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Définition combinatoire des polynômes de Kazhdan-Lusztig

Abstract : The theory of Coxeter groups, originating in the study
of isometry groups, provides a connection
between many areas in algebra and geometry, ranging
from representation theory (of Coxeter groups, Lie
groups, Lie algebras and Hecke algebras) and algebraic geometry (Schubert varieties)
to combinatorics (the Bruhat order).
Kazhdan-Lusztig polynomials appear in rather different
forms in several of those fields : those polynomials can be defined as the
coordinates of a remarkable basis of the Hecke algebra,
(which yields nontrivial representations
of this algebra),
their value at 1 appears in the decomposition
of certain Verma modules, and their coefficients can be
interpreted as dimensions of certain local homology spaces.
The original definition of the Kazhdan-Lusztig polynomials
translates into an inductive (and involved) computational definition,
which naturally leads to the open question about the possibility of a
purely combinatorial definition. This report tries to show
some of the latest efforts that have been made towards answering
this question. Our main result is
that an isomorphism between two lower Bruhat intervals preserves
Kazhdan-Lusztig polynomials. We also provide heuristic
arguments (both theoretical and computational) that support
the conjecture that this also holds for an isomorphism
between completely compressible
intervals in finite Coxeter groups.\newline

Keywords : Coxeter group, Kazhdan-Lusztig polynomial,
reflection subgroup, Bruhat interval, special matching,
completely compressible interval
Document type :
Theses
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https://tel.archives-ouvertes.fr/tel-00137528
Contributor : Ewan Delanoy <>
Submitted on : Tuesday, March 20, 2007 - 2:53:26 PM
Last modification on : Wednesday, July 8, 2020 - 12:43:13 PM
Long-term archiving on: : Friday, September 21, 2012 - 1:05:34 PM

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  • HAL Id : tel-00137528, version 1

Citation

Ewan Delanoy. Définition combinatoire des polynômes de Kazhdan-Lusztig. Mathematics [math]. Université Claude Bernard - Lyon I, 2006. English. ⟨tel-00137528⟩

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