Kazhdan–Lusztig Polynomial
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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 P_(q) 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 \ell-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 \ell(w) 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 T_w for w\in W over the ring \mathbb ^, q^/math>, with multiplication defined by :\begin T_y T_w &= T_, && \mbox\ell(yw) = \ell(y) + \ell(w) \\ (T_s + 1)(T_s - q) &= 0, && \mboxs \in S. \end The quadratic second relation implies that each generator is invertible in the Hecke algebra, with inverse . These inverses satisfy the relation (obtained by multiplying the quadratic relation for by −2''q''−1), and also the braid relations. From this it follows that the Hecke algebra has an automorphism ''D'' that sends ''q''1/2 to ''q''−1/2 and each to . More generally one has D(T_w)=T_^; also ''D'' can be seen to be an involution. The Kazhdan–Lusztig polynomials ''P''''yw''(''q'') are indexed by a pair of elements ''y'', ''w'' of ''W'', and uniquely determined by the following properties. *They are 0 unless ''y'' ≤ ''w'' (in the
Bruhat order In mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or Bruhat–Chevalley order or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion o ...
of ''W''), 1 if ''y'' = ''w'', and for ''y'' < ''w'' their degree is at most (''ℓ''(''w'') − ''ℓ''(''y'') − 1)/2. *The elements ::C'_w=q^\sum_ P_T_y :are invariant under the involution ''D'' of the Hecke algebra. The elements C'_w form a basis of the Hecke algebra as a \mathbb ^, q^/math>-module, called the Kazhdan–Lusztig basis. To establish existence of the Kazhdan–Lusztig polynomials, Kazhdan and Lusztig gave a simple recursive procedure for computing the polynomials ''Pyw''(''q'') in terms of more elementary polynomials denoted ''R''''yw''(''q''). defined by :T_^ = \sum_xD(R_)q^T_x. They can be computed using the recursion relations :R_ = \begin 0, & \mbox x \not\le y \\ 1, & \mbox x = y \\ R_, & \mbox sx < x \mbox sy < y \\ R_, & \mbox xs < x \mbox ys < y \\ (q-1)R_ + qR_, & \mbox sx > x \mbox sy < y \end The Kazhdan–Lusztig polynomials can then be computed recursively using the relation :q^D(P_) - q^P_ = \sum_(-1)^q^D(R_)P_ using the fact that the two terms on the left are polynomials in ''q''1/2 and ''q''−1/2 without
constant term In mathematics, a constant term is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial :x^2 + 2x + 3,\ the 3 is a constant term. After like terms are com ...
s. These formulas are tiresome to use by hand for rank greater than about 3, but are well adapted for computers, and the only limit on computing Kazhdan–Lusztig polynomials with them is that for large rank the number of such polynomials exceeds the storage capacity of computers.


Examples

*If ''y'' ≤ ''w'' then ''P''''y'',''w'' has constant term 1. *If ''y'' ≤ ''w'' and then ''P''''y'',''w'' = 1. *If ''w'' = ''w''0 is the longest element of a finite Coxeter group then ''P''''y'',''w'' = 1 for all ''y''. *If ''W'' is the Coxeter group ''A''1 or ''A''2 (or more generally any Coxeter group of rank at most 2) then ''P''''y'',''w'' is 1 if ''y''≤''w'' and 0 otherwise. *If ''W'' is the Coxeter group ''A''3 with generating set ''S'' = with ''a'' and ''c'' commuting then ''P''''b'',''bacb'' = 1 + ''q'' and ''P''''ac'',''acbca'' = 1 + ''q'', giving examples of non-constant polynomials. *The simple values of Kazhdan–Lusztig polynomials for low rank groups are not typical of higher rank groups. For example, for the split form of E8 th
most complicated Lusztig–Vogan polynomial
(a variation of Kazhdan–Lusztig polynomials: see below) is ::\begin 152 q^ &+ 3,472 q^ + 38,791 q^ + 293,021 q^ + 1,370,892 q^ + 4,067,059 q^ + 7,964,012 q^\\ &+ 11,159,003 q^ + 11,808,808 q^ + 9,859,915 q^ + 6,778,956 q^ + 3,964,369 q^ + 2,015,441 q^\\ &+ 906,567 q^9 + 363,611 q^8 + 129,820 q^7 + 41,239 q^6 + 11,426 q^5 + 2,677 q^4 + 492 q^3 + 61 q^2 + 3 q \end * showed that any polynomial with constant term 1 and non-negative integer coefficients is the Kazhdan–Lusztig polynomial for some pair of elements of some symmetric group.


