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 Z := Z [ v , v − 1 ] with its two subrings Z ± := Z [ v ± 1 ] and the automorphism ⋅ ¯ that is defined by v ¯ := v − 1 .
A precanonical structure on a free Z -module F consists of
A standard basis ( t i ) i ∈ I of F ,A partial order on I that is interval finite, i.e. ( − ∞ , i ] := { j ∈ I ∣ j ≤ i } is finite for all i ∈ I ,A dualization operation, i.e. a bijection F → F of order two that is ⋅ ¯ -semilinear and will be denoted by ⋅ ¯ as well.If a precanonical structure is given, then one can define the Z ± submodule F ± := ∑ Z ± t j of F .
A canonical basis at v = 0 of the precanonical structure is then a Z -basis ( c i ) i ∈ I of F that satisfies:
c i ¯ = c i and c i ∈ ∑ j ≤ i Z + t j and c i ≡ t i mod v F + for all i ∈ I . A canonical basis at v = ∞ is analogously defined to be a basis ( c ~ i ) i ∈ I that satisfies
c ~ i ¯ = c ~ i and c ~ i ∈ ∑ j ≤ i Z − t j and c ~ i ≡ t i mod v − 1 F − for all i ∈ I . The naming "at v = ∞ " alludes to the fact lim v → ∞ v − 1 = 0 and hence the "specialization" v ↦ ∞ corresponds to quotienting out the relation v − 1 = 0 .
One can show that there exists at most one canonical basis at v = 0 (and at most one at v = ∞ ) for each precanonical structure. A sufficient condition for existence is that the polynomials r i j ∈ Z defined by t j ¯ = ∑ i r i j t i satisfy r i i = 1 and r i j ≠ 0 ⟹ i ≤ j .
A canonical basis at v = 0 ( v = ∞ ) induces an isomorphism from F + ∩ F + ¯ = ∑ i Z c i to F + / v F + ( F − ∩ F − ¯ = ∑ i Z c ~ i → F − / v − 1 F − respectively).
The canonical basis of quantum groups in the sense of Lusztig and Kashiwara are canonical basis at v = 0 .
Let ( W , S ) be a Coxeter group. The corresponding Iwahori-Hecke algebra H has the standard basis ( T w ) w ∈ W , the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by T w ¯ := T w − 1 − 1 . This is a precanonical structure on H that satisfies the sufficient condition above and the corresponding canonical basis of H at v = 0 is the Kazhdan–Lusztig basis
C w ′ = ∑ y ≤ w P y , w ( v 2 ) T w with P y , w being the Kazhdan–Lusztig polynomials.
If we are given an n × n matrix A and wish to find a matrix J in Jordan normal form, similar to A , 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 D 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 A possesses n linearly independent generalized eigenvectors. Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent. If λ is an eigenvalue of A of algebraic multiplicity μ , then A will have μ linearly independent generalized eigenvectors corresponding to λ .
For any given n × n matrix A , 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 A 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 x m − 1 , x m − 2 , … , x 1 that are in the Jordan chain generated by x m are also in the canonical basis.
Let λ i be an eigenvalue of A of algebraic multiplicity μ i . First, find the ranks (matrix ranks) of the matrices ( A − λ i I ) , ( A − λ i I ) 2 , … , ( A − λ i I ) m i . The integer m i is determined to be the first integer for which ( A − λ i I ) m i has rank n − μ i (n being the number of rows or columns of A , that is, A is n × n).
Now define
ρ k = rank ( A − λ i I ) k − 1 − rank ( A − λ i I ) k ( k = 1 , 2 , … , m i ) . The variable ρ k designates the number of linearly independent generalized eigenvectors of rank k (generalized eigenvector rank; see generalized eigenvector) corresponding to the eigenvalue λ i that will appear in a canonical basis for A . Note that
rank ( A − λ i I ) 0 = rank ( I ) = n . 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).
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
A = ( 4 1 1 0 0 − 1 0 4 2 0 0 1 0 0 4 1 0 0 0 0 0 5 1 0 0 0 0 0 5 2 0 0 0 0 0 4 ) has eigenvalues λ 1 = 4 and λ 2 = 5 with algebraic multiplicities μ 1 = 4 and μ 2 = 2 , but geometric multiplicities γ 1 = 1 and γ 2 = 1 .
For λ 1 = 4 , we have n − μ 1 = 6 − 4 = 2 ,
( A − 4 I ) has rank 5,
( A − 4 I ) 2 has rank 4,
( A − 4 I ) 3 has rank 3,
( A − 4 I ) 4 has rank 2.
Therefore m 1 = 4.
ρ 4 = rank ( A − 4 I ) 3 − rank ( A − 4 I ) 4 = 3 − 2 = 1 , ρ 3 = rank ( A − 4 I ) 2 − rank ( A − 4 I ) 3 = 4 − 3 = 1 , ρ 2 = rank ( A − 4 I ) 1 − rank ( A − 4 I ) 2 = 5 − 4 = 1 , ρ 1 = rank ( A − 4 I ) 0 − rank ( A − 4 I ) 1 = 6 − 5 = 1. Thus, a canonical basis for A will have, corresponding to λ 1 = 4 , one generalized eigenvector each of ranks 4, 3, 2 and 1.
For λ 2 = 5 , we have n − μ 2 = 6 − 2 = 4 ,
( A − 5 I ) has rank 5,
( A − 5 I ) 2 has rank 4.
Therefore m 2 = 2.
ρ 2 = rank ( A − 5 I ) 1 − rank ( A − 5 I ) 2 = 5 − 4 = 1 , ρ 1 = rank ( A − 5 I ) 0 − rank ( A − 5 I ) 1 = 6 − 5 = 1. Thus, a canonical basis for A will have, corresponding to λ 2 = 5 , one generalized eigenvector each of ranks 2 and 1.
A canonical basis for A is
{ x 1 , x 2 , x 3 , x 4 , y 1 , y 2 } = { ( − 4 0 0 0 0 0 ) ( − 27 − 4 0 0 0 0 ) ( 25 − 25 − 2 0 0 0 ) ( 0 36 − 12 − 2 2 − 1 ) ( 3 2 1 1 0 0 ) ( − 8 − 4 − 1 0 1 0 ) } . x 1 is the ordinary eigenvector associated with λ 1 . x 2 , x 3 and x 4 are generalized eigenvectors associated with λ 1 . y 1 is the ordinary eigenvector associated with λ 2 . y 2 is a generalized eigenvector associated with λ 2 .
A matrix J in Jordan normal form, similar to A is obtained as follows:
M = ( x 1 x 2 x 3 x 4 y 1 y 2 ) = ( − 4 − 27 25 0 3 − 8 0 − 4 − 25 36 2 − 4 0 0 − 2 − 12 1 − 1 0 0 0 − 2 1 0 0 0 0 2 0 1 0 0 0 − 1 0 0 ) , J = ( 4 1 0 0 0 0 0 4 1 0 0 0 0 0 4 1 0 0 0 0 0 4 0 0 0 0 0 0 5 1 0 0 0 0 0 5 ) , where the matrix M is a generalized modal matrix for A and A M = M J .
