### Abstract

Let G be a finite group of Lie type in characteristic p. This paper addresses the problem of describing the irreducible complex (or p-adic) representations of G that remain absolutely irreducible under the Brauer reduction modulo p. An efficient approach to solve this problem for p > 3 has been elaborated in earlier papers by the authors. In this paper, we use arithmetical properties of character degrees to solve this problem for the groups G ∈ {^{2}B_{2}(q), ^{2}G_{2}(q), G_{2}(q), ^{2}F_{4}(q), F_{4}(q), ^{3}D_{4}(q)} provided that p ≤ 3. We also prove an asymptotical result, which solves the problem for all finite groups of Lie type over double-struck F sign_{q} with q large enough.

Original language | English (US) |
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Pages (from-to) | 3177-3184 |

Number of pages | 8 |

Journal | Proceedings of the American Mathematical Society |

Volume | 130 |

Issue number | 11 |

DOIs | |

State | Published - Nov 1 2002 |

Externally published | Yes |

### Fingerprint

### Keywords

- Finite groups of Lie type
- Reduction modulo p
- Steinberg representation

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Reducibility modulo p of complex representations of finite groups of lie type : Asymptotical result and small characteristic cases.** / Tiep, Pham Huu; Zalesskii, A. E.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 130, no. 11, pp. 3177-3184. https://doi.org/10.1090/S0002-9939-02-06459-6

}

TY - JOUR

T1 - Reducibility modulo p of complex representations of finite groups of lie type

T2 - Asymptotical result and small characteristic cases

AU - Tiep, Pham Huu

AU - Zalesskii, A. E.

PY - 2002/11/1

Y1 - 2002/11/1

N2 - Let G be a finite group of Lie type in characteristic p. This paper addresses the problem of describing the irreducible complex (or p-adic) representations of G that remain absolutely irreducible under the Brauer reduction modulo p. An efficient approach to solve this problem for p > 3 has been elaborated in earlier papers by the authors. In this paper, we use arithmetical properties of character degrees to solve this problem for the groups G ∈ {2B2(q), 2G2(q), G2(q), 2F4(q), F4(q), 3D4(q)} provided that p ≤ 3. We also prove an asymptotical result, which solves the problem for all finite groups of Lie type over double-struck F signq with q large enough.

AB - Let G be a finite group of Lie type in characteristic p. This paper addresses the problem of describing the irreducible complex (or p-adic) representations of G that remain absolutely irreducible under the Brauer reduction modulo p. An efficient approach to solve this problem for p > 3 has been elaborated in earlier papers by the authors. In this paper, we use arithmetical properties of character degrees to solve this problem for the groups G ∈ {2B2(q), 2G2(q), G2(q), 2F4(q), F4(q), 3D4(q)} provided that p ≤ 3. We also prove an asymptotical result, which solves the problem for all finite groups of Lie type over double-struck F signq with q large enough.

KW - Finite groups of Lie type

KW - Reduction modulo p

KW - Steinberg representation

UR - http://www.scopus.com/inward/record.url?scp=0036842544&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036842544&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-02-06459-6

DO - 10.1090/S0002-9939-02-06459-6

M3 - Article

VL - 130

SP - 3177

EP - 3184

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 11

ER -