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, ${\displaystyle (X^{i})_{i))$.
• 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 ${\displaystyle A}$, if the set is composed entirely of Jordan chains.^[1]
• 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 ${\displaystyle ADE}$ and also for the plus part of that algebra was introduced by Lusztig ^[2] by two methods: an algebraic one (using a braid group action and PBW bases) and a topological one (using intersection cohomology). Specializing the parameter ${\displaystyle q}$ to ${\displaystyle q=1}$ yields a canonical basis for the irreducible representations of the corresponding simple Lie algebra, which was not known earlier. Specializing the parameter ${\displaystyle q}$ to ${\displaystyle q=0}$ 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;^[3] it is sometimes called the crystal basis. The definition of the canonical basis was extended to the Kac-Moody setting by Kashiwara ^[4] (by an algebraic method) and by Lusztig ^[5] (by a topological method).

There is a general concept underlying these bases:

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

A precanonical structure on a free ${\displaystyle {\mathcal {Z))}$-module ${\displaystyle F}$ consists of

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

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

A canonical basis of the precanonical structure is then a ${\displaystyle {\mathcal {Z))}$-basis ${\displaystyle (c_{i})_{i\in I))$ of ${\displaystyle F}$ that satisfies:

• ${\displaystyle {\overline {c_{i))}=c_{i))$ and
• ${\displaystyle c_{i}\in \sum _{j\leq i}{\mathcal {Z))^{+}t_{j}{\text{ and ))c_{i}\equiv t_{i}\mod vF^{+))$

for all ${\displaystyle i\in I}$.

One can show that there exists at most one canonical basis for each precanonical structure.^[6] A sufficient condition for existence is that the polynomials ${\displaystyle r_{ij}\in {\mathcal {Z))}$ defined by ${\textstyle {\overline {t_{j))}=\sum _{i}r_{ij}t_{i))$ satisfy ${\displaystyle r_{ii}=1}$ and ${\displaystyle r_{ij}\neq 0\implies i\leq j}$.

A canonical basis induces an isomorphism from ${\displaystyle \textstyle F^{+}\cap {\overline {F^{+))}=\sum _{i}\mathbb {Z} c_{i))$ to ${\displaystyle F^{+}/vF^{+))$.

Hecke algebras

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

${\displaystyle C_{w}'=\sum _{y\leq w}P_{y,w}(v^{2})T_{w))$

with ${\displaystyle P_{y,w))$ being the Kazhdan–Lusztig polynomials.

Linear algebra

If we are given an n × n matrix ${\displaystyle A}$ and wish to find a matrix ${\displaystyle J}$ in Jordan normal form, similar to ${\displaystyle A}$, 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 ${\displaystyle D}$ 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 ${\displaystyle A}$ possesses n linearly independent generalized eigenvectors. Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent. If ${\displaystyle \lambda }$ is an eigenvalue of ${\displaystyle A}$ of algebraic multiplicity ${\displaystyle \mu }$, then ${\displaystyle A}$ will have ${\displaystyle \mu }$ linearly independent generalized eigenvectors corresponding to ${\displaystyle \lambda }$.

For any given n × n matrix ${\displaystyle A}$, 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 ${\displaystyle A}$ 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 ${\displaystyle \mathbf {x} _{m-1},\mathbf {x} _{m-2},\ldots ,\mathbf {x} _{1))$ that are in the Jordan chain generated by ${\displaystyle \mathbf {x} _{m))$ are also in the canonical basis.^[7]

Computation

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

Now define

${\displaystyle \rho _{k}=\operatorname {rank} (A-\lambda _{i}I)^{k-1}-\operatorname {rank} (A-\lambda _{i}I)^{k}\qquad (k=1,2,\ldots ,m_{i}).}$

The variable ${\displaystyle \rho _{k))$ designates the number of linearly independent generalized eigenvectors of rank k (generalized eigenvector rank; see generalized eigenvector) corresponding to the eigenvalue ${\displaystyle \lambda _{i))$ that will appear in a canonical basis for ${\displaystyle A}$. Note that

${\displaystyle \operatorname {rank} (A-\lambda _{i}I)^{0}=\operatorname {rank} (I)=n.}$

