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 Kronecker delta.
- In a polynomial ring, it refers to its standard basis given by the monomials, .
- For finite extension fields, it means the polynomial basis.
In representation theory there are several basis that are called "canonical", e.g. Lusztig's canonical basis and closely related Kashiwara's crystal basis in quantum groups and their representations. There is a general concept underlying these basis:
Consider the ring of integral Laurent polynomials with its two subrings and the automorphism that is defined by .
A precanonical structure on a free -module consists of
- A standard basis of ,
- A partial order on that is interval finite, i.e. is finite for all ,
- A dualization operation, i.e. a bijection of order two that is -semilinear and will be denoted by as well.
If a precanonical structure is given, then one can define the submodule of .
A canonical basis at of the precanonical structure is then a -basis of that satisfies:
for all . A canonical basis at is analogously defined to be a basis that satisfies
for all . The naming "at " alludes to the fact and hence the "specialization" corresponds to quotienting out the relation .
One can show that there exists at most one canonical basis at v=0 (and at most one at ) for each precanonical structure. A sufficient condition for existence is that the polynomials defined by satisfy and .
A canonical basis at v=0 () induces an isomorphism from to ( respectively).
The canonical basis of quantum groups in the sense of Lusztig and Kashiwara are canonical basis at .
Let be a Coxeter group. The corresponding Iwahori-Hecke algebra has the standard basis , the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by . This is a precanonical structure on that satisfies the sufficient condition above and the corresponding canonical basis of at is the Kazhdan-Lusztig basis with being the Kazhdan-Lusztig polynomials.