Wednesday, December 3, 2008

Weights & Roots

There is a one-to-one correspondence between the finite-dimensional complex representations Π of SU(3) and the finite-dimensional complex-linear representation π of sl(3;C). This correspondence is determined by the property that Π(eX) = eπ(X) for all X su(3) sl(3;C). The representation Π is irreducible if and only if the representation π is irreducible. 


Simultaneous Diagonalization

Suppose that V is a vector space and A is some collection of linear operators on V. Then a simultaneous eigenvector for A is a nonzero vector v V such that for all A A, there exists a constant λA with Av = λAv. The numbers λA are the simultaneous eigenvalues associated to v. For example, consider the space D of all diagonal nxn matrices. For each k = 1,...,n, the standard basis element ek is a simultaneous eigenvector for D. For each diagonal matrix A, the simultaneous eigenvalue associated to ek is the k-th diagonal entry of A.


If A is a simultaneously diagonalisable family of linear operators on a finite-dimensional vector space V, then the elements of A commute.

If A is commuting collection of linear operators on a finite-dimensional vector space V and each A A is diagonalizable, then the elements of A are simultaneously diagonalizable. 


Basis for sl(3;C)

Every finite-dimensional represetation of sl(2;C) or sl(3;C) decomposes as a direct sum of irreducible invariant subspaces. Consider the following basis for sl(3;C):

H1 = (1,0,0 : 0,-1,0 : 0,0,0), H1 = (0,0,0 : 0,1,0 : 0,0,-1),

X1 = (0,0,0 : 1,0,0 : 0,0,0), X2 = (0,0,0 : 0,0,0 : 0,1,0), X3 = (0,0,0 : 0,0,0 : 1,0,0),

Y1 = (0,1,0 : 0,0,0 : 0,0,0), Y2 = (0,0,0 : 0,0,1 : 0,0,0), Y3 = (0,0,1 : 0,0,0 : 0,0,0).


The span of {H1, X1, Y1} is a subalgebra of sl(3;C) which is isomorphic to sl(2;C) (can be seen by ignoring the third row and the third column in each matrix). Similarly for  {H2, X2, Y2}. Thus, the following commutation relations exists:

[X1, Y1] = H1, [X2, Y2] = H2,

[H1, X1] = 2X1, [H2, X2] = 2X2,

[H1, Y1] = -2Y1, [H2, Y2] = -2Y2.

Other commute relations among the basis elements which involve at least one H1 and H2:

[H1, H2] = 0;

[H1, X1] = 2X1, [H1, Y1] = -2Y1,

[H2, X1] = -X1, [H2, Y1] = Y1;

[H1, X2] = -X2, [H1, Y2] = Y2,

[H2, X2] = 2X2, [H2, Y2] = -2Y2;

[H1, X3] = X3, [H1, Y3] = -Y3,

[H2, X3] = X3, [H2, Y3] = -Y3;

Adding all of the remaining commutation relations:

[X1, Y1] = H1,

[X2, Y2] = H2,

[X3, Y3] = H1 + H2;

[X1, X2] = X3, [Y1, Y2] = -Y3,

[X1, Y2] = 0, [X2, Y1] = 0;

[X1, X3] = 0, [Y1, Y3] = 0,

[X2, X3] = 0, [Y2, Y3] = 0;

[X2, Y3] = Y1, [X3, Y2] = X1,

[X1, Y3] = -Y2, [X3, Y1] = -X2.


Weights of sl(3;C)

A strategy to classify the representation sl(3;C) is to simultaneously diagonalize π(H1) and π(H2). Since H1 and H2 commute, π(H1) and π(H2) will also commute and so there is at least a chance that π(H1) and π(H2) can be simultaneously diagonalized. 

If (π, V) is a representation of sl(3;C), then an ordered pair μ = (m1, m2) C2 is called a weight of π if there exists v ≠ 0 in V such that π(H1)v = m1v, π(H2)v = m2v. A nonzero vector v satisfying this is called a weight vector corresponding to the weight μ. If μ = (m1, m2) is a weight, then the space of all vectors v satisfying π(H1)v = m1v, π(H2)v = m2v is the weight space corresponding to the weight μ. The multiplicity of a weight is the dimension of the corresponding weight space. Equivalent representations have the same weights and multiplicities. 


If π is a representation of sl(3;C), then all of the weights of π are of the form μ = (m1, m2) with m1 and m2 being integers.


Roots of sl(3;C)

An ordered pair α = (a1, a2) C2 is called a root if 

1. a1 and a2 are not both zero, and

2. there exists a nonzero Z sl(3;C) such that [H1, Z] = a1Z, [H2, Z] = a1Z. The element Z is called a root vector corresponding to the root α. 


Recall that adX(Y) = [X, Y], eadx = Ad(eX), and the adjoint mapping AdA(X) = AXA-1. Condition 2 of above says that Z is a simultaneous eigenvector for adH1 and adH2.  This means that Z is a weight vector for the adjoint representation and weight (a1, a2). By condition 1 the roots are precisely the nonzero weights of the adjoint representation. 


There are six roots of sl(3;C). They form a "root system", called A2.

α            Z

(2, -1)    X1

(-1, 2)    X2

(1, 1)     X3

(-2, 1)    Y1

(1, -2)    Y2

(-1, -1)   Y3

It is convenient to single out the two roots corresponding to X1 and X2 and given them special names: α1 = (2, -1), α2 = (-1,2). α1 and α2 are called the positive simple roots. They have the property that all of the roots can be expressed as linear combinations of α1 and α2 with integer coefficients :

(2, -1) = α1

(-1, 2) = α2

(1, 1) = α1 + α2

(-2, 1) = -α1

(1, -2) = -α2

(-1, -1) = -α1 - α2 .

Let α = (a1, a2) be a root and Zα a corresponding root vector in sl(3;C). Let π be the representation of sl(3;C), μ = (m1, m2) a weight for π, and v ≠ 0 a corresponding weight vector. Then

π(H1)π(Zα)v = (m1+ a1)π(Zα)v,

π(H2)π(Zα)v = (m2+ a2)π(Zα)v.

Thus, either π(Zα)v = 0 or π(Zα)v is a new weight vector with weight

μ + α = (m1+ a1, m2+ a2) .

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