Analogue of the Bruhat–Chevalley Order for Reductive Monoids

Pennell, Edwin A.; Putcha, Mohan S.; Renner, Lex E.

doi:10.1006/jabr.1997.7111

Cited by 38 publications

(32 citation statements)

References 8 publications

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“…We have studied this order in [12] (with Pennell and Renner). In particular we found an algebraic description of this order that we now describe.…”

Section: Bruhat-chevalley Ordermentioning

confidence: 99%

Shellability in reductive monoids

Putcha

2001

Trans. Amer. Math. Soc.

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Abstract. The purpose of this paper is to extend to monoids the work of Björner, Wachs and Proctor on the shellability of the Bruhat-Chevalley order on Weyl groups. Let M be a reductive monoid with unit group G, Borel subgroup B and Weyl group W . We study the partially ordered set of B × Borbits (with respect to Zariski closure inclusion) within a G × G-orbit of M . This is the same as studying a W × W -orbit in the Renner monoid R. Such an orbit is the retract of a 'universal orbit', which is shown to be lexicograhically shellable in the sense of Björner and Wachs.

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“…We have studied this order in [12] (with Pennell and Renner). In particular we found an algebraic description of this order that we now describe.…”

Section: Bruhat-chevalley Ordermentioning

confidence: 99%

Shellability in reductive monoids

Putcha

2001

Trans. Amer. Math. Soc.

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“…The description of (1a)(iii) is similar to the result for reductive algebraic monoids obtained in [9].…”

Section: Proofmentioning

confidence: 59%

“…Our first aim is to determine these relations explicitly. To this end we introduce three normal forms of the elements of W, which are similar to the standard form of an element of a Renner monoid introduced in [9].…”

Section: Remarkmentioning

confidence: 99%

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Extending the Bruhat order and the length function from the Weyl group to the Weyl monoid

Mokler¹

2004

Journal of Algebra

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For a symmetrizable Kac-Moody algebra the category of admissible representations of the category O is an analogue of the category of finite-dimensional representations of a semisimple Lie algebra. The monoid which is associated to this category and the category of restricted duals by a Tannaka-Krein reconstruction contains the Kac-Moody group as open dense unit group. It has similar properties as a reductive algebraic monoid. In particular, there are Bruhat and Birkhoff decompositions, the Weyl group replaced by the Weyl monoid [C. Mokler, An analogue of a reductive algebraic monoid, whose unit group is a Kac-Moody group, arXiv: math.AG/0204246, Mem. Amer. Math. Soc., in press]. We determine the closure relations of the Bruhat and Birkhoff cells, which give extensions of the Bruhat order from the Weyl group to the Weyl monoid. We show that the Bruhat and Birkhoff cells are irreducible and principal open in their closures. We give product decompositions of the Bruhat and Birkhoff cells. We define extended length functions, which are compatible with the extended Bruhat orders. We show a generalization of some of the Tits axioms for twin BN-pairs.

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“…The order has been determined in [5] and more explicitly in [11,12]. In particular for e, e ∈ , x, y ∈ W , e ≤ e ⇒ xey ≤ xe y (12) and for y ∈ D(e) −1 , y ∈ D(e ) −1 , ey ≤ e y ⇐⇒ y ≤ zy for some z ∈ W (e)…”

Section: Preliminariesmentioning

confidence: 99%

Nilpotent variety of a reductive monoid

Putcha

2007

J Algebr Comb

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In this paper we study the variety M nil of nilpotent elements of a reductive monoid M. In general this variety has a completely different structure than the variety G uni of unipotent elements of the unit group G of M. When M has a unique non-trivial minimal or maximal G × G-orbit, we find a precise description of the irreducible components of M nil via the combinatorics of the Renner monoid of M and the Weyl group of G. In particular for a semisimple monoid M, we find necessary and sufficient conditions for the variety M nil to be irreducible.

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Analogue of the Bruhat–Chevalley Order for Reductive Monoids

Cited by 38 publications

References 8 publications

Shellability in reductive monoids

Shellability in reductive monoids

Extending the Bruhat order and the length function from the Weyl group to the Weyl monoid

Nilpotent variety of a reductive monoid

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