Finding possible permutation characters.

*(English)*Zbl 0921.20013The authors describe three different methods to compute all those characters of a finite group that have certain properties of transitive permutation characters. It is assumed that all rationally irreducible characters of the group are known in advance. All methods check which sums of integral multiples of these are possible permutation characters.

First, a combinatorial approach can be used to enumerate the vectors of multiplicities. Secondly, these characters can be found as integral solutions of a system of inequalities. Thirdly, they are calculated via Gaussian elimination after prescribing a permutation character value on a certain conjugacy class.

The methods are used to determine these characters for some finite groups and runtimes of GAP [M. Schönert, et al., GAP – Groups, Algorithms and Programming. Lehrstuhl D für Mathematik, RWTH Aachen (1994)], version 3.4 are listed. Often the second method is faster than the first. Gaussian elimination appears to be more efficient only for permutation characters of larger degree.

In the final section, the permutation character of the Lyons group operating on cosets of a maximal subgroup of type \(3^{2+4}:2A_5.D_8\) is constructed. The authors explain the interaction between their methods to find possible permutation characters and further reduction of these by additional knowledge in detail in this example and construct the character.

First, a combinatorial approach can be used to enumerate the vectors of multiplicities. Secondly, these characters can be found as integral solutions of a system of inequalities. Thirdly, they are calculated via Gaussian elimination after prescribing a permutation character value on a certain conjugacy class.

The methods are used to determine these characters for some finite groups and runtimes of GAP [M. Schönert, et al., GAP – Groups, Algorithms and Programming. Lehrstuhl D für Mathematik, RWTH Aachen (1994)], version 3.4 are listed. Often the second method is faster than the first. Gaussian elimination appears to be more efficient only for permutation characters of larger degree.

In the final section, the permutation character of the Lyons group operating on cosets of a maximal subgroup of type \(3^{2+4}:2A_5.D_8\) is constructed. The authors explain the interaction between their methods to find possible permutation characters and further reduction of these by additional knowledge in detail in this example and construct the character.

Reviewer: M.Weller (Essen)

##### MSC:

20C40 | Computational methods (representations of groups) (MSC2010) |

20C15 | Ordinary representations and characters |

20C34 | Representations of sporadic groups |

68W30 | Symbolic computation and algebraic computation |

20-04 | Software, source code, etc. for problems pertaining to group theory |