Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0701125

On non-projective block components of Lefschetz characters for sporadic geometries

R. Philip Grizzard
View original source ↗Read source document ↗Export metadata ↓

Abstract

This work examines the possible projectivity of 2-modular block parts of non-projective Lefschetz characters over 2-local geometries of several sporadic groups. Previously known results on $M_{12}$, $J_2$, and $HS$ are mentioned for completeness. The main new results are on the sporadic groups $Suz$, $Co_3$, $Ru$, $O'N$, and $He$. For each group, the Lefschetz character is calculated, and the 2-modular blocks are examined for projectivity. In each case it is confirmed that a non-principal block contains a non-projective summand. The case of $O'N$ is additionally found to have a non-projective summand in its principal block part.

Explore related subjects

math.GRmath.CO

Keep this discovery

BibTeXRIS

R. Philip Grizzard. 2007-05-10. On non-projective block components of Lefschetz characters for sporadic geometries. https://arxiv.org/abs/math/0701125

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR↗
arXiv2026

Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

math.GR↗
arXiv2026

The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

math.GR↗