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:
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while if is a group, then also
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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) |
Intertwining number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intertwining_number&oldid=13682