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:
Representation theory.
In representation theory there are several bases that are called "canonical", for example, Lusztig's canonical basis and closely related Kashiwara's crystal basis in quantum groups and their representations. There is a general concept underlying these bases:
Consider the ring of integral Laurent polynomials formula_3 with its two subrings formula_4 and the automorphism formula_5 defined by formula_6.
A "precanonical structure" on a free formula_7-module formula_8 consists of
If a precanonical structure is given, then one can define the formula_17 submodule formula_18 of formula_8.
A "canonical basis at formula_20" of the precanonical structure is then a formula_7-basis formula_22 of formula_8 that satisfies:
for all formula_13. A "canonical basis at formula_27" is analogously defined to be a basis formula_28 that satisfies
for all formula_13. The naming "at formula_27" alludes to the fact formula_33 and hence the "specialization" formula_34 corresponds to quotienting out the relation formula_35.
One can show that there exists at most one canonical basis at "v" = 0 (and at most one at formula_27) for each precanonical structure. A sufficient condition for existence is that the polynomials formula_37 defined by formula_38 satisfy formula_39 and formula_40.
A canonical basis at "v" = 0 (formula_27) induces an isomorphism from formula_42 to formula_43 (formula_44 respectively).
Examples.
Quantum groups.
The canonical basis of quantum groups in the sense of Lusztig and Kashiwara are canonical basis at formula_20.
Hecke algebras.
Let formula_46 be a Coxeter group. The corresponding Iwahori-Hecke algebra formula_47 has the standard basis formula_48, the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by formula_49. This is a precanonical structure on formula_47 that satisfies the sufficient condition above and the corresponding canonical basis of formula_47 at formula_20 is the Kazhdan–Lusztig basis
with formula_54 being the Kazhdan–Lusztig polynomials.
Linear algebra.
If we are given an "n" × "n" matrix formula_2 and wish to find a matrix formula_56 in Jordan normal form, similar to formula_2, we are interested only in sets of linearly independent generalized eigenvectors. A matrix in Jordan normal form is an "almost diagonal matrix," that is, as close to diagonal as possible. A diagonal matrix formula_58 is a special case of a matrix in Jordan normal form. An ordinary eigenvector is a special case of a generalized eigenvector.
Every "n" × "n" matrix formula_2 possesses "n" linearly independent generalized eigenvectors. Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent. If formula_60 is an eigenvalue of formula_2 of algebraic multiplicity formula_62, then formula_2 will have formula_62 linearly independent generalized eigenvectors corresponding to formula_60.
For any given "n" × "n" matrix formula_2, there are infinitely many ways to pick the "n" linearly independent generalized eigenvectors. If they are chosen in a particularly judicious manner, we can use these vectors to show that formula_2 is similar to a matrix in Jordan normal form. In particular,
Definition: A set of "n" linearly independent generalized eigenvectors is a canonical basis if it is composed entirely of Jordan chains.
Thus, once we have determined that a generalized eigenvector of rank "m" is in a canonical basis, it follows that the "m" − 1 vectors formula_68 that are in the Jordan chain generated by formula_69 are also in the canonical basis.
Computation.
Let formula_70 be an eigenvalue of formula_2 of algebraic multiplicity formula_72. First, find the ranks (matrix ranks) of the matrices formula_73. The integer formula_74 is determined to be the "first integer" for which formula_75 has rank formula_76 ("n" being the number of rows or columns of formula_2, that is, formula_2 is "n" × "n").
Now define
The variable formula_80 designates the number of linearly independent generalized eigenvectors of rank "k" (generalized eigenvector rank; see generalized eigenvector) corresponding to the eigenvalue formula_70 that will appear in a canonical basis for formula_2. Note that
Once we have determined the number of generalized eigenvectors of each rank that a canonical basis has, we can obtain the vectors explicitly (see generalized eigenvector).
Example.
This example illustrates a canonical basis with two Jordan chains. Unfortunately, it is a little difficult to construct an interesting example of low order.
The matrix
has eigenvalues formula_85 and formula_86 with algebraic multiplicities formula_87 and formula_88, but geometric multiplicities formula_89 and formula_90.
For formula_91 we have formula_92
Therefore formula_97
Thus, a canonical basis for formula_2 will have, corresponding to formula_91 one generalized eigenvector each of ranks 4, 3, 2 and 1.
For formula_104 we have formula_105
Therefore formula_108
Thus, a canonical basis for formula_2 will have, corresponding to formula_104 one generalized eigenvector each of ranks 2 and 1.
A canonical basis for formula_2 is
formula_115 is the ordinary eigenvector associated with formula_116. 
formula_117 and formula_118 are generalized eigenvectors associated with formula_116. 
formula_120 is the ordinary eigenvector associated with formula_121. 
formula_122 is a generalized eigenvector associated with formula_121.
A matrix formula_56 in Jordan normal form, similar to formula_2 is obtained as follows:
where the matrix formula_128 is a generalized modal matrix for formula_2 and formula_130.