining operator) for two mappings and of a set into topological vector spaces and , respectively. The concept of the intertwining number is especially fruitful in the case when is a group or an algebra and are representations of . Even for finite-dimensional representations, in general, but for finite-dimensional representations , , the following relations hold:
while if is a group, then also
If and are irreducible and finite dimensional or unitary, then is equal to 1 or 0, depending on whether and 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).
|||A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)|
|||A.I. Shtern, "Theory of group representations" , Springer (1982) (Translated from Russian)|
Intertwining number. A.I. Shtern (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intertwining_number&oldid=13682