Rendiconti del Seminario Matematico della Università di Padova

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Volume 136, 2016, pp. 35–50
DOI: 10.4171/RSMUP/136-4

Irreducible characters of finite simple groups constant at the $p$-singular elements

Marco A. Pellegrini[1] and Alexandre Zalesski[2]

(1) Università Cattolica del Sacro Cuore, Brescia, Italy
(2) University of East Anglia, Norwich, UK

In representation theory of finite groups an important role is played by irreducible characters of $p$-defect 0, for a prime $p$ dividing the group order. These are exactly those vanishing at the $p$-singular elements. In this paper we generalize this notion investigating the irreducible characters that are constant at the $p$-singular elements. We determine all such characters of non-zero defect for alternating, symmetric and sporadic simple groups.

We also classify the irreducible characters of quasi-simple groups of Lie type that are constant at the non-identity unipotent elements.In particular, we show that for groups of BN-pair rank greater than 2 the Steinberg and the trivial characters are the only characters in question. Additionally, we determine all irreducible characters whose degrees differ by 1 from the degree of the Steinberg character.

Keywords: Chevalley groups, alternating groups, irreducible characters, principal block

Pellegrini Marco, Zalesski Alexandre: Irreducible characters of finite simple groups constant at the $p$-singular elements. Rend. Sem. Mat. Univ. Padova 136 (2016), 35-50. doi: 10.4171/RSMUP/136-4