报告题目：Designs and Representation Theory
        报 告 人: Prof. Johannes Siemons
        报告摘要: It is well-known that often combinatorial structures can be represented as f0; 1g-solutions of a suitable system of algebraic equations. For instance, the t (n; k; ) designs on the set V = f1:ng correspond one-to-one to the solutions X of the matrix equation in which M is the incidence matrix between the t- and the k-element subsets of V; and J is a column of 1s. Formally the condition Xi 2 f0; 1g can be replaced by X2i = Xi for all i; and so at least in principle, algebra is su±cient to express the combinatorial requirements. A second feature is that the symmetric group Sym(V ) permutes the k-element sub-sets of V and hence the solutions of (E): It is therefore natural to consider this equation also from the view point of the representation theory of the symmetric group. The solutions of the homogeneous equation MX = 0 form the module of null-designs on V which has been studied extensively. We discuss new and old results concerning the intriguing structure of null-designs as Specht modules. In particular, certain standard polytope bases for null-designs arise naturally from the standard polytabloids of Specht theory. We prove a spectral theorem for the module of all k-element subsets of a set and give some applications to Smith canonical forms and Delsarte's inner distribution of subset-families.

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