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On the combinatorics of B×B-orbits on group compactifications

✍ Scribed by Yu Chen; Matthew Dyer

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
169 KB
Volume
263
Category
Article
ISSN
0021-8693

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✦ Synopsis

It is shown that there is an order isomorphism ϕ from the poset V of B × B-orbits on the wonderful compactification of a semi-simple adjoint group G with Weyl group W to an interval in reverse Chevalley-Bruhat order on a non-canonically associated Coxeter group W (in general neither finite nor affine). Moreover, ϕ preserves the corresponding Kazhdan-Lusztig polynomials. Springer's (partly conjectural) construction of Kazhdan-Lusztig polynomials for the analogues of V for general Coxeter groups W is completed by reducing it by a similar order isomorphism to known results involving a "twisted" Chevalley-Bruhat order on W .

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Let S be a finite subset of a group G, |S| = n, and let g ∈ S • S. Then g induces a partial function λ g : S → S by λ g (s) = t if and only if st = g and λ g (s) is not defined if g ∈ sS. For every g ∈ S • S, λ g is a one-to-one mapping. In this note we describe the groups which have a finite genera