Studies in the representation theory of finite semigroups

Author:
Yechezkel Zalcstein

Journal:
Trans. Amer. Math. Soc. **161** (1971), 71-87

MSC:
Primary 20.90

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 of *U*, 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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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