Beijing Institute of Mathematical Sciences and Applications

Lin Zhe Huang , Zheng Wei Liu , Sébastien Palcoux , Yi Long Wang , Jin Song Wu

Pavel Etingof

Wednesday, February 22, 2023 10:30 AM - 12:00 PM

A3-3-301

Zoom 559 700 6085 (BIMSA)

Let $G$ be a group and $k$ an algebraically closed field of positive characteristic $p$. Let $V$ be a finite dimensional representation of $G$ over $k$. Then by the classical Krull-Schmidt theorem, the tensor power $V^{\otimes n}$ 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 $G = (Z/2)^3$ for $p = 2$ or $G = (Z/3)^2$ for $p = 3$. For example, what can we say about the number $d_n(V)$ of such summands of dimension coprime to $p$? It is easy to show that there exists a finite limit $d(V ) := \lim_{n \to \infty} d_n(V )^{1/n}$, but what kind of number is it? For example, is it algebraic or transcendental? Until recently, there was 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 $n_j(V) \in \mathbb{N}$, $q := \exp(\pi i/p)$, and $[j]_q := (q^j-q^{-j})/(q-q^{-1})$. Moreover, $d$ is a character of the Green ring of $G$ over $k$. Furthermore, $$d_n(V) \leq C_V d(V)^n$$ for some positive $C_V \leq 1$ gre 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. References: https://arxiv.org/abs/2107.02372 (to appear in Annals of Math) https://arxiv.org/abs/2301.09804

Professor Pavel Etingof is a professor of mathematics at MIT. He works on many aspects of representation theory and mathematical physics, and has published many influential papers in top journals. In 1994, he received his PhD in mathematics at Yale University and became Benjamin Peirce Assistant Professor at Harvard University after that. In 1998, he was an assistant professor at MIT, and became a professor there since 2005. In 1999, he was a Fellow of the Clay Mathematics Institute. In 2002, he was an invited speaker at the International Congress of Mathematicians in Beijing. He is a Fellow of the American Mathematical Society, and in 2016, he became a fellow of the American Academy of Arts and Sciences. His editorial work includes Editor-in-Chief of Selecta Mathematica and Managing Editor of Journal of the AMS.