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The combinatorics of twisted involutions in Coxeter groups

Author: Axel Hultman
Journal: Trans. Amer. Math. Soc. 359 (2007), 2787-2798
MSC (2000): Primary 20F55; Secondary 06A07
Published electronically: January 26, 2007
MathSciNet review: 2286056
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Abstract: The open intervals in the Bruhat order on twisted involutions in a Coxeter group are shown to be PL spheres. This implies results conjectured by F. Incitti and sharpens the known fact that these posets are Gorenstein$ ^*$ over $ \mathbb{Z}_2$.

We also introduce a Boolean cell complex which is an analogue for twisted involutions of the Coxeter complex. Several classical Coxeter complex properties are shared by our complex. When the group is finite, it is a shellable sphere, shelling orders being given by the linear extensions of the weak order on twisted involutions. Furthermore, the $ h$-polynomial of the complex coincides with the polynomial counting twisted involutions by descents. In particular, this gives a type-independent proof that the latter is symmetric.

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Additional Information

Axel Hultman
Affiliation: Department of Mathematics, KTH, SE-100 44 Stockholm, Sweden

Keywords: Twisted involutions, Bruhat order, weak order, Coxeter complexes
Received by editor(s): November 23, 2004
Received by editor(s) in revised form: April 25, 2005
Published electronically: January 26, 2007
Additional Notes: This work was supported by the European Commission’s IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe”.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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