### Abstract

We discuss the focal subgroup of the defect group D of a p-block B, which we refer to as the focal defect group, and denote by Do. We note that (the character group) of D/D-0 acts (in a defect (or height) preserving fashion) on irreducible characters in B, and prove that the action on irreducible characters of height zero is semi-regular. We also prove that all orbits under this action have length divisible by [Z(D) : D-0 boolean AND Z(D)]. As applications, we prove that all Cartan invariants for B are divisible by [Z(D) : D-0 boolean AND Z(D)], that if Out(D) is a p-group (and D 0 1), then the number of irreducible characters of height zero in B is divisible by p and that if Z(D)not less than or equal to Do, then the block B is of Lefschetz type (see [R. Knorr, G.R. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. (2) 39 (1) (1989) 48-60]). (C) 2008 Elsevier Inc. All rights reserved.

Original language | English |
---|---|

Pages (from-to) | 2624-2628 |

Number of pages | 5 |

Journal | Journal of Algebra |

Volume | 320 |

Issue number | 6 |

Early online date | 16 Jun 2008 |

DOIs | |

Publication status | Published - 15 Sep 2008 |

### Keywords

- modular representations
- group characters

### Cite this

**On the focal defect group of a block, characters of height zero, and lower defect group multiplicities.** / Robinson, Geoffrey R.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 320, no. 6, pp. 2624-2628. https://doi.org/10.1016/j.jalgebra.2008.04.032

}

TY - JOUR

T1 - On the focal defect group of a block, characters of height zero, and lower defect group multiplicities

AU - Robinson, Geoffrey R.

PY - 2008/9/15

Y1 - 2008/9/15

N2 - We discuss the focal subgroup of the defect group D of a p-block B, which we refer to as the focal defect group, and denote by Do. We note that (the character group) of D/D-0 acts (in a defect (or height) preserving fashion) on irreducible characters in B, and prove that the action on irreducible characters of height zero is semi-regular. We also prove that all orbits under this action have length divisible by [Z(D) : D-0 boolean AND Z(D)]. As applications, we prove that all Cartan invariants for B are divisible by [Z(D) : D-0 boolean AND Z(D)], that if Out(D) is a p-group (and D 0 1), then the number of irreducible characters of height zero in B is divisible by p and that if Z(D)not less than or equal to Do, then the block B is of Lefschetz type (see [R. Knorr, G.R. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. (2) 39 (1) (1989) 48-60]). (C) 2008 Elsevier Inc. All rights reserved.

AB - We discuss the focal subgroup of the defect group D of a p-block B, which we refer to as the focal defect group, and denote by Do. We note that (the character group) of D/D-0 acts (in a defect (or height) preserving fashion) on irreducible characters in B, and prove that the action on irreducible characters of height zero is semi-regular. We also prove that all orbits under this action have length divisible by [Z(D) : D-0 boolean AND Z(D)]. As applications, we prove that all Cartan invariants for B are divisible by [Z(D) : D-0 boolean AND Z(D)], that if Out(D) is a p-group (and D 0 1), then the number of irreducible characters of height zero in B is divisible by p and that if Z(D)not less than or equal to Do, then the block B is of Lefschetz type (see [R. Knorr, G.R. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. (2) 39 (1) (1989) 48-60]). (C) 2008 Elsevier Inc. All rights reserved.

KW - modular representations

KW - group characters

U2 - 10.1016/j.jalgebra.2008.04.032

DO - 10.1016/j.jalgebra.2008.04.032

M3 - Article

VL - 320

SP - 2624

EP - 2628

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 6

ER -