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:=\mathbb\left ,v^\right/math> with its two subrings \mathcal^:=\mathbb\left ^\right/math> and the automorphism $\overline$ defined by $\overline:=v^$.

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

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

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

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

A canonical basis induces an isomorphism from $\textstyle F^+\cap \overline = \sum_i \mathbbc_i$ to $F^+/vF^+$.

Hecke algebras

Let $(W,S)$ be a Coxeter group. The corresponding Iwahori-Hecke algebra $H$ has the standard basis $(T_w)_$, the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by $\overline:=T_^$. 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

: $C_w' = \sum_ P_(v^2) T_w$

with $P_$ 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 $\mathbf x_, \mathbf x_, \ldots , \mathbf x_1$ 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 $(A - \lambda_i I), (A - \lambda_i I)^2, \ldots , (A - \lambda_i I)^$. The integer $m_i$ is determined to be the ''first integer'' for which $(A - \lambda_i I)^$ has rank $n - \mu_i$ (''n'' being the number of rows or columns of $A$, that is, $A$ is ''n'' × ''n'').

Now define

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

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

:$\operatorname(A - \lambda_i I)^0 = \operatorname(I) = n .$

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

:$A = \begin 4 & 1 & 1 & 0 & 0 & -1 \\ 0 & 4 & 2 & 0 & 0 & 1 \\ 0 & 0 & 4 & 1 & 0 & 0 \\ 0 & 0 & 0 & 5 & 1 & 0 \\ 0 & 0 & 0 & 0 & 5 & 2 \\ 0 & 0 & 0 & 0 & 0 & 4 \end$

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 $m_1 = 4.$

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

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 $m_2 = 2.$

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

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

:$\left\ = \left\.$

$\mathbf x_1$ 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:

:$M = \begin \mathbf x_1 & \mathbf x_2 & \mathbf x_3 & \mathbf x_4 & \mathbf y_1 & \mathbf y_2 \end = \begin -4 & -27 & 25 & 0 & 3 & -8 \\ 0 & -4 & -25 & 36 & 2 & -4 \\ 0 & 0 & -2 & -12 & 1 & -1 \\ 0 & 0 & 0 & -2 & 1 & 0 \\ 0 & 0 & 0 & 2 & 0 & 1 \\ 0 & 0 & 0 & -1 & 0 & 0 \end,$
:$J = \begin 4 & 1 & 0 & 0 & 0 & 0 \\ 0 & 4 & 1 & 0 & 0 & 0 \\ 0 & 0 & 4 & 1 & 0 & 0 \\ 0 & 0 & 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 5 & 1 \\ 0 & 0 & 0 & 0 & 0 & 5 \end,$

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

See also

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

Notes

References

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* {{ citation , first1 = Evar D. , last1 = Nering , year = 1970 , title = Linear Algebra and Matrix Theory , edition = 2nd , publisher = Wiley , location = New York , lccn = 76091646

Linear algebra
Abstract algebra
Lie algebras
Representation theory
Quantum groups