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Abstract We prove character ratio bounds for finite exceptional groups $G(q)$ of Lie type. These take the form $$\dfrac{|\chi (g)|}{\chi (1)} \le \dfrac{c}{q^k}$$ for all nontrivial irreducible characters $$\chi$$ and nonidentity elements $$g$$, where $$c$$ is an absolute constant, and $$k$$ is a positive integer. Applications are given to bounding mixing times for random walks on these groups and also diameters of their McKay graphs.
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Representations and Tensor Product Growth
Abstract The deep theory of approximate subgroups establishes three-step product growth for subsets of finite simple groups $$G$$ of Lie type of bounded rank. In this paper, we obtain two-step growth results for representations of such groups $$G$$ (including those of unbounded rank), where products of subsets are replaced by tensor products of representations. Let $$G$$ be a finite simple group of Lie type and $$\chi $$ a character of $$G$$. Let $$|\chi |$$ denote the sum of the squares of the degrees of all (distinct) irreducible characters of $$G$$ that are constituents of $$\chi $$. We show that for all $$\delta>0$$, there exists $$\epsilon>0$$, independent of $$G$$, such that if $$\chi $$ is an irreducible character of $$G$$ satisfying $$|\chi | \le |G|^{1-\delta }$$, then $$|\chi ^2| \ge |\chi |^{1+\epsilon }$$. We also obtain results for reducible characters and establish faster growth in the case where $$|\chi | \le |G|^{\delta }$$. In another direction, we explore covering phenomena, namely situations where every irreducible character of $$G$$ occurs as a constituent of certain products of characters. For example, we prove that if $$|\chi _1| \cdots |\chi _m|$$ is a high enough power of $|G|$, then every irreducible character of $$G$$ appears in $$\chi _1\cdots \chi _m$$. Finally, we obtain growth results for compact semisimple Lie groups.
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- Award ID(s):
- 2200850
- PAR ID:
- 10501233
- Publisher / Repository:
- International Mathematics Research Notices
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2023
- Issue:
- 14
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 12477-12511
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnac172
