The Maximal Chains of the Extended Bruhat Orders on the  × -Orbits of an Infinite Renner Monoid

Abstract

Let (, S) be a Coxeter system. For ∊, δ ∊ {+, −} we introduce and investigate combinatorially certain partial orders ≤ _∊δ, called extended Bruhat orders, on a  × -set (N, C), which depends on , a subset N ⊆ S, and a component C ⊆ N. We determine the length of the maximal chains between two elements x, y ∊ (N, C), x ≤ _∊δ y.

These posets generalize  equipped with its Bruhat order. They include the  × -orbits of the Renner monoids of reductive algebraic monoids and of some infinite-dimensional generalizations which are equipped with the partial orders obtained by the closure relations of the Bruhat and Birkhoff cells. They also include the  × -orbits of certain posets obtained by generalizing the closure relation of the Bruhat cells of the wonderful compactification.

Key Words:

• Bruhat–Chevalley order
• Extended Bruhat order
• Renner monoid

Mathematics Subject Classification 2000:

• Primary 06A07
• Secondary 20G99, 22E65

ACKNOWLEDGMENTS

I want to thank the Marie Curie Research Training Network “Liegrits” of the European Community (project number MRTN-CT 2003-505078) for the possibility of attending training conferences and workshops in various parts of Europe.

In particular, I want to thank the Deutsche Forschungsgemeinschaft for providing my financial support when the results of this article have been found out and written down.

Notes

Communicated by V. Gould.