41 research outputs found

    Bijection between Conjugacy Classes and Irreducible Representations of Finite Inverse Semigroups

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    In this paper we show that the irreducible representations of a finite inverse semigroup SS over an algebraically closed field FF are in bijection with the conjugacy classes of SS if the characteristic of FF is zero or a prime number that does not divide the order of any maximal subgroup of SS

    Conjugacy Classes of Renner Monoids

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    In this paper we describe conjugacy classes of a Renner monoid RR with unit group WW, the Weyl group. We show that every element in RR is conjugate to an element ueue where u∈Wu\in W and ee is an idempotent in a cross section lattice. Denote by W(e)W(e) and Wβˆ—(e)W_*(e) the centralizer and stabilizer of eβˆˆΞ›e\in \Lambda in WW, respectively. Let W(e)W(e) act by conjugation on the set of left cosets of Wβˆ—(e)W_*(e) in WW. We find that ueue and veve (u,v∈Wu, v\in W) are conjugate if and only if uWβˆ—(e)uW_*(e) and vWβˆ—(e)vW_*(e) are in the same orbit. As consequences, there is a one-to-one correspondence between the conjugacy classes of RR and the orbits of this action. We then obtain a formula for calculating the number of conjugacy classes of RR, and describe in detail the conjugacy classes of the Renner monoid of some J\cal J-irreducible monoids. We then generalize the Munn conjugacy on a rook monoid to any Renner monoid and show that the Munn conjugacy coincides with the semigroup conjugacy, action conjugacy, and character conjugacy. We also show that the number of inequivalent irreducible representations of RR over an algebraically closed field of characteristic zero equals the number of the Munn conjugacy classes in RR.Comment: A reference ([13]) and Corollary 4.5 are added to show the connection between the result in Theorem 4.4 of the previous version and the results in [13]. A paragraph on page 12 is new to show that Theorem 4.4 can also be deduced from the results in [13]. Two necessary concepts from [13] to describe the connection are inserted in Section 2.

    The Renner Monoids and Cell Decompositions of the Classical Algebraic Monoids

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    The Renner monoids, cross section lattices and cell decompositions of the classical algebraic monoids are studied. The Renner monoid is extremely important in the theory of reductive algebraic monoids. It is well know that the Renner monoid [Special characters omitted.] of Mn (K ) is the monoid of all zero-one matrices which have at most one entry equal to one in each row and column, i.e., [Special characters omitted.] consists of injective partial maps on a set of n elements. We obtain that the Renner monoids of the symplectic algebraic monoids and special orthogonal algebraic monoids turn out to be submonoids of [Special characters omitted.] consisting of symplectic and special orthogonal 1-1 partial maps, respectively. The cardinalities of the Renner monoids are obtained, as well. The cross section lattice is another very important concept in the theory of irreducible algebraic monoids. The cross section lattices of the symplectic and special orthogonal algebraic monoids are explicitly characterized. The cell decompositions of symplectic algebraic monoids and special orthogonal monoids are explicitly determined. Each cell here turns out to be an intersection of the monoid with some cell of Mn ( K )
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