Iwahori–Hecke Algebra

In mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a deformation of the group algebra of a Coxeter group.

Hecke algebras are quotients of the group rings of Artin braid groups. This connection found a spectacular application in Vaughan Jones' construction of new invariants of knots. Representations of Hecke algebras led to discovery of quantum groups by Michio Jimbo. Michael Freedman proposed Hecke algebras as a foundation for topological quantum computation.

Hecke algebras of Coxeter groups

Start with the following data:

* (''W'', ''S'') is a Coxeter system with the Coxeter matrix ''M'' = (''m''_''st''),
* ''R'' is a commutative ring with identity.
* is a family of units of ''R'' such that ''q_s'' = ''q_t'' whenever ''s'' and ''t'' are conjugate in ''W''
* ''A'' is the ring of Laurent polynomials over Z with indeterminates ''q_s'' (and the above restriction that ''q_s'' = ''q_t'' whenever ''s'' and ''t'' are conjugated), that is ''A'' = Z 'q''

Multiparameter Hecke Algebras

The ''multiparameter Hecke algebra'' ''H_R(W,S,q)'' is a unital, associative ''R''-algebra with generators ''T_s'' for all ''s'' ∈ ''S'' and relations:
* Braid Relations: ''T_s T_t T_s'' ... = ''T_t T_s T_t'' ..., where each side has ''m_st'' < ∞ factors and ''s,t'' belong to ''S''.
* Quadratic Relation: For all ''s'' in ''S'' we have: (''T_s'' - ''q_s'')(''T_s'' + 1) = 0.

Warning: in later books and papers, Lusztig used a modified form of the quadratic relation that reads $(T_s-q_s^)(T_s+q_s^)=0.$ After extending the scalars to include the half integer powers ''q'' the resulting Hecke algebra is isomorphic to the previously defined one (but the ''T_s'' here corresponds to ''q'' ''T''_s in our notation). While this does not change the general theory, many formulae look different.

Generic Multiparameter Hecke Algebras

''H_A(W,S,q)'' is the ''generic'' multiparameter Hecke algebra. This algebra is universal in the sense that every other multiparameter Hecke algebra can be obtained from it via the (unique) ring homomorphism ''A'' → ''R'' which maps the indeterminate ''q_s'' ∈ ''A'' to the unit ''q_s'' ∈ ''R''. This homomorphism turns ''R'' into a ''A''-algebra and the scalar extension ''H_A(W,S)'' ⊗_''A'' ''R'' is canonically isomorphic to the Hecke algebra ''H_R(W,S,q)'' as constructed above. One calls this process ''specialization'' of the generic algebra.

One-parameter Hecke Algebras

If one specializes every indeterminate ''q_s'' to a single indeterminate ''q'' over the integers (or ''q'' to ''q''^½ respectively), then one obtains the so-called generic one-parameter Hecke algebra of ''(W,S)''.

Since in Coxeter groups with single laced Dynkin diagrams (for example groups of type A and D) every pair of Coxeter generators is conjugated, the above-mentioned restriction of ''q_s'' being equal ''q_t'' whenever ''s'' and ''t'' are conjugated in ''W'' forces the multiparameter and the one-parameter Hecke algebras to be equal. Therefore, it is also very common to only look at one-parameter Hecke algebras.

Coxeter groups with weights

If an integral weight function is defined on ''W'' (i.e. a map ''L:W'' → Z with ''L(vw)=L(v)+L(w)'' for all ''v,w'' ∈ ''W'' with ''l(vw)=l(v)+l(w)''), then a common specialization to look at is the one induced by the homomorphism ''q_s'' ↦ ''q^L(s)'', where ''q'' is a single indeterminate over Z.

If one uses the convention with half-integer powers, then weight function ''L:W'' → ½Z may be permitted as well. For technical reasons it is also often convenient only to consider positive weight functions.

Properties

1. The Hecke algebra has a basis $(T_w)_$ over ''A'' indexed by the elements of the Coxeter group ''W''. In particular, ''H'' is a free ''A''-module. If $w=s_1 s_2 \ldots s_n$ is a reduced decomposition of ''w'' ∈ ''W'', then $T_w=T_T_\ldots T_$. This basis of Hecke algebra is sometimes called the natural basis. The neutral element of ''W'' corresponds to the identity of ''H'': ''T_e'' = 1.

2. The elements of the natural basis are ''multiplicative'', namely, ''T''_yw=''T''_y ''T''_w whenever ''l(yw)=l(y)+l(w)'', where ''l'' denotes the length function on the Coxeter group ''W''.

3. Elements of the natural basis are invertible. For example, from the quadratic relation we conclude that ''T'' = ''q'' ''T_s'' + (''q''-1).

4. Suppose that ''W'' is a finite group and the ground ring is the field C of complex numbers. Jacques Tits has proved that if the indeterminate ''q'' is specialized to any complex number outside of an explicitly given list (consisting of roots of unity), then the resulting one parameter Hecke algebra is semisimple and isomorphic to the complex group algebra C 'W''(which also corresponds to the specialization ''q'' ↦ 1) .

5. More generally, if ''W'' is a finite group and the ground ring ''R'' is a field of characteristic zero, then the one parameter Hecke algebra is a semisimple associative algebra over ''R'' 'q''^±1 Moreover, extending earlier results of Benson and Curtis, George Lusztig provided an explicit isomorphism between the Hecke algebra and the group algebra after the extension of scalars to the quotient field of ''R'' 'q''^±½

Canonical basis

A great discovery of Kazhdan and Lusztig was that a Hecke algebra admits a ''different'' basis, which in a way controls representation theory of a variety of related objects.

The generic multiparameter Hecke algebra, ''H_A(W,S,q)'', has an involution ''bar'' that maps ''q''^½ to ''q''^−½ and acts as identity on Z. Then ''H'' admits a unique ring automorphism ''i'' that is semilinear with respect to the bar involution of ''A'' and maps ''T_s'' to ''T''. It can further be proved that this automorphism is involutive (has order two) and takes any ''T_w'' to $T^_.$

Kazhdan - Lusztig Theorem: For each ''w'' ∈ ''W'' there exists a unique element $C^_w$ which is invariant under the involution ''i'' and if one writes its expansion in terms of the natural basis:
::$C'_w= \left (q^ \right )^\sum_P_T_y,$
one has the following:
* ''P''_w,w=1,
* ''P''_y,w in Z 'q''has degree less than or equal to ½''(l(w)-l(y)-1)'' if ''yBruhat order