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:


 * 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, $$(X^i)_i$$.
 * For finite extension fields, it means the polynomial basis.
 * In linear algebra, it refers to a set of n linearly independent generalized eigenvectors of an n×n matrix $$A$$, if the set is composed entirely of Jordan chains.
 * In representation theory, it refers to the basis of the quantum groups introduced by Lusztig.

Representation theory
The canonical basis for the irreducible representations of a quantized enveloping algebra of type $$ADE$$ and also for the plus part of that algebra was introduced by Lusztig by two methods: an algebraic one (using a braid group action and PBW bases) and a topological one (using intersection cohomology). Specializing the parameter $$q$$ to $$q=1$$ yields a canonical basis for the irreducible representations of the corresponding simple Lie algebra, which was not known earlier. Specializing the parameter $$q$$ to $$q=0$$ yields something like a shadow of a basis. This shadow (but not the basis itself) for the case of irreducible representations was considered independently by Kashiwara; it is sometimes called the crystal basis. The definition of the canonical basis was extended to the Kac-Moody setting by Kashiwara (by an algebraic method) and by Lusztig (by a topological method).

There is a general concept underlying these bases:

Consider the ring of integral Laurent polynomials $$\mathcal{Z}:=\mathbb{Z}\left[v,v^{-1}\right]$$ with its two subrings $$\mathcal{Z}^{\pm}:=\mathbb{Z}\left[v^{\pm 1}\right]$$ and the automorphism $$\overline{\cdot}$$ defined by $$\overline{v}:=v^{-1}$$.

A precanonical structure on a free $$\mathcal{Z}$$-module $$F$$ consists of
 * A standard basis $$(t_i)_{i\in I}$$ of $$F$$,
 * An interval finite partial order on $$I$$, that is, $$(-\infty,i] := \{j\in I \mid j\leq i\}$$ is finite for all $$i\in I$$,
 * A dualization operation, that is, a bijection $$F\to F$$ of order two that is $$\overline{\cdot}$$-semilinear and will be denoted by $$\overline{\cdot}$$ as well.

If a precanonical structure is given, then one can define the $$\mathcal{Z}^{\pm}$$ submodule $F^{\pm} := \sum \mathcal{Z}^{\pm} t_j$ of $$F$$.

A ''canonical basis of the precanonical structure is then a $$\mathcal{Z}$$-basis $$(c_i)_{i\in I}$$ of $$F$$ that satisfies: for all $$i\in I$$.
 * $$\overline{c_i}=c_i$$ and
 * $$c_i \in \sum_{j\leq i} \mathcal{Z}^+ t_j \text{ and } c_i \equiv t_i \mod vF^+$$

One can show that there exists at most one canonical basis for each precanonical structure. A sufficient condition for existence is that the polynomials $$r_{ij}\in\mathcal{Z}$$ defined by $\overline{t_j}=\sum_i r_{ij} t_i$ satisfy $$r_{ii}=1$$ and $$r_{ij}\neq 0 \implies i\leq j$$.

A canonical basis induces an isomorphism from $$\textstyle F^+\cap \overline{F^+} = \sum_i \mathbb{Z}c_i$$ to $$F^+/vF^+$$.

Hecke algebras
Let $$(W,S)$$ be a Coxeter group. The corresponding Iwahori-Hecke algebra $$H$$ has the standard basis $$(T_w)_{w\in W}$$, the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by $$\overline{T_w}:=T_{w^{-1}}^{-1}$$. This is a precanonical structure on $$H$$ that satisfies the sufficient condition above and the corresponding canonical basis of $$H$$ is the Kazhdan–Lusztig basis


 * $$C_w' = \sum_{y\leq w} P_{y,w}(v^2) T_w$$

with $$P_{y,w}$$ being the Kazhdan–Lusztig polynomials.

Linear algebra
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 $$\lambda$$ is an eigenvalue of $$A$$ of algebraic multiplicity $$\mu$$, then $$A$$ will have $$\mu$$ linearly independent generalized eigenvectors corresponding to $$\lambda$$.

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 $$ \mathbf x_{m-1}, \mathbf x_{m-2}, \ldots, \mathbf x_1 $$ that are in the Jordan chain generated by $$ \mathbf x_m $$ are also in the canonical basis.

Computation
Let $$ \lambda_i $$ be an eigenvalue of $$A$$ of algebraic multiplicity $$ \mu_i $$. First, find the ranks (matrix ranks) of the matrices $$ (A - \lambda_i I), (A - \lambda_i I)^2, \ldots, (A - \lambda_i I)^{m_i} $$. The integer $$m_i$$ is determined to be the first integer for which $$ (A - \lambda_i I)^{m_i} $$ has rank $$n - \mu_i $$ (n being the number of rows or columns of $$A$$, that is, $$A$$ is n × n).

