Finite geometry and character theory.

*(English)*Zbl 0818.05001
Lecture Notes in Mathematics. 1601. Berlin: Springer-Verlag. vii, 181 p. (1995).

In this excellent monograph (which grew out of the author’s “Habilitationsschrift”), the author discusses the application of methods from algebraic number theory and character theory to finite geometry, in particular to difference sets, their corresponding codes and associated periodic sequences (and similar, more general objects). In all cases, the existence problem for the object in question (e.g., a finite projective plane with a certain type of collineation group \(G\)) is first translated into an equation in the integral group ring \(\mathbb{Z} G\) (which usually has the form \(DD^{-1}= M\) for a suitable \(M\); e.g., in the classical case of difference sets, one has \(M= n+ \lambda G)\).

Then, if \(G\) is Abelian (or if it has a large Abelian factor group), one may apply characters to this equation (or to a projected version of the original equation) and use results from algebraic number theory to obtain information about the character sums \(\chi(D)\). This quite often leads to non-existence results or constructions for the desired objects. While this approach goes back to a classical paper of R. J. Turyn [“Character sums and difference sets”, Pac. J. Math. 15, No. 1, 319-346 (1965; Zbl 0135.05403)], there has been no place in the literature where one could find such a systematic and wide-ranging treatment. Many of the results presented come from the author’s own research; but there are also many new or simplified proofs for well-known results which are central to the theory.

In Chapter 1, after giving basic definitions and results from design theory and finite geometries and reviewing the algebraic tools needed in what follows (which include standard results like the Fourier inversion formula and Kronecker’s theorem as well as results specifically designed for the study of group ring equations, e.g. Ma’s lemma), the author concentrates on multipliers, one of the most useful concepts in the study of difference set like objects. In Chapter 2, he presents series of examples for difference sets, relative difference sets and divisible difference sets as well as the most important non-existence results (e.g., the Mann test) for divisible difference sets (the most general of the three classes of objects mentioned before). In Chapter 3, difference sets with “classical” parameters are studied; this includes Singer difference sets, a nice treatment of the Gordon-Mills-Welsh difference sets (and a generalization of their construction) and the associated binary sequences and the complete solution of the Waterloo problem (which asked for “lifting” the Singer difference sets and their complements to certain relative difference sets). The next two chapters are mainly concerned with relative difference sets (which can always be thought of as “liftings” of ordinary difference sets). Chapter 4 deals with the semi-regular case (liftings of \((m,m,m)\)-difference sets) and mainly considers parameters of the form \((p^ a, p^ b, p^ a, p^{a-b})\) for primes \(p\). Chapter 5 studies quasiregular collineation groups of projective planes; here the two most important cases are affine difference sets (parameters \((n+1,n-1,n,1)\)) and the case corresponding to planar functions (parameters \((n,n,n,1)\)). A third case does not lead to relative difference sets but to a very similar object, i.e. direct product difference sets. In all these cases, restrictions on the parameters are obtained. Two particularly interesting results concern direct product difference sets; here it is shown that the order \(n\) has to be a prime power or a perfect square and that the only examples of order \(p\) (\(p\) a prime) come from Desarguesian planes. Finally, in Chapter 6, the connections to codes (e.g., Abelian difference set codes and their dimensions) and periodic sequences (in particular, perfect and almost perfect sequences) are studied in detail.

While the monograph under review is not intended as a textbook (indeed, many results are presented without proof and no encyclopaedic treatment is attempted), it should be an ideal preparation for anybody who wants to do serious work in difference sets and their generalizations and to see the connections to codes and sequences which are usually not mentioned in textbooks. After studying Pott’s text (which is, though very nicely presented, no easy reading), the reader will be well prepared to cope with the (by now usually quite considerable) technical difficulties of original papers in the area. This monograph would also provide an excellent basis for an advanced course on algebraic design theory.

Then, if \(G\) is Abelian (or if it has a large Abelian factor group), one may apply characters to this equation (or to a projected version of the original equation) and use results from algebraic number theory to obtain information about the character sums \(\chi(D)\). This quite often leads to non-existence results or constructions for the desired objects. While this approach goes back to a classical paper of R. J. Turyn [“Character sums and difference sets”, Pac. J. Math. 15, No. 1, 319-346 (1965; Zbl 0135.05403)], there has been no place in the literature where one could find such a systematic and wide-ranging treatment. Many of the results presented come from the author’s own research; but there are also many new or simplified proofs for well-known results which are central to the theory.

In Chapter 1, after giving basic definitions and results from design theory and finite geometries and reviewing the algebraic tools needed in what follows (which include standard results like the Fourier inversion formula and Kronecker’s theorem as well as results specifically designed for the study of group ring equations, e.g. Ma’s lemma), the author concentrates on multipliers, one of the most useful concepts in the study of difference set like objects. In Chapter 2, he presents series of examples for difference sets, relative difference sets and divisible difference sets as well as the most important non-existence results (e.g., the Mann test) for divisible difference sets (the most general of the three classes of objects mentioned before). In Chapter 3, difference sets with “classical” parameters are studied; this includes Singer difference sets, a nice treatment of the Gordon-Mills-Welsh difference sets (and a generalization of their construction) and the associated binary sequences and the complete solution of the Waterloo problem (which asked for “lifting” the Singer difference sets and their complements to certain relative difference sets). The next two chapters are mainly concerned with relative difference sets (which can always be thought of as “liftings” of ordinary difference sets). Chapter 4 deals with the semi-regular case (liftings of \((m,m,m)\)-difference sets) and mainly considers parameters of the form \((p^ a, p^ b, p^ a, p^{a-b})\) for primes \(p\). Chapter 5 studies quasiregular collineation groups of projective planes; here the two most important cases are affine difference sets (parameters \((n+1,n-1,n,1)\)) and the case corresponding to planar functions (parameters \((n,n,n,1)\)). A third case does not lead to relative difference sets but to a very similar object, i.e. direct product difference sets. In all these cases, restrictions on the parameters are obtained. Two particularly interesting results concern direct product difference sets; here it is shown that the order \(n\) has to be a prime power or a perfect square and that the only examples of order \(p\) (\(p\) a prime) come from Desarguesian planes. Finally, in Chapter 6, the connections to codes (e.g., Abelian difference set codes and their dimensions) and periodic sequences (in particular, perfect and almost perfect sequences) are studied in detail.

While the monograph under review is not intended as a textbook (indeed, many results are presented without proof and no encyclopaedic treatment is attempted), it should be an ideal preparation for anybody who wants to do serious work in difference sets and their generalizations and to see the connections to codes and sequences which are usually not mentioned in textbooks. After studying Pott’s text (which is, though very nicely presented, no easy reading), the reader will be well prepared to cope with the (by now usually quite considerable) technical difficulties of original papers in the area. This monograph would also provide an excellent basis for an advanced course on algebraic design theory.

Reviewer: D.Jungnickel (Augsburg)

##### MSC:

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

05B10 | Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) |

05B25 | Combinatorial aspects of finite geometries |

51-02 | Research exposition (monographs, survey articles) pertaining to geometry |

05B05 | Combinatorial aspects of block designs |

05E20 | Group actions on designs, etc. (MSC2000) |

11R04 | Algebraic numbers; rings of algebraic integers |

11T24 | Other character sums and Gauss sums |

20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |

51E15 | Finite affine and projective planes (geometric aspects) |

51E30 | Other finite incidence structures (geometric aspects) |

94A55 | Shift register sequences and sequences over finite alphabets in information and communication theory |

94B15 | Cyclic codes |

05B30 | Other designs, configurations |