Christopher French

Functors from association schemes

We construct a wide subcategory of the category of finite association schemes with a collection of desirable properties. Our subcategory has a first isomorphism theorem analogous to that of groups. Also, standard constructions taking schemes to groups (thin radicals and thin quotients) or algebras (adjacency algebras) become functorial when restricted to our category. We use our category to give a more conceptual account for a result of Hanaki concerning products of characters of association schemes; i.e. we show that the virtual representations of an association scheme form a module over the representation ring of the thin quotient of the association scheme.

On the Normal Structure of NonCommutative Association Schemes of Rank 6

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. One of these generalizations is the observation that, similar to groups, association schemes of finite order are commutative if they have at most five elements and not necessarily commutative if they have six elements. While there is (up to isomorphism) only one noncommutative group of order 6, there are infinitely many pairwise non-isomorphic noncommutative association schemes of finite order with six elements. (Each finite projective plane provides such a scheme, and non-isomorphic projective planes yield non-isomorphic schemes.) In this note, we investigate noncommutative schemes of finite order with six elements which have a symmetric normal closed subset with three elements. We take advantage of the classification of the finite simple groups.

On a Class of Non-commutative Imprimitive Association Schemes of Rank 6

Many facts about group theory can be generalized to the context of the theory of association schemes. In particular, association schemes with fewer than 6 elements are all commutative. While there is a nonabelian group with 6 elements which is unique up to isomorphism, there are infinitely many isomorphism classes of non-commutative association schemes with 6 elements. All examples previously known to us are imprimitive, and fall into three classes which are reasonably well understood. In this paper, we construct a fourth class of noncommutative, imprimitive association schemes of rank 6.

The Jacobi orientation and the two-variable elliptic genus

We explain the relationship between the sigma orientation and Witten genus on the one hand and the two-variable elliptic genus on the other. We show that if E is an elliptic spectrum, then the Theorem of the Cube implies the existence of canonical SU-orientation of the associated spectrum of Jacobi forms. In the case of the elliptic spectrum associated to the Tate curve, this gives the two-variable elliptic genus. We also show that the two-variable genus arises as an instance of the circle-equivariant sigma orientation.