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]**E. Artin, C. J. Nesbitt and R. M. Thrall,*Rings with minimum condition*, Univ. of Michigan Publ. Math., no. 1, Univ. of Michigan Press, Ann Arbor, Mich., 1944. MR**6**, 33. MR**0010543 (6:33e)****[3]**W. Burnside,*Theory of groups of finite order*, Dover, New York, 1950. MR**0069818 (16:1086c)****[4]**A. H. Clifford and G. B. Preston,*The algebraic theory of semigroups*. Vol. 1, Math. Surveys, no. 7, Amer. Math. Soc., Providence, R. I., 1961. MR**24**#A2627. MR**0132791 (24:A2627)****[5]**C. W. Curtis and I. Reiner,*Representation theory of finite groups and associative algebras*, Pure and Appl. Math., vol. XI, Interscience, New York, 1962. MR**26**#2519. MR**0144979 (26:2519)****[6]**W. Feit,*Characters of finite groups*, Benjamin, New York, 1967. MR**36**#2715. MR**0219636 (36:2715)****[7]**G. Frobenius,*Über die Charaktere der mehrfach transitiven Gruppen*, Sitzungsbericht Preuss. Akad. Berlin**1904**, 558-571.**[8]**Kenneth Krohn, John Rhodes and Bret Tilson,*Algebraic theory of machines, languages and semigroups*(M. A. Arbib, editor), Academic Press, New York, 1968, chaps. 1, 7, 8. MR**38**#1198. MR**0232875 (38:1198)****[9]**Dudley E. Littlewood,*The theory of group characters and matrix representations of groups*, Oxford Univ. Press, New York, 1940. MR**2**, 3. MR**0002127 (2:3a)****[10]**F. D. Murnaghan,*Theory of representations of groups*, Johns Hopkins Press, Baltimore, Md., 1938; reprint, Dover, New York, 1963. MR**31**#258. MR**0175982 (31:258)****[11]**John Rhodes,*Some results on finite semigroups*, J. Algebra**4**(1966), 471-504. MR**34**#1428. MR**0201546 (34:1428)****[12]**-,*Characters and complexity of finite semigroups*, J. Combinatorial Theory**6**(1969), 67-85. MR**38**#4590. MR**0236293 (38:4590)****[13]**John Rhodes and Bret Tilson,*Lower bounds for complexity of finite semigroups*, J. Pure Appl. Algebra**1**(1971), 79-95. MR**0285649 (44:2867)****[14]**John Rhodes and Y. Zalcstein,*Elementary representation and character theory of finite semigroups and its applications*, Advances in Math. (to appear). MR**1142387 (92k:20129)****[15]**Bret Tilson,*Complexity of two*-*class semigroups*, Advances in Math. (to appear).**[16]**T. Tsuzuku,*On multiple transitivity of permutation groups*, Nagoya Math. J.**18**(1961), 93-109. MR**23**#A1732. MR**0124420 (23:A1732)****[17]**H. Wielandt,*Finite permutation groups*, Academic Press, New York, 1964. MR**32**#1252. MR**0183775 (32:1252)****[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