[论文解读] The eigenvalue one property of finite groups, II

Gerhard Hiss, Rafał Lutowski|arXiv (Cornell University)|Jan 14, 2026

Geometric and Algebraic Topology被引用 0

一句话总结

该论文证明了关于有限群在奇维实模表示下存在特征值1的猜想，在平坦流形成为R∞-流形的条件方面提供了依据。

ABSTRACT

We prove a conjecture of Dekimpe, De Rock and Penninckx concerning the existence of eigenvalues one in certain elements of finite groups acting irreducibly on a real vector space of odd dimension. This yields a sufficient condition for a closed flat manifold to be an $R_{\infty}$-manifold.

研究动机与目标

• Motivate and complete the proof of a conjecture by Dekimpe, De Rock and Penninckx relating eigenvalue one to irreducible real representations of finite groups.
• Establish that finite simple groups of Lie type in even characteristic are not minimal counterexamples to the cited theorem.
• Provide a framework to identify real characters among high-degree irreducible characters using Deligne-Lusztig theory and Lusztig's generalized Jordan decomposition.

提出的方法

• Apply the large degree method to rule out most candidates with large irreducible character degrees.
• Use the restriction method for characters of smaller degree and apply Deligne-Lusztig theory to identify real characters among odd-degree irreducibles.
• Employ Lusztig’s generalized Jordan decomposition to relate irreducible characters to semisimple elements and their centralizers.
• Compute centralizer orders and automorphism actions to bound possibilities for minimal counterexamples.
• Utilize tables and computational tools (GAP, Chevie) to handle numerous exceptional cases and small q, Lie rank, or small degrees.
• Develop and apply a sigma-setup for groups of Lie type to manage twisted and untwisted cases.

实验结果

研究问题

• RQ1 Does every finite simple group of Lie type in even characteristic fail to be a minimal counterexample to the eigenvalue one property in odd-dimensional real representations?
• RQ2 Can Deligne-Lusztig theory and Lusztig’s Jordan decomposition determine which real irreducible characters have odd degree and correspond to eigenvalue one in the considered setting?
• RQ3 What structural conditions on centralizers and automorphisms prevent minimal counterexamples among the listed families (including PSL, PSU, E6, and PΩ8 groups)?
• RQ4 How do restrictions to unipotent and Levi subgroups (via BN-pair structure) assist in establishing the E1-property for broad classes of groups?
• RQ5 Under what automorphism configurations do specific groups fail to produce eigenvalue one elements that would contradict the conjecture?

主要发现

• Theorem: If G is a finite simple group of Lie type of even characteristic, then G is not a minimal counterexample to the eigenvalue one conjecture analyzed in the paper.
• For many groups with trivial Schur multiplier and no graph automorphism of order 3, the E1-property holds, ruling out minimal counterexamples in broad families.
• Application of Deligne-Lusztig theory and Lusztig’s generalized Jordan decomposition identifies when odd-degree characters are real and corresponds to eigenvalue one conditions.
• Bounds on centralizers and orders of automorphisms demonstrate that certain groups (notably PΩ8^+(q)) cannot be minimal counterexamples; in some cases tighter bounds exclude specific automorphisms.
• The analysis combines large-degree arguments, restriction techniques, and computational aids to handle exceptional cases and aligned subcases across several families like SL_d(q), SU_d(q), E6(q), and 2E6(q).