By L. Loomis, S. Sternberg

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K}. Hence Ri∗ R is of the ∗ ... ∗ 0 . . ∗ .. . . . 0 ... 0 Altogether, it becomes evident that χS (Ri∗ R) = 0. 21 we have χS (Ri∗ R) = λi |Ri |. From the assumption λi = 0 it now follows that |Ri | = 0. But this implies χF divides |Ri |. An immediate consequence of the previous theorem is that abstract coherent algebras over fields of characteristic zero are always semisimple. Henceforth, we will work in the following setting: H = (X, {Ri }i∈I ) is a coherent configuration, F is a field such that ∀ i ∈ I : χF | |Ri |, and F(H) = A1 ⊕ · · · ⊕ Ak where the Ai are the simple ideals of F(H).

Completely independent origins of coherent configurations may be found in India – to be precise from the Statistical Laboratory of Calcutta. C. R. Nair introduced the notion of partially balanced incomplete block designs (also called PBIBD, cf. [BosN-39]). These are incidence structures between points and blocks that are used for the design of statistical experiments. The set of pairs of points are distributed into binary relations, called associate classes, which obey several regularity conditions.

Let us conclude this section by showing one more interrelation between Srings and centralizer algebras. Let (G, X) be a transitive permutation group which contains a regular subgroup (H, X). For x ∈ X let U = Gx = {g ∈ G | xg = x}. Then G acts on the set U \G of right cosets of U in G according to (U g)g = U gg . It is well known that this action is equivalent to the action of (G, X). Under this equivalence, H acts regularly on U \G. Hence each right coset contains precisely one element of H. Moreover, the mapping U → 1H , Uh → h defines an action of G on H that is equivalent to the action of (G, U \G).