In the mathematical field of
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, a Kazhdan–Lusztig polynomial
is a member of a family of
integral polynomials introduced by . They are indexed by pairs of elements ''y'', ''w'' of a
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
''W'', which can in particular be the
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
of a
Lie group.
Motivation and history
In the spring of 1978 Kazhdan and Lusztig were studying
Springer representations of the Weyl group of an algebraic group on
-adic cohomology groups related to unipotent conjugacy classes. They found a new construction of these representations over the complex numbers . The representation had two natural bases, and the transition matrix between these two bases is essentially given by the Kazhdan–Lusztig polynomials. The actual Kazhdan–Lusztig construction of their polynomials is more elementary. Kazhdan and Lusztig used this to construct a
canonical basis
In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:
* In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the ...
in the
Hecke algebra
In mathematics, the Hecke algebra is the algebra generated by Hecke operators.
Properties
The algebra is a commutative ring.
In the classical elliptic modular form theory, the Hecke operators ''T'n'' with ''n'' coprime to the level acting o ...
of the Coxeter group and its representations.
In their first paper Kazhdan and Lusztig mentioned that their polynomials were related to the failure of local
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
for
Schubert varieties. In they reinterpreted this in terms of the
intersection cohomology In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them ov ...
of
Mark Goresky
Robert Mark Goresky is a Canadian mathematician who invented intersection homology with his advisor and life partner Robert MacPherson.
Career
Goresky received his Ph.D. from Brown University in 1976. His thesis, titled ''Geometric Cohomology a ...
and
Robert MacPherson, and gave another definition of such a basis in terms of the dimensions of certain intersection cohomology groups.
The two bases for the Springer representation reminded Kazhdan and Lusztig of the two bases for the
Grothendieck group of certain infinite dimensional representations of semisimple Lie algebras, given by
Verma module Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.
Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Spe ...
s and
simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every cy ...
s. This analogy, and the work of
Jens Carsten Jantzen
Jens Carsten Jantzen (born 18 October 1948, in Störtewerkerkoog, Nordfriesland) is a mathematician working on representation theory and algebraic groups, who introduced the Jantzen filtration, the Jantzen sum formula, and translation funct ...
and
Anthony Joseph
Anthony Joseph (born 12 November 1966 in Port of Spain, Trinidad and Tobago) is a British/Trinidadian poet, novelist, musician and academic.
Biography
Joseph was born in Port of Spain, Trinidad, where he was raised by his grandparents. He b ...
relating
primitive ideal
In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals.
Primitive ideals ar ...
s of
enveloping algebras to representations of Weyl groups, led to the Kazhdan–Lusztig conjectures.
Definition
Fix a Coxeter group ''W'' with generating set ''S'', and write
for the length of an element ''w'' (the smallest length of an expression for ''w'' as a product of elements of ''S''). The
Hecke algebra
In mathematics, the Hecke algebra is the algebra generated by Hecke operators.
Properties
The algebra is a commutative ring.
In the classical elliptic modular form theory, the Hecke operators ''T'n'' with ''n'' coprime to the level acting o ...
of ''W'' has a basis of elements
for
over the ring