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 basis: Consider the ring of integral Laurent polynomials with its two subrings and the automorphism defined by. A precanonical structure on a free -module consists of
A dualization operation, that is, 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:
and
for all. A canonical basis at is analogously defined to be a basis that satisfies
and
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 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 .
Examples
Quantum groups
The canonical basis of quantum groups in the sense of Lusztig and Kashiwara are canonical basis at.
If we are given an n × n matrix and wish to find a matrix in Jordan normal form, similar to, 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 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 possesses n linearly independent generalized eigenvectors. Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent. If is an eigenvalue of of algebraic multiplicity, then will have linearly independent generalized eigenvectors corresponding to. For any given n × n matrix, 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 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 rankm is in a canonical basis, it follows that the m − 1 vectors that are in the Jordan chain generated by are also in the canonical basis.
Computation
Let be an eigenvalue of of algebraic multiplicity. First, find the ranks of the matrices. The integer is determined to be the first integer for which has rank . Now define The variable designates the number of linearly independent generalized eigenvectors of rank k corresponding to the eigenvalue that will appear in a canonical basis for. 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.
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 and with algebraic multiplicities and, but geometric multiplicities and. For we have Therefore Thus, a canonical basis for will have, corresponding to one generalized eigenvector each of ranks 4, 3, 2 and 1. For we have Therefore Thus, a canonical basis for will have, corresponding to one generalized eigenvector each of ranks 2 and 1. A canonical basis for is is the ordinary eigenvector associated with. and are generalized eigenvectors associated with. is the ordinary eigenvector associated with. is a generalized eigenvector associated with. A matrix in Jordan normal form, similar to is obtained as follows: where the matrix is a generalized modal matrix for and.
