Let $G$ be a group and $k$ an algebraically closed field of characteristic $p$. If $V$ is a finite-dimensional representation of $G$ over $k$, then by the classical Krull–Schmidt theorem, the $n$th tensor power of $V$ can be uniquely decomposed into a direct sum of indecomposable representations. But we know very little about this decomposition, even for very small groups, such as $G = (\Bbb Z/2)^3$ for $p = 2$ or $G = (\Bbb Z/3)^2$ for $p = 3$.

For example, what can we say about the number $d_n(V)$ of summands with dimension coprime to $p$? It is easy to show that there is a finite limit $d(V) := \lim_{n \to \infty} d_n(V)^{1/n}$, but what kind of number is this? Is it algebraic or transcendental? Until recently, there were no techniques to solve such questions (and in particular the same question about the sum of dimensions of these summands is still wide open). Remarkably, a new subject which may be called "Lie theory in tensor categories" gives methods to show that $d(V)$ is indeed an algebraic number, which moreover has the form \[ d(V) = \sum_{1 \leq j \leq p/2} n_j(V)[j]_q, \] where $n_j(V)$ is a natural number, $q := \exp(\pi i/p)$ is a particular root of unity, and $[j]_q := \frac{q^j-q^{-j}}{q-q^{-1}}$ is a $q$-number. Moreover, $d(V \oplus W) = d(V) + d(W)$ and $d(V \otimes W) = d(V) d(W)$, so $d$ is a character of the Green ring of $G$ over $k$. Finally, $d_n(V) \geq C_V d(V)^n$, for some $0 < C_V \leq 1$, and we can give lower bounds for $C_V$. In the talk, I will explain what Lie theory in tensor categories is and how it can be applied to such problems. This is joint work with K. Coulembier and V. Ostrik.