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 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.

**[1]**Dennis Allen, Jr.,*Some relationships between local and global structure of finite semigroups*, Ph.D. Thesis, University of California, Berkeley, 1968.**[2]**Emil Artin, Cecil J. Nesbitt, and Robert M. Thrall,*Rings with Minimum Condition*, University of Michigan Publications in Mathematics, no. 1, University of Michigan Press, Ann Arbor, Mich., 1944. MR**0010543****[3]**W. Burnside,*Theory of groups of finite order*, Dover Publications, Inc., New York, 1955. 2d ed. MR**0069818****[4]**A. H. Clifford and G. B. Preston,*The algebraic theory of semigroups. Vol. I*, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR**0132791****[5]**Charles W. Curtis and Irving Reiner,*Representation theory of finite groups and associative algebras*, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR**0144979****[6]**Walter Feit,*Characters of finite groups*, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR**0219636****[7]**G. Frobenius,*Über die Charaktere der mehrfach transitiven Gruppen*, Sitzungsbericht Preuss. Akad. Berlin**1904**, 558-571.**[8]***Algebraic theory of machines, languages, and semigroups*, Edited by Michael A. Arbib. With a major contribution by Kenneth Krohn and John L. Rhodes, Academic Press, New York-London, 1968. MR**0232875****[9]**Dudley E. Littlewood,*The Theory of Group Characters and Matrix Representations of Groups*, Oxford University Press, New York, 1940. MR**0002127****[10]**Francis D. Murnaghan,*The theory of group representations*, Dover Publications, Inc., New York, 1963. MR**0175982****[11]**John Rhodes,*Some results on finite semigroups*, J. Algebra**4**(1966), 471–504. MR**201546**, https://doi.org/10.1016/0021-8693(66)90035-4**[12]**John Rhodes,*Characters and complexity of finite semigroups*, J. Combinatorial Theory**6**(1969), 67–85. MR**236293****[13]**John Rhodes and Bret R. Tilson,*Lower bounds for complexity of finite semigroups*, J. Pure Appl. Algebra**1**(1971), no. 1, 79–95. MR**285649**, https://doi.org/10.1016/0022-4049(71)90012-0**[14]**John Rhodes and Yechezkel Zalcstein,*Elementary representation and character theory of finite semigroups and its application*, Monoids and semigroups with applications (Berkeley, CA, 1989) World Sci. Publ., River Edge, NJ, 1991, pp. 334–367. MR**1142387****[15]**Bret Tilson,*Complexity of two*-*class semigroups*, Advances in Math. (to appear).**[16]**Tosiro Tsuzuku,*On multiple transitivity of permutation groups*, Nagoya Math. J.**18**(1961), 93–109. MR**124420****[17]**Helmut Wielandt,*Finite permutation groups*, Translated from the German by R. Bercov, Academic Press, New York-London, 1964. MR**0183775****[18]**Y. Zalcstein,*Complexity and character theory of finite semigroups*, Ph.D. Thesis, University of California, Berkeley, 1968.**[19]**-,*On the group-complexity of finite semigroups*, Advances in Math. (to appear).

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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