Kazhdan–Lusztig conjectures

The Kazhdan–Lusztig polynomials arise as transition coefficients between their canonical basis and the natural basis of the Hecke algebra. The ''Inventiones'' paper also put forth two equivalent conjectures, known now as Kazhdan–Lusztig conjectures, which related the values of their polynomials at 1 with representations of complex
semisimple Lie group In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
s and
Lie algebras In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
, addressing a long-standing problem in representation theory. Let ''W'' be a finite
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 ...
. For each w ∈ ''W'' denote by be the
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 ...
of highest weight where ρ is the half-sum of positive roots (or
Weyl vector In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the c ...
), and let be its irreducible quotient, the simple highest weight module of highest weight . Both and are locally-finite weight modules over the complex semisimple Lie algebra ''g'' with the Weyl group ''W'', and therefore admit an
algebraic character An algebraic character is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite-dimensional representation and is analogous to the Harish-Chandra character of the re ...
. Let us write ch(''X'') for the character of a ''g''-module ''X''. The Kazhdan–Lusztig conjectures state: : \operatorname(L_w)=\sum_(-1)^P_(1)\operatorname(M_y) : \operatorname(M_w)=\sum_P_(1)\operatorname(L_y) where is the element of maximal length of the Weyl group. These conjectures were proved over characteristic 0 algebraically closed fields independently by and by . The methods introduced in the course of the proof have guided development of representation theory throughout the 1980s and 1990s, under the name ''geometric representation theory''.


Remarks

1. The two conjectures are known to be equivalent. Moreover, Borho–Jantzen's translation principle implies that can be replaced by for any dominant integral weight . Thus, the Kazhdan–Lusztig conjectures describe the Jordan–Hölder multiplicities of Verma modules in any regular integral block of Bernstein–Gelfand–Gelfand
category O In the representation theory of semisimple Lie algebras, Category O (or category \mathcal) is a category (mathematics), category whose Object (category theory), objects are certain Lie algebra representation, representations of a Semisimple Lie alge ...
. 2. A similar interpretation of ''all'' coefficients of Kazhdan–Lusztig polynomials follows from the ''Jantzen conjecture'', which roughly says that individual coefficients of are multiplicities of in certain subquotient of the Verma module determined by a canonical filtration, the Jantzen filtration. The Jantzen conjecture in regular integral case was proved in a later paper of . 3.
David Vogan David Alexander Vogan, Jr. (born September 8, 1954) is a mathematician at the Massachusetts Institute of Technology who works on unitary representations of simple Lie groups. While studying at the University of Chicago, he became a Putnam Fellow ...
showed as a consequence of the conjectures that :P_(q) = \sum_ q^i \dim(\operatorname^(M_y,L_w)) and that vanishes if is odd, so the dimensions of all such Ext groups in category ''O'' are determined in terms of coefficients of Kazhdan–Lusztig polynomials. This result demonstrates that all coefficients of the Kazhdan–Lusztig polynomials of a finite Weyl group are non-negative integers. However, positivity for the case of a finite Weyl group ''W'' was already known from the interpretation of coefficients of the Kazhdan–Lusztig polynomials as the dimensions of intersection cohomology groups, irrespective of the conjectures. Conversely, the relation between Kazhdan–Lusztig polynomials and the Ext groups theoretically can be used to prove the conjectures, although this approach to proving them turned out to be more difficult to carry out. 4. Some special cases of the Kazhdan–Lusztig conjectures are easy to verify. For example, ''M''1 is the antidominant Verma module, which is known to be simple. This means that ''M''1 = ''L''1, establishing the second conjecture for ''w'' = 1, since the sum reduces to a single term. On the other hand, the first conjecture for ''w'' = ''w''0 follows from the
Weyl character formula In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the ch ...
and the formula for the character of a Verma module, together with the fact that all Kazhdan–Lusztig polynomials P_ are equal to 1. 5. Kashiwara (1990) proved a generalization of the Kazhdan–Lusztig conjectures to symmetrizable
Kac–Moody algebra In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a g ...
s.


Relation to intersection cohomology of Schubert varieties

By the
Bruhat decomposition In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) ''G'' = ''BWB'' of certain algebraic groups ''G'' into cells can be regarded as a general expression of the principle ...
the space ''G''/''B'' of the algebraic group ''G'' with Weyl group ''W'' is a disjoint union of affine spaces ''X''''w'' parameterized by elements ''w'' of ''W''. The closures of these spaces are called Schubert varieties, and Kazhdan and Lusztig, following a suggestion of Deligne, showed how to express Kazhdan–Lusztig polynomials in terms of intersection cohomology groups of Schubert varieties. More precisely, the Kazhdan–Lusztig polynomial ''P''''y'',''w''(''q'') is equal to :P_(q) = \sum_iq^i\dim IH^_(\overline) where each term on the right means: take the complex IC of sheaves whose hyperhomology is the
intersection homology 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 the
Schubert variety In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces ''V'', specified using line ...
of ''w'' (the closure of the cell ), take its cohomology of degree , and then take the dimension of the stalk of this sheaf at any point of the cell whose closure is the Schubert variety of ''y''. The odd-dimensional cohomology groups do not appear in the sum because they are all zero. This gave the first proof that all coefficients of Kazhdan–Lusztig polynomials for finite Weyl groups are non-negative integers.


