Quasi-symmetry and Representation Theory – 541 –

— Quasi-symmetry is a model for square contingency tables with rows and columns indexed by the same set. In a log-linear model, quasi-symmetry is associated with a vector subspace QSn of certain realvalued functions on n × n. Permutation of factor levels induces a linear transformation QSn → QSn, making this into a representation of the symmetric group Sn. However, there are many group representations that are totally unsatisfactory as linear or generalized linear models. Quasisymmetry is also a hereditary group representation, which is to say that the restriction of QSn+1 to the leading n× n sub-array is equal to QSn. Another way of saying the same thing is that, as a sequence of vector spaces, QS is a representation of the category of injective maps on finite sets. This property is of fundamental importance for statistical models. This paper sets out to list all hereditary sub-representations by real-valued square matrices, and to explain how these may used in model construction. We conclude that there are exactly six non-trivial log-linear models for a square contingency table. (∗) Reçu le 18 septembre 2001, accepté le 18 septembre 2002 (1) Department of Statistics, University of Chicago, Chicago, Il 60637, U.S.A. E-mail: [email protected] – 541 –