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 edit

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.

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 edit

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 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.^[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]

• Canonical form
• Change of basis
• Normal basis
• Normal form (disambiguation)
• Polynomial basis

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