Canonical basis

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Short description: Basis of a type of algebraic structure

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

The canonical basis for the irreducible representations of a quantized enveloping algebra of type [math]\displaystyle{ ADE }[/math] 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 [math]\displaystyle{ q }[/math] to [math]\displaystyle{ q=1 }[/math] yields a canonical basis for the irreducible representations of the corresponding simple Lie algebra, which was not known earlier. Specializing the parameter [math]\displaystyle{ q }[/math] to [math]\displaystyle{ q=0 }[/math] 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 [math]\displaystyle{ \mathcal{Z}:=\mathbb{Z}\left[v,v^{-1}\right] }[/math] with its two subrings [math]\displaystyle{ \mathcal{Z}^{\pm}:=\mathbb{Z}\left[v^{\pm 1}\right] }[/math] and the automorphism [math]\displaystyle{ \overline{\cdot} }[/math] defined by [math]\displaystyle{ \overline{v}:=v^{-1} }[/math].

A precanonical structure on a free [math]\displaystyle{ \mathcal{Z} }[/math]-module [math]\displaystyle{ F }[/math] consists of

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

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

A canonical basis of the precanonical structure is then a [math]\displaystyle{ \mathcal{Z} }[/math]-basis [math]\displaystyle{ (c_i)_{i\in I} }[/math] of [math]\displaystyle{ F }[/math] that satisfies:

  • [math]\displaystyle{ \overline{c_i}=c_i }[/math] and
  • [math]\displaystyle{ c_i \in \sum_{j\leq i} \mathcal{Z}^+ t_j \text{ and } c_i \equiv t_i \mod vF^+ }[/math]

for all [math]\displaystyle{ i\in I }[/math].

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 [math]\displaystyle{ r_{ij}\in\mathcal{Z} }[/math] defined by [math]\displaystyle{ \overline{t_j}=\sum_i r_{ij} t_i }[/math] satisfy [math]\displaystyle{ r_{ii}=1 }[/math] and [math]\displaystyle{ r_{ij}\neq 0 \implies i\leq j }[/math].

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

Hecke algebras

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

[math]\displaystyle{ C_w' = \sum_{y\leq w} P_{y,w}(v^2) T_w }[/math]

with [math]\displaystyle{ P_{y,w} }[/math] being the Kazhdan–Lusztig polynomials.

Linear algebra

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

For any given n × n matrix [math]\displaystyle{ A }[/math], 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 [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ \mathbf x_{m-1}, \mathbf x_{m-2}, \ldots , \mathbf x_1 }[/math] that are in the Jordan chain generated by [math]\displaystyle{ \mathbf x_m }[/math] are also in the canonical basis.[7]

Computation

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

Now define

[math]\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). }[/math]

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

[math]\displaystyle{ \operatorname{rank}(A - \lambda_i I)^0 = \operatorname{rank}(I) = n . }[/math]

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

[math]\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} }[/math]

has eigenvalues [math]\displaystyle{ \lambda_1 = 4 }[/math] and [math]\displaystyle{ \lambda_2 = 5 }[/math] with algebraic multiplicities [math]\displaystyle{ \mu_1 = 4 }[/math] and [math]\displaystyle{ \mu_2 = 2 }[/math], but geometric multiplicities [math]\displaystyle{ \gamma_1 = 1 }[/math] and [math]\displaystyle{ \gamma_2 = 1 }[/math].

For [math]\displaystyle{ \lambda_1 = 4, }[/math] we have [math]\displaystyle{ n - \mu_1 = 6 - 4 = 2, }[/math]

[math]\displaystyle{ (A - 4I) }[/math] has rank 5,
[math]\displaystyle{ (A - 4I)^2 }[/math] has rank 4,
[math]\displaystyle{ (A - 4I)^3 }[/math] has rank 3,
[math]\displaystyle{ (A - 4I)^4 }[/math] has rank 2.

Therefore [math]\displaystyle{ m_1 = 4. }[/math]

[math]\displaystyle{ \rho_4 = \operatorname{rank}(A - 4I)^3 - \operatorname{rank}(A - 4I)^4 = 3 - 2 = 1, }[/math]
[math]\displaystyle{ \rho_3 = \operatorname{rank}(A - 4I)^2 - \operatorname{rank}(A - 4I)^3 = 4 - 3 = 1, }[/math]
[math]\displaystyle{ \rho_2 = \operatorname{rank}(A - 4I)^1 - \operatorname{rank}(A - 4I)^2 = 5 - 4 = 1, }[/math]
[math]\displaystyle{ \rho_1 = \operatorname{rank}(A - 4I)^0 - \operatorname{rank}(A - 4I)^1 = 6 - 5 = 1. }[/math]

Thus, a canonical basis for [math]\displaystyle{ A }[/math] will have, corresponding to [math]\displaystyle{ \lambda_1 = 4, }[/math] one generalized eigenvector each of ranks 4, 3, 2 and 1.

For [math]\displaystyle{ \lambda_2 = 5, }[/math] we have [math]\displaystyle{ n - \mu_2 = 6 - 2 = 4, }[/math]

[math]\displaystyle{ (A - 5I) }[/math] has rank 5,
[math]\displaystyle{ (A - 5I)^2 }[/math] has rank 4.

Therefore [math]\displaystyle{ m_2 = 2. }[/math]

[math]\displaystyle{ \rho_2 = \operatorname{rank}(A - 5I)^1 - \operatorname{rank}(A - 5I)^2 = 5 - 4 = 1, }[/math]
[math]\displaystyle{ \rho_1 = \operatorname{rank}(A - 5I)^0 - \operatorname{rank}(A - 5I)^1 = 6 - 5 = 1. }[/math]

Thus, a canonical basis for [math]\displaystyle{ A }[/math] will have, corresponding to [math]\displaystyle{ \lambda_2 = 5, }[/math] one generalized eigenvector each of ranks 2 and 1.

A canonical basis for [math]\displaystyle{ A }[/math] is

[math]\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\}. }[/math]

[math]\displaystyle{ \mathbf x_1 }[/math] is the ordinary eigenvector associated with [math]\displaystyle{ \lambda_1 }[/math]. [math]\displaystyle{ \mathbf x_2, \mathbf x_3 }[/math] and [math]\displaystyle{ \mathbf x_4 }[/math] are generalized eigenvectors associated with [math]\displaystyle{ \lambda_1 }[/math]. [math]\displaystyle{ \mathbf y_1 }[/math] is the ordinary eigenvector associated with [math]\displaystyle{ \lambda_2 }[/math]. [math]\displaystyle{ \mathbf y_2 }[/math] is a generalized eigenvector associated with [math]\displaystyle{ \lambda_2 }[/math].

A matrix [math]\displaystyle{ J }[/math] in Jordan normal form, similar to [math]\displaystyle{ A }[/math] is obtained as follows:

[math]\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}, }[/math]
[math]\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}, }[/math]

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

See also

Notes

  1. ↑ (Bronson 1970)
  2. ↑ (Lusztig 1990)
  3. ↑ (Kashiwara 1990)
  4. ↑ (Kashiwara 1991)
  5. ↑ (Lusztig 1991)
  6. ↑ (Lusztig 1993)
  7. ↑ (Bronson 1970)
  8. ↑ (Bronson 1970)
  9. ↑ (Nering 1970)
  10. ↑ (Bronson 1970)

References



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