# Intertwining number

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.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Intertwining_number&oldid=13682