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).^[8]

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.^[9] The matrix

${\displaystyle A={\begin{pmatrix}4&1&1&0&0&-1\\0&4&2&0&0&1\\0&0&4&1&0&0\\0&0&0&5&1&0\\0&0&0&0&5&2\\0&0&0&0&0&4\end{pmatrix))}$

has eigenvalues ${\displaystyle \lambda _{1}=4}$ and ${\displaystyle \lambda _{2}=5}$ with algebraic multiplicities ${\displaystyle \mu _{1}=4}$ and ${\displaystyle \mu _{2}=2}$, but geometric multiplicities ${\displaystyle \gamma _{1}=1}$ and ${\displaystyle \gamma _{2}=1}$.

For ${\displaystyle \lambda _{1}=4,}$ we have ${\displaystyle n-\mu _{1}=6-4=2,}$

${\displaystyle (A-4I)}$ has rank 5,

${\displaystyle (A-4I)^{2))$ has rank 4,

${\displaystyle (A-4I)^{3))$ has rank 3,

${\displaystyle (A-4I)^{4))$ has rank 2.

Therefore ${\displaystyle m_{1}=4.}$

${\displaystyle \rho _{4}=\operatorname {rank} (A-4I)^{3}-\operatorname {rank} (A-4I)^{4}=3-2=1,}$

${\displaystyle \rho _{3}=\operatorname {rank} (A-4I)^{2}-\operatorname {rank} (A-4I)^{3}=4-3=1,}$

${\displaystyle \rho _{2}=\operatorname {rank} (A-4I)^{1}-\operatorname {rank} (A-4I)^{2}=5-4=1,}$

${\displaystyle \rho _{1}=\operatorname {rank} (A-4I)^{0}-\operatorname {rank} (A-4I)^{1}=6-5=1.}$

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

For ${\displaystyle \lambda _{2}=5,}$ we have ${\displaystyle n-\mu _{2}=6-2=4,}$

${\displaystyle (A-5I)}$ has rank 5,

${\displaystyle (A-5I)^{2))$ has rank 4.

Therefore ${\displaystyle m_{2}=2.}$

${\displaystyle \rho _{2}=\operatorname {rank} (A-5I)^{1}-\operatorname {rank} (A-5I)^{2}=5-4=1,}$

${\displaystyle \rho _{1}=\operatorname {rank} (A-5I)^{0}-\operatorname {rank} (A-5I)^{1}=6-5=1.}$

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

A canonical basis for ${\displaystyle A}$ is

${\displaystyle \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\}.}$

${\displaystyle \mathbf {x} _{1))$ is the ordinary eigenvector associated with ${\displaystyle \lambda _{1))$. ${\displaystyle \mathbf {x} _{2},\mathbf {x} _{3))$ and ${\displaystyle \mathbf {x} _{4))$ are generalized eigenvectors associated with ${\displaystyle \lambda _{1))$. ${\displaystyle \mathbf {y} _{1))$ is the ordinary eigenvector associated with ${\displaystyle \lambda _{2))$. ${\displaystyle \mathbf {y} _{2))$ is a generalized eigenvector associated with ${\displaystyle \lambda _{2))$.

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

${\displaystyle M={\begin{pmatrix}\mathbf {x} _{1}&\mathbf {x} _{2}&\mathbf {x} _{3}&\mathbf {x} _{4}&\mathbf {y} _{1}&\mathbf {y} _{2}\end{pmatrix))={\begin{pmatrix}-4&-27&25&0&3&-8\\0&-4&-25&36&2&-4\\0&0&-2&-12&1&-1\\0&0&0&-2&1&0\\0&0&0&2&0&1\\0&0&0&-1&0&0\end{pmatrix)),}$

${\displaystyle J={\begin{pmatrix}4&1&0&0&0&0\\0&4&1&0&0&0\\0&0&4&1&0&0\\0&0&0&4&0&0\\0&0&0&0&5&1\\0&0&0&0&0&5\end{pmatrix)),}$

where the matrix ${\displaystyle M}$ is a generalized modal matrix for ${\displaystyle A}$ and ${\displaystyle AM=MJ}$.^[10]