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Séminaire Lotharingien de Combinatoire, B49b (2003), 30 pp.

# Francesco Brenti

#
Kazhdan-Lusztig Polynomials: History, Problems, and Combinatorial Invariance

**Abstract.**
This is an expository paper on Kazhdan-Lusztig polynomials,
and particularly on the recent result concerning their combinatorial
invariance,
based on my lectures at the 49th Seminaire Lotharingien de
Combinatoire. The paper consists of three parts entitled: History; Problems;
and Combinatorial Invariance. In the first one we give the main definitions
and facts about the Bruhat order and graph, and about the Kazhdan-Lusztig
and *R*-polynomials. In the second one we present, as a sample, two results,
one on the *R*-polynomials and one on the Kazhdan-Lusztig polynomials,
which in the author's opinion illustrate very well the rich combinatorics
that hides in these polynomials. Finally, in the third part, we explain
the recent result that the Kazhdan-Lusztig and *R*-polynomials depend only
on a certain poset, and mention some open problems and conjectures related
to this.

Received: February 7, 2003.
Revised: July 25, 2003.
Accepted: August 2, 2003.
Final version: August 12, 2003.

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