On Imprimitive NonCommutative Association Schemes of Order 6

Abstract

The concept of an association scheme is a far-reaching generalization of the notion of a group. Many group theoretic facts have found a natural generalization in scheme theory; cf. [8] and [7]. One of these generalizations is the observation that, similar to groups, association schemes are commutative if they have at most five elements and not necessarily commutative if they have six elements; cf. [6, (4.1)]. (In this article, all association schemes are assumed to have finite valency). While any two noncommutative groups of order 6 are isomorphic to each other, there exist infinitely many isomorphism classes of noncommutative association schemes of order 6 (among them all finite projective planes). The present article is a first attempt to obtain insight into the structure of noncommutative schemes of order 6.

Key Words:

• Association schemes
• Groups
• Irreducible characters
• Representation theory

2010 Mathematics Subject Classification:

• 20N20
• 16G10
• 05E30

ACKNOWLEDGMENT

This research was initiated at the Mathematisches Forschungsinstitut Oberwolfach during a stay within the Research in Pairs Programme from November 29 to December 19, 2009. The authors gratefully acknowledge the kind hospitality and the comfortable working environment at Oberwolfach. The second author acknowledges the support of the Max-Planck-Gesellschaft zur Förderung der Wissenschaften e.V. He finished this work while he was visiting the Max-Planck-Institut für Mathematik at Bonn.

Notes

¹Coxeter schemes are schemes which are defined on the flags of a building. In the case where a Coxeter scheme has six elements, the corresponding building is a projective plane. Coxeter schemes have been investigated in general in [10] and in the last two chapters of [11].

²Semidirect products of association schemes are defined in [11, Section 7.3]. They were also considered in [1]. Note that the complement in a semidirect product is always thin.

³Thin closed subsets of schemes can be considered as groups; cf. [11, Section 5.5].

In general, if e is an idempotent element of ℂS, then we have e ∈ Z(ℂS) if and only if χ(e) ∈ {0, χ(1)} for all elements χ in Irr(S). This is easy to see. In fact, let e ₁, ⋅, e _n be the identity elements of the Wedderburn components of ℂS and assume first that e ∈ Z(ℂS). Then there exist elements c ₁, ⋅, c _n in ℂ such that

Since e is assumed to be idempotent, c _i  ∈ {0, 1} for all elements i in {1, ⋅, n}. Thus, χ(e) ∈ {0, χ(1)} for all elements χ in Irr(S).

Conversely, assume that χ(e) ∈ {0, χ(1)} for all elements χ in Irr(S). Then, as e is assumed to be idempotent, there exist elements c ₁, ⋅, c _n in {0, 1} such that the above equation is satisfied. This implies e ∈ Z(ℂS).

Communicated by S. Sehgal.