Generalization to real groups

Lusztig–Vogan polynomials (also called Kazhdan–Lusztig polynomials or Kazhdan–Lusztig–Vogan polynomials) were introduced in . They are analogous to Kazhdan–Lusztig polynomials, but are tailored to representations of ''real'' semisimple Lie groups, and play major role in the conjectural description of their
unitary dual In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G'' ...
s. Their definition is more complicated, reflecting relative complexity of representations of real groups compared to complex groups. The distinction, in the cases directly connection to representation theory, is explained on the level of
double coset In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let be a group, and let and be subgroups. Let act on by left mult ...
s; or in other terms of actions on analogues of complex
flag manifold In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smo ...
s ''G''/''B'' where ''G'' is a complex Lie group and ''B'' a
Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
. The original (K-L) case is then about the details of decomposing :B\backslash G/ B, a classical theme of the
Bruhat decomposition In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) ''G'' = ''BWB'' of certain algebraic groups ''G'' into cells can be regarded as a general expression of the principle ...
, and before that of Schubert cells in a
Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...
. The L-V case takes a
real form In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is the complexification of ''g''0: : \mathf ...
of ''G'', a
maximal compact subgroup In mathematics, a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the class ...
in that
semisimple group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...
, and makes the
complexification In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...
''K'' of . Then the relevant object of study is :K\backslash G/ B. In March 2007, it was announced that the L–V polynomials had been calculated for the split form of ''E''8.


Generalization to other objects in representation theory

The second paper of Kazhdan and Lusztig established a geometric setting for definition of Kazhdan–Lusztig polynomials, namely, the
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
of singularities of Schubert varieties in the
flag variety In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smo ...
. Much of the later work of Lusztig explored analogues of Kazhdan–Lusztig polynomials in the context of other natural singular algebraic varieties arising in representation theory, in particular, closures of
nilpotent orbit In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important role in representation theory of real and complex semisimple Lie groups and semisimple Lie algebras. Definition An element ''X'' of a semisimple Lie ...
s and quiver varieties. It turned out that the representation theory of
quantum groups In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...
, modular Lie algebras and
affine Hecke algebra In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials. Definition Let V be a Euclidean space of a finite dimension and \ ...
s are all tightly controlled by appropriate analogues of Kazhdan–Lusztig polynomials. They admit an elementary description, but the deeper properties of these polynomials necessary for representation theory follow from sophisticated techniques of modern algebraic geometry and homological algebra, such as the use of
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 ...
,
perverse sheaves The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space ''X'', which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was intro ...
and Beilinson–Bernstein–Deligne decomposition. The coefficients of the Kazhdan–Lusztig polynomials are conjectured to be the dimensions of some homomorphism spaces in Soergel's bimodule category. This is the only known positive interpretation of these coefficients for arbitrary Coxeter groups.


Combinatorial theory

Combinatorial properties of Kazhdan–Lusztig polynomials and their generalizations are a topic of active current research. Given their significance in representation theory and algebraic geometry, attempts have been undertaken to develop the theory of Kazhdan–Lusztig polynomials in purely combinatorial fashion, relying to some extent on geometry, but without reference to intersection cohomology and other advanced techniques. This has led to exciting developments in algebraic combinatorics, such as ''pattern-avoidance phenomenon''. Some references are given in the textbook of . A research monograph on the subject is . , there is no known combinatorial interpretation of all the coefficients of the Kazhdan–Lusztig polynomials (as the cardinalities of some natural sets) even for the symmetric groups, though explicit formulas exist in many special cases.


Inequality

Kobayashi (2013) proved that values of Kazhdan–Lusztig polynomials at q=1 for crystallographic Coxeter groups satisfy certain strict inequality: Let (W, S) be a crystallographic Coxeter system and its Kazhdan–Lusztig polynomials. If u and P_(1)>1, then there exists a reflection t such that P_(1)>P_(1)>0.


References

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External links


Readings
from Spring 2005 course on Kazhdan–Lusztig Theory at
U.C. Davis The University of California, Davis (UC Davis, UCD, or Davis) is a public land-grant research university near Davis, California. Named a Public Ivy, it is the northernmost of the ten campuses of the University of California system. The institut ...
by Monica Vazirani * *The GA
programs
for computing Kazhdan–Lusztig polynomials. *Fokko du Cloux'
Coxeter
software for computing Kazhdan–Lusztig polynomials for any Coxeter group *Atla
software
for computing Kazhdan–Lusztig-Vogan polynomials. {{DEFAULTSORT:Kazhdan-Lusztig polynomial Polynomials Representation theory of Lie groups Representation theory of Lie algebras Algebraic combinatorics