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Intertwining number

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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).

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. A.I. Shtern (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intertwining_number&oldid=13682
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098