On eigenvalues of the Cartan matrices for $ p $-constrained groups

Manal H. Algreagri; Manal H. Algreagri

doi:10.3934/math.2026776

AIMS Mathematics

2026, Volume 11, Issue 6: 19046-19057. doi: 10.3934/math.2026776

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Research article

On eigenvalues of the Cartan matrices for $ p $-constrained groups

Manal H. Algreagri ^,

Department of Mathematics, Jamoum University College, Umm Al-Qura University, Makkah, Saudi Arabia

Received: 15 May 2026 Revised: 18 June 2026 Accepted: 25 June 2026 Published: 29 June 2026
MSC : 20C15, 20C20

Let $ p $ be a prime number. For $ p $-constrained groups, we demonstrated that the Perron–Frobenius eigenvalue of the Cartan matrix of a $ p $-block $ B $ was bounded above by the order of its defect group. The proof employed block-theoretic methods, spectral bounds in modular representation theory, and structural $ p' $-reduction techniques. This conclusion extended a previously established inequality for $ p $-solvable groups to the larger class of $ p $-constrained groups. Finally, we presented computational results regarding the relationship between elementary divisors and eigenvalues of Cartan matrices for $ p $-constrained groups that were not $ p $-solvable.
- finite group,
- $ p $-constrained groups,
- block,
- defect group,
- eigenvalues,
- nonnegative matrices
Citation: Manal H. Algreagri. On eigenvalues of the Cartan matrices for $ p $-constrained groups[J]. AIMS Mathematics, 2026, 11(6): 19046-19057. doi: 10.3934/math.2026776

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Abstract

Let $ p $ be a prime number. For $ p $-constrained groups, we demonstrated that the Perron–Frobenius eigenvalue of the Cartan matrix of a $ p $-block $ B $ was bounded above by the order of its defect group. The proof employed block-theoretic methods, spectral bounds in modular representation theory, and structural $ p' $-reduction techniques. This conclusion extended a previously established inequality for $ p $-solvable groups to the larger class of $ p $-constrained groups. Finally, we presented computational results regarding the relationship between elementary divisors and eigenvalues of Cartan matrices for $ p $-constrained groups that were not $ p $-solvable.

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