The system of idempotents and the lattice of I-classes of reductive algebraic monoids

Putcha, Mohan S.; Renner, Lex E.

doi:10.1016/0021-8693(88)90225-6

Cited by 64 publications

(69 citation statements)

References 13 publications

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“…where P, P − are opposite parabolic subgroups, L = P ∩ P − , K L. These monoids have been classified by the author [18], [20] using the theory of buildings [30] and the ideas of Renner and the author [21] for reductive monoids. Renner and the author [22] obtained an analogue of the canonical monoid M for any finite group G of Lie type.…”

Section: Monoid Hecke Algebras 3519mentioning

confidence: 99%

Monoid Hecke algebras

Putcha

1997

Trans. Amer. Math. Soc.

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Abstract. This paper concerns the monoid Hecke algebras H introduced by Louis Solomon. We determine explicitly the unities of the orbit algebras associated with the two-sided action of the Weyl group W . We use this to:1. find a description of the irreducible representations of H, 2. find an explicit isomorphism between H and the monoid algebra of the Renner monoid R, 3. extend the Kazhdan-Lusztig involution and basis to H, and 4. prove, for a W × W orbit of R, the existence (conjectured by Renner) of generalized Kazhdan-Lusztig polynomials.

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Section: Monoid Hecke Algebras 3519mentioning

confidence: 99%

Monoid Hecke algebras

Putcha

1997

Trans. Amer. Math. Soc.

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“…Let 7 = max{7 3 |7 3 < use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700031748 [6] …”

Section: If / C T Let W = (I)mentioning

confidence: 99%

Maps into Dynkin diagrams arising from regular monoids

Augustine

Putcha

1989

J Aust Math Soc A

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It has been shown by one of the authors that the system of idempotents of monoids on a group G of Lie type with Dynkin diagram T can be classified by the following data: a partially ordered set 2/ with maximum element 1 and a map A: ^ -> 2 r with A(l) = V and with the property that for all J\, J 2 , J} €%f with J\ < J 2 < J3, any connected component of X(J 2 ) is contained in either X(J\) or A(J$). In this paper we show that X comes from a regular monoid if and only if the following conditions are satisfied:(1) % is a A-semilattice;If J\,J 2 e 2^with J\ < J 2 and if A" is a two element discrete subset of k{J\) U X(J 2 ), then X C X(J) for some J e % with /, < J < J 2 .1980 Mathematics subject classification {Amer. Math. Soc.) (1985 Revision): 20 G 99, 20 M 17. By a Coxeter group W = (W, F) is meant a group W generated by a subset F of elements of order 2, such that W has a presentation by the relations ( a e)m{o,e) = i, for a, (9 e T. We assume that the rank \T\ < 00. If a, 6 e F, define a -6 if m = m{a,d) > 3. In this way T becomes a graph, called the Coxeter graph of W. Note that a, 6 are not adjacent in the graph if and only if ad = 0a. It is customary to write a -6 to mean m -3, a -6 to mean w = 4 and

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“…Conversely, Aff = {eeA| a(e) = e} = {e £ A\ cr(CG(e)) = CG(e)} . Hence ACT\{0} -> 2r is bijective, and order preserving since by [13,Lemma 4.12], A\{0} -y {P | P 2 F} is order preserving. This proves (a).…”

Section: Reductive Monoids and Finite Monoidsmentioning

confidence: 99%

“…Here Y is the set of simple involutions relative to Ta and Ba , and 1(e) CY is such that CG (e) = BaWI{e)Ba . By the results of [13] one can obtain reductive algebraic monoids with these properties by choosing a high weight in general position. The resulting monoid is a cone on the canonical compactification.…”

Section: Reductive Monoids and Finite Monoidsmentioning

confidence: 99%

See 1 more Smart Citation

The canonical compactification of a finite group of Lie type

Putcha¹,

Renner²

1993

Trans. Amer. Math. Soc.

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Abstract.Let G be a finite group of Lie type. We construct a finite monoid J? having G as the group of units. JÍ has properties analogous to the canonical compactification of a reductive group. The complex representation theory of J! yields Harish-Chandra's philosophy of cuspidal representations of G . The main purpose of this paper is to determine the irreducible modular representations of J! . We then show that all the irreducible modular representations of G come (via the 1942 work of Clifford) from the one-dimensional representations of the maximal subgroups of J! . This yields a semigroup approach to the modular representation theory of G, via the full rank factorizations of the 'sandwich matrices' of Ji. We then determine the irreducible modular representations of any finite monoid of Lie type.

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The system of idempotents and the lattice of I-classes of reductive algebraic monoids

Cited by 64 publications

References 13 publications

Monoid Hecke algebras

Monoid Hecke algebras

Maps into Dynkin diagrams arising from regular monoids

The canonical compactification of a finite group of Lie type

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