18,158 research outputs found

    On relative tt-designs in polynomial association schemes

    Full text link
    Motivated by the similarities between the theory of spherical tt-designs and that of tt-designs in QQ-polynomial association schemes, we study two versions of relative tt-designs, the counterparts of Euclidean tt-designs for PP- and/or QQ-polynomial association schemes. We develop the theory based on the Terwilliger algebra, which is a noncommutative associative semisimple C\mathbb{C}-algebra associated with each vertex of an association scheme. We compute explicitly the Fisher type lower bounds on the sizes of relative tt-designs, assuming that certain irreducible modules behave nicely. The two versions of relative tt-designs turn out to be equivalent in the case of the Hamming schemes. From this point of view, we establish a new algebraic characterization of the Hamming schemes.Comment: 17 page

    Distance-regular graphs

    Get PDF
    This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

    A characterization of Q-polynomial association schemes

    Get PDF
    We prove a necessary and sufficient condition for a symmetric association scheme to be a Q-polynomial scheme.Comment: 8 pages, no figur

    Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems

    Get PDF
    Inspired by some intriguing examples, we study uniform association schemes and uniform coherent configurations, including cometric Q-antipodal association schemes. After a review of imprimitivity, we show that an imprimitive association scheme is uniform if and only if it is dismantlable, and we cast these schemes in the broader context of certain --- uniform --- coherent configurations. We also give a third characterization of uniform schemes in terms of the Krein parameters, and derive information on the primitive idempotents of such a scheme. In the second half of the paper, we apply these results to cometric association schemes. We show that each such scheme is uniform if and only if it is Q-antipodal, and derive results on the parameters of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We revisit the correspondence between uniform indecomposable three-class schemes and linked systems of symmetric designs, and show that these are cometric Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class schemes in terms of only a few parameters, and show that any strongly regular graph with a ("non-exceptional") strongly regular decomposition gives rise to such a scheme. Hemisystems in generalized quadrangles provide interesting examples of such decompositions. We finish with a short discussion of five-class schemes as well as a list of all feasible parameter sets for cometric Q-antipodal four-class schemes with at most six fibres and fibre size at most 2000, and describe the known examples. Most of these examples are related to groups, codes, and geometries.Comment: 42 pages, 1 figure, 1 table. Published version, minor revisions, April 201
    • …
    corecore