This talk is based on joint work with K. Coulembier, P. Etingof,
D. Tubbenhauer. Let ?? be any group and let V be a nite dimensional
representation of ?? over arbitrary eld. We consider tensor powers
V
n of V and their decompositions into indecomposable summands.
Let bn(V ) be the total number of indecomposable summands in V
n.
We prove that
lim
n!1
n p
bn(V ) = dim(V ):
Similarly let dn(V ) be the number of indecomposable summands of
V
n with dimension not divisible by the characteristic of the eld.
Then we de ne
(V ) := lim
n!1
n p
dn(V ):
The real number (V ) is an interesting invariant of the representation
V . Using theory of tensor categories we show that this invariant is ad-
ditive (under direct sums), multiplicative and takes values in algebraic
integers.
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