Fixed Point Free Actions of Group Elements in Linear Groups and Finite Group Representations

2025/09/28 by Zalesski, Alexandre · 2 citations
#11F80 #11G10 #20C33 #20H30 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2509.24018

Abstract

This is an expository paper aimed to outline the current situation with problems related with the occurrence of eigenvalue 1 of elements in linear groups and group representations.

Citations

• Minimal polynomials of p-elements of finite groups of Lie type with cyclic Sylow p-subgroups
• On eigenvalues of permutations in irreducible representations of symmetric and alternating groups
• Derangements in non-Frobenius groups
• Cyclic Characters of Alternating Groups
• On arithmetical properties and arithmetical characterizations of finite groups
• On the Existence of Elementwise Invariant Vectors in Representations of Symmetric Groups
• Upper bounds for eigenvalue multiplicities of almost cyclic elements in irreducible representations of simple algebraic groups
• On characterization by Gruenberg-Kegel graph of finite simple exceptional groups of Lie type
• Rational elements in representations of simple algebraic groups, I
• Realizations of Unisingular Representations by Hyperelliptic Jacobians
• Representations of dimensions (pⁿ± 1)/2 of the symplectic group of degree 2n over a field of characteristic p
• Hall-Higman type theorems for exceptional groups of Lie type, I
• Unisingular representations in arithmetic and Lie theory
• Weight zero in tensor-decomposable irreducible representations of simple algebraic groups
• Almost cyclic elements in Weil representations of finite classical groups
• On conjugacy classes of S_n containing all irreducibles
• Subgroups of simple algebraic groups containing regular tori, and irreducible representations with multiplicity 1 non-zero weights
• Products of conjugacy classes and fixed point spaces
• Hall-Higman type theorems for semisimple elements of finite classical groups

Cited by

• On the Positivity of Dihedral Branching Coefficients of the Symmetric and Alternating Groups
• Unisingular representations of rank 1 finite simple groups of Lie type

Related

• Dynamics of primitive elements under group actions
• Elements in finite classical groups whose powers have large\n 1-Eigenspaces
• Some irreducible free group representations in which a linear combination of the generators has an eigenvalue
• Representations of finite groups
• Derangements in Subspace Actions of Finite Classical Groups