Canonical basis

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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[edit]

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:

  • and
  • and

for all . A canonical basis at is analogously defined to be a basis that satisfies

  • and
  • 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 (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).

Examples[edit]

Quantum groups[edit]

The canonical basis of quantum groups in the sense of Lusztig and Kashiwara are canonical basis at .

Hecke algebras[edit]

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.

Linear algebra[edit]

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 rank m 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.[2]

Computation[edit]

Let be an eigenvalue of of algebraic multiplicity . First, find the ranks (matrix ranks) of the matrices . The integer is determined to be the first integer for which has rank (n being the number of rows or columns of , that is, is n × n).

Now define

The variable designates the number of linearly independent generalized eigenvectors of rank k (generalized eigenvector rank; see generalized eigenvector) 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 (see generalized eigenvector).[3]

Example[edit]

This example illustrates a canonical basis with two Jordan chains. Unfortunately, it is a little difficult to construct an interesting example of low order.[4] The matrix

has eigenvalues and with algebraic multiplicities and , but geometric multiplicities and .

For we have

has rank 5,
has rank 4,
has rank 3,
has rank 2.

Therefore

Thus, a canonical basis for will have, corresponding to one generalized eigenvector each of ranks 4, 3, 2 and 1.

For we have

has rank 5,
has rank 4.

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 .[5]

See also[edit]

Notes[edit]

  1. ^ Bronson (1970, p. 196)
  2. ^ Bronson (1970, pp. 196,197)
  3. ^ Bronson (1970, pp. 197,198)
  4. ^ Nering (1970, pp. 122,123)
  5. ^ Bronson (1970, p. 203)

References[edit]