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Primitivity of permutation groups, coherent algebras and matrices. (English) Zbl 1011.20003
The term “primitive” is used in many senses in algebra. In particular, a permutation group is primitive if it preserves no non-trivial equivalence relation; a coherent algebra (such as the centralizer algebra of a permutation group) is primitive if it does not contain the matrix of such a relation; and a real square matrix is primitive if some power of the matrix has all entries strictly positive. There is a natural relationship between the first two uses of the term primitive. The main result of the present paper is the following relationship between the last two uses. A coherent algebra $$W$$ has a basis consisting of square $$\{0,1\}$$-matrices. Suppose that $$W$$ is not the centralizer algebra of a regular permutation group of prime degree. Then $$W$$ is a primitive coherent algebra if and only if each nonidentity basis element is a primitive matrix. This result is applied to give necessary and sufficient conditions for the exponentiation $$W\uparrow G$$ of a coherent algebra $$W$$ by a permutation group $$G$$ to be primitive.
Reviewer: J.D.Dixon (Ottawa)

MSC:
 20B15 Primitive groups 15A30 Algebraic systems of matrices 15B48 Positive matrices and their generalizations; cones of matrices 05E30 Association schemes, strongly regular graphs
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References:
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