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Studies in the representation theory of finite semigroups


Author: Yechezkel Zalcstein
Journal: Trans. Amer. Math. Soc. 161 (1971), 71-87
MSC: Primary 20.90
DOI: https://doi.org/10.1090/S0002-9947-1971-0283104-2
MathSciNet review: 0283104
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Abstract: This paper is a continuation of [14], developing the representation theory of finite semigroups further. The main result, Theorem 1.24, states that if the group of units U of a mapping semigroup (X, S) is multiply transitive with a sufficiently high degree of transitivity, then for certain irreducible characters $ \chi $ of U, $ \chi $ can be ``extended'' formally to an irreducible character of S. This yields a partial generalization of a well-known theorem of Frobenius on the characters of multiply-transitive groups and provides the first nontrivial explicit formula for an irreducible character of a finite semigroup. The paper also contains preliminary results on the ``spectrum'' (i.e., the set of ranks of the various elements) of a mapping semigroup.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0283104-2
Keywords: Finite semigroup, irreducible representation, character, K-transitive
Article copyright: © Copyright 1971 American Mathematical Society