In representation theory there are several basis that are called "canonical", e.g. Lusztig's canonical basis and closely related Kashiwara's crystal basis in quantum groups and their representations. There is a general concept underlying these basis:
Consider the ring of integral Laurent polynomials
Z
:=
Z
[
v
,
v
−
1
]
with its two subrings
Z
±
:=
Z
[
v
±
1
]
and the automorphism
⋅
¯
that is defined by
v
¯
:=
v
−
1
.
A precanonical structure on a free
Z
-module
F
consists of
A standard basis
(
t
i
)
i
∈
I
of
F
,
A partial order on
I
that is interval finite, i.e.
(
−
∞
,
i
]
:=
{
j
∈
I
∣
j
≤
i
}
is finite for all
i
∈
I
,
A dualization operation, i.e. a bijection
F
→
F
of order two that is
⋅
¯
-semilinear and will be denoted by
⋅
¯
as well.
If a precanonical structure is given, then one can define the
Z
±
submodule
F
±
:=
∑
Z
±
t
j
of
F
.
A canonical basis at
v
=
0
of the precanonical structure is then a
Z
-basis
(
c
i
)
i
∈
I
of
F
that satisfies:
c
i
¯
=
c
i
and
c
i
∈
∑
j
≤
i
Z
+
t
j
and
c
i
≡
t
i
mod
v
F
+
for all
i
∈
I
. A canonical basis at
v
=
∞
is analogously defined to be a basis
(
c
~
i
)
i
∈
I
that satisfies
c
~
i
¯
=
c
~
i
and
c
~
i
∈
∑
j
≤
i
Z
−
t
j
and
c
~
i
≡
t
i
mod
v
−
1
F
−
for all
i
∈
I
. The naming "at
v
=
∞
" alludes to the fact
lim
v
→
∞
v
−
1
=
0
and hence the "specialization"
v
↦
∞
corresponds to quotienting out the relation
v
−
1
=
0
.
One can show that there exists at most one canonical basis at v = 0 (and at most one at
v
=
∞
) for each precanonical structure. A sufficient condition for existence is that the polynomials
r
i
j
∈
Z
defined by
t
j
¯
=
∑
i
r
i
j
t
i
satisfy
r
i
i
=
1
and
r
i
j
≠
0
⟹
i
≤
j
.
A canonical basis at v = 0 (
v
=
∞
) induces an isomorphism from
F
+
∩
F
+
¯
=
∑
i
Z
c
i
to
F
+
/
v
F
+
(
F
−
∩
F
−
¯
=
∑
i
Z
c
~
i
→
F
−
/
v
−
1
F
−
respectively).
The canonical basis of quantum groups in the sense of Lusztig and Kashiwara are canonical basis at
v
=
0
.
Let
(
W
,
S
)
be a Coxeter group. The corresponding Iwahori-Hecke algebra
H
has the standard basis
(
T
w
)
w
∈
W
, the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by
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
at
v
=
0
is the Kazhdan–Lusztig basis
C
w
′
=
∑
y
≤
w
P
y
,
w
(
v
2
)
T
w
with
P
y
,
w
being the Kazhdan–Lusztig polynomials.
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
λ
is an eigenvalue of
A
of algebraic multiplicity
μ
, then
A
will have
μ
linearly independent generalized eigenvectors corresponding to
λ
.
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
x
m
−
1
,
x
m
−
2
,
…
,
x
1
that are in the Jordan chain generated by
x
m
are also in the canonical basis.
Let
λ
i
be an eigenvalue of
A
of algebraic multiplicity
μ
i
. First, find the ranks (matrix ranks) of the matrices
(
A
−
λ
i
I
)
,
(
A
−
λ
i
I
)
2
,
…
,
(
A
−
λ
i
I
)
m
i
. The integer
m
i
is determined to be the first integer for which
(
A
−
λ
i
I
)
m
i
has rank
n
−
μ
i
(n being the number of rows or columns of
A
, that is,
A
is n × n).
Now define
ρ
k
=
rank
(
A
−
λ
i
I
)
k
−
1
−
rank
(
A
−
λ
i
I
)
k
(
k
=
1
,
2
,
…
,
m
i
)
.
The variable
ρ
k
designates the number of linearly independent generalized eigenvectors of rank k (generalized eigenvector rank; see generalized eigenvector) corresponding to the eigenvalue
λ
i
that will appear in a canonical basis for
A
. Note that
rank
(
A
−
λ
i
I
)
0
=
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).
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
=
(
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
)
has eigenvalues
λ
1
=
4
and
λ
2
=
5
with algebraic multiplicities
μ
1
=
4
and
μ
2
=
2
, but geometric multiplicities
γ
1
=
1
and
γ
2
=
1
.
For
λ
1
=
4
,
we have
n
−
μ
1
=
6
−
4
=
2
,
(
A
−
4
I
)
has rank 5,
(
A
−
4
I
)
2
has rank 4,
(
A
−
4
I
)
3
has rank 3,
(
A
−
4
I
)
4
has rank 2.
Therefore
m
1
=
4.
ρ
4
=
rank
(
A
−
4
I
)
3
−
rank
(
A
−
4
I
)
4
=
3
−
2
=
1
,
ρ
3
=
rank
(
A
−
4
I
)
2
−
rank
(
A
−
4
I
)
3
=
4
−
3
=
1
,
ρ
2
=
rank
(
A
−
4
I
)
1
−
rank
(
A
−
4
I
)
2
=
5
−
4
=
1
,
ρ
1
=
rank
(
A
−
4
I
)
0
−
rank
(
A
−
4
I
)
1
=
6
−
5
=
1.
Thus, a canonical basis for
A
will have, corresponding to
λ
1
=
4
,
one generalized eigenvector each of ranks 4, 3, 2 and 1.
For
λ
2
=
5
,
we have
n
−
μ
2
=
6
−
2
=
4
,
(
A
−
5
I
)
has rank 5,
(
A
−
5
I
)
2
has rank 4.
Therefore
m
2
=
2.
ρ
2
=
rank
(
A
−
5
I
)
1
−
rank
(
A
−
5
I
)
2
=
5
−
4
=
1
,
ρ
1
=
rank
(
A
−
5
I
)
0
−
rank
(
A
−
5
I
)
1
=
6
−
5
=
1.
Thus, a canonical basis for
A
will have, corresponding to
λ
2
=
5
,
one generalized eigenvector each of ranks 2 and 1.
A canonical basis for
A
is
{
x
1
,
x
2
,
x
3
,
x
4
,
y
1
,
y
2
}
=
{
(
−
4
0
0
0
0
0
)
(
−
27
−
4
0
0
0
0
)
(
25
−
25
−
2
0
0
0
)
(
0
36
−
12
−
2
2
−
1
)
(
3
2
1
1
0
0
)
(
−
8
−
4
−
1
0
1
0
)
}
.
x
1
is the ordinary eigenvector associated with
λ
1
.
x
2
,
x
3
and
x
4
are generalized eigenvectors associated with
λ
1
.
y
1
is the ordinary eigenvector associated with
λ
2
.
y
2
is a generalized eigenvector associated with
λ
2
.
A matrix
J
in Jordan normal form, similar to
A
is obtained as follows:
M
=
(
x
1
x
2
x
3
x
4
y
1
y
2
)
=
(
−
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
)
,
J
=
(
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
)
,
where the matrix
M
is a generalized modal matrix for
A
and
A
M
=
M
J
.