Now define


 * $$ \rho_k = \operatorname{rank}(A - \lambda_i I)^{k-1} - \operatorname{rank}(A - \lambda_i I)^k \qquad (k = 1, 2, \ldots, m_i).$$

The variable $$ \rho_k $$ designates the number of linearly independent generalized eigenvectors of rank k (generalized eigenvector rank; see generalized eigenvector) corresponding to the eigenvalue $$ \lambda_i $$ that will appear in a canonical basis for $$A$$. Note that


 * $$ \operatorname{rank}(A - \lambda_i I)^0 = \operatorname{rank}(I) = n .$$

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).

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


 * $$A = \begin{pmatrix}

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 \end{pmatrix}$$

has eigenvalues $$ \lambda_1 = 4 $$ and $$ \lambda_2 = 5 $$ with algebraic multiplicities $$ \mu_1 = 4 $$ and $$ \mu_2 = 2 $$, but geometric multiplicities $$ \gamma_1 = 1 $$ and $$ \gamma_2 = 1 $$.

For $$ \lambda_1 = 4,$$ we have $$ n - \mu_1 = 6 - 4 = 2, $$


 * $$ (A - 4I) $$ has rank 5,
 * $$ (A - 4I)^2 $$ has rank 4,
 * $$ (A - 4I)^3 $$ has rank 3,
 * $$ (A - 4I)^4 $$ has rank 2.

Therefore $$m_1 = 4.$$


 * $$ \rho_4 = \operatorname{rank}(A - 4I)^3 - \operatorname{rank}(A - 4I)^4 = 3 - 2 = 1,$$
 * $$ \rho_3 = \operatorname{rank}(A - 4I)^2 - \operatorname{rank}(A - 4I)^3 = 4 - 3 = 1,$$
 * $$ \rho_2 = \operatorname{rank}(A - 4I)^1 - \operatorname{rank}(A - 4I)^2 = 5 - 4 = 1,$$
 * $$ \rho_1 = \operatorname{rank}(A - 4I)^0 - \operatorname{rank}(A - 4I)^1 = 6 - 5 = 1.$$

Thus, a canonical basis for $$A$$ will have, corresponding to $$ \lambda_1 = 4,$$ one generalized eigenvector each of ranks 4, 3, 2 and 1.

For $$ \lambda_2 = 5,$$ we have $$ n - \mu_2 = 6 - 2 = 4, $$


 * $$ (A - 5I) $$ has rank 5,
 * $$ (A - 5I)^2 $$ has rank 4.

Therefore $$m_2 = 2.$$


 * $$ \rho_2 = \operatorname{rank}(A - 5I)^1 - \operatorname{rank}(A - 5I)^2 = 5 - 4 = 1,$$
 * $$ \rho_1 = \operatorname{rank}(A - 5I)^0 - \operatorname{rank}(A - 5I)^1 = 6 - 5 = 1.$$

Thus, a canonical basis for $$A$$ will have, corresponding to $$ \lambda_2 = 5,$$ one generalized eigenvector each of ranks 2 and 1.

A canonical basis for $$A$$ is



\left\{ \mathbf x_1, \mathbf x_2, \mathbf x_3, \mathbf x_4, \mathbf y_1, \mathbf y_2 \right\} = \left\{ \begin{pmatrix} -4 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} -27 \\ -4 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 25 \\ -25 \\ -2 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 36 \\ -12 \\ -2 \\ 2 \\ -1 \end{pmatrix}, \begin{pmatrix} 3 \\ 2 \\ 1 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} -8 \\ -4 \\ -1 \\ 0 \\ 1 \\ 0 \end{pmatrix} \right\}. $$

$$ \mathbf x_1 $$ is the ordinary eigenvector associated with $$ \lambda_1 $$. $$ \mathbf x_2, \mathbf x_3 $$ and $$ \mathbf x_4 $$ are generalized eigenvectors associated with $$ \lambda_1 $$. $$ \mathbf y_1 $$ is the ordinary eigenvector associated with $$ \lambda_2 $$. $$ \mathbf y_2 $$ is a generalized eigenvector associated with $$ \lambda_2 $$.

A matrix $$J$$ in Jordan normal form, similar to $$A$$ is obtained as follows:



M = \begin{pmatrix} \mathbf x_1 & \mathbf x_2 & \mathbf x_3 & \mathbf x_4 & \mathbf y_1 & \mathbf y_2 \end{pmatrix} = \begin{pmatrix} -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 \end{pmatrix}, $$
 * $$ J = \begin{pmatrix}

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 \end{pmatrix}, $$

where the matrix $$M$$ is a generalized modal matrix for $$A$$ and $$AM = MJ$$.