The characters of the symmetric inverse semigroup

Munn, W. D.

doi:10.1017/s0305004100031947

Cited by 61 publications

(56 citation statements)

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“…The monoid ~ has been studied in semigroup theory under the name symmetric inverse semigroup ( [7], [16]) but it has not been studied in the spirit of the combinatorics of Coxeter groups.…”

Section: Introductionmentioning

confidence: 99%

The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field

Solomon

1990

Geom Dedicata

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AnSTRACT. Let G = GLn(Fq) be the finite general linear group and let M = M,(Fq) be the monoid of all n x n matrices over Fq. Let B be a Borel subgroup of G, let W be the subgroup of permutation matrices, and let ~ D W be the monoid of all zero-one matrices which have at most one non-zero entry in each row and each column. The monoid ~ plays the same role for M that the Weyl group W does for G. In particular there is a length function on ~ which extends the length function on W and a C-algebra Hc(M, B) which includes lwahori's 'Hecke algebra' Hc(G, B) and shares many of its properties. Added in proof.Concerning the remarks which precede Theorem 4.41 and the related footnote: About a week ago Putcha informed me that there is a gap in my argument for the semisimplicity of C [M]. Thus, at this writing, the only proof of semisimplicity is the one by Oknifiski and Putcha.

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“…The monoid ~ has been studied in semigroup theory under the name symmetric inverse semigroup ( [7], [16]) but it has not been studied in the spirit of the combinatorics of Coxeter groups.…”

Section: Introductionmentioning

confidence: 99%

The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field

Solomon

1990

Geom Dedicata

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“…Linear representation theory of semigroups begins with the 1942 paper of Clifford [2] on the irreducible representations of completely 0-simple semigroups. Clifford's results became applicable to arbitrary finite semigroups via the work of Munn [9] and Ponizovskii [13]. In all these and subsequent works, the concept of a 'sandwich matrix' of a completely 0-simple semigroup plays a critical role.…”

Section: Introductionmentioning

confidence: 99%

“…[3]. Clifford [2] showed that the irreducible representations of J° can be obtained from the irreducible representations <f> of H via a full rank factorization of d)(S) over C. Munn [9] and Ponizovskii [13] showed that Co[/°] is semisimple if and only if S is invertible over C[H], cf. [3].…”

Section: Preliminariesmentioning

confidence: 99%

Sandwich Matrices, Solomon Algebras, and Kazhdan-Lusztig Polynomials

Putcha¹

1993

Transactions of the American Mathematical Society

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Abstract. Sandwich matrices have proved to be of importance in semigroup theory for the last 50 years. The work of the author on algebraic monoids leads to sandwich matrices in group theory. In this paper, we find some connections between sandwich matrices and the Hecke algebras (for monoids) introduced recently by Louis Solomon. At the local level we then obtain an explicit isomorphism between Solomon's Hecke algebra and the complex monoid algebra of the Renner monoid. In the simplest case of monoids associated with a Borel subgroup, we find that the entries of the inverse of the sandwich matrix, as well as those of the related structure matrix of Solomon's Hecke algebra are 'almost' the polynomials Rx, y associated with the Kazhdan-Lusztig polynomials.

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“…For any field K , we consider the rook monoid algebra R n . This has been studied (mostly in the case where K has characteristic zero) by, amongst others, Munn [13], Grood [6] and Solomon [14]. More generally, Solomon [15] defined a q-analogue of the rook monoid algebra and its representation theory was investigated by various authors in [1,7,8].…”

Section: Introductionmentioning

confidence: 99%