[Paper Review] On characters of Chevalley groups vanishing at the non-semisimple elements

Marco Antonio Pellegrini, Alexandre Zalesski|arXiv (Cornell University)|Apr 18, 2013

Finite Group Theory Research7 references3 citations

TL;DR

This paper investigates p-vanishing characters of degree |G|_p in finite simple groups of Lie type, generalizing the Steinberg character. Using character theory and explicit decomposition of p-vanishing characters, it proves that for most groups of small rank, the Steinberg character is the only such character, while identifying specific exceptions where reducible p-vanishing characters exist under number-theoretic conditions on q.

ABSTRACT

Let G be a finite simple group of Lie type. In this paper we study characters of G that vanish at the non-semisimple elements and whose degree is equal to the order of a maximal unipotent subgroup of G. Such characters can be viewed as a natural generalization of the Steinberg character. For groups G of small rank we also determine the characters of this degree vanishing only at the non-identity unipotent elements.

Motivation & Objective

• To generalize the Steinberg character by studying p-vanishing characters of degree equal to the order of a maximal unipotent subgroup in finite simple groups of Lie type.
• To determine which groups admit reducible p-vanishing characters of degree |G|_p, beyond the Steinberg character.
• To provide explicit decomposition of p-vanishing characters into irreducible constituents for small-rank groups.
• To establish uniform results across all q, avoiding reliance on p-modular representation theory.
• To explore implications for projective modules and decomposition matrices in modular representation theory.

Proposed method

• The authors use the contemporary theory of characters of groups of Lie type to analyze p-vanishing characters.
• They decompose p-vanishing characters into irreducible constituents using character tables and known decomposition rules.
• For groups like PSL(6,q), they derive character decompositions from those of PSL(5,q), enabling comparison with the Steinberg character.
• They compute character values at semisimple and unipotent elements, showing that non-Steinberg p-vanishing characters do not vanish at semisimple elements.
• They apply uniform arguments valid for all q, avoiding dependence on p-modular representation theory.
• They tabulate explicit character decompositions for exceptional cases, such as groups with q+1 divisible by 3 or 7.

Experimental results

Research questions

• RQ1 For which finite simple groups of Lie type is the Steinberg character the only p-vanishing character of degree |G|_p?
• RQ2 Under what conditions on q do reducible p-vanishing characters of degree |G|_p exist for groups of small rank?
• RQ3 How can p-vanishing characters be explicitly decomposed into irreducible characters across different families of groups?
• RQ4 What is the relationship between p-vanishing characters and projective modules in the context of decomposition matrices?
• RQ5 Do all p-vanishing characters of degree |G|_p have real values and non-vanishing values at semisimple elements?

Key findings

• The Steinberg character is the unique p-vanishing character of degree |G|_p for all groups of Lie type except B3(q), B4(q), B5(q), D4(q), and D5(q), for which the result remains open.
• For A1(q), A2(q), 2B2(q²), and 2G2(q²), reducible p-vanishing characters of degree |G|_p exist for all q.
• For 2A2(q), A3(q), and A4(q), such reducible characters exist if and only if q+1 is divisible by 3.
• For C2(q) ≅ B2(q), reducible p-vanishing characters exist iff q+1 is divisible by 7.
• All p-vanishing characters of degree |G|_p have real values and do not vanish at semisimple elements.
• In groups satisfying Theorem 1.1, the dimension difference between characters of projective modules Ψ and Φ with Ψ > Φ is either |G|_p (only if Ψ − Φ = St) or at least 2|G|_p.