Representation of sl(3,C)Representation of sl(3,C)

Dongseok KIMDongseok KIM

OutlineOutline

• Adjoint representation of sl(3,C).• Eigenspace decomposition with respect to H, the max

imal abelian subalgebra.• Classify irreducible representations of sl(3,C).• Tensor rules (weight, Brauer, honeycomb, hives)• Invariant spaces, web spaces

• The Lie algebra sl(3,C) is generated by

Adjoint representation of sl(3,C).Adjoint representation of sl(3,C).

• We consider the general setup for the adjoint representation. First, we find L0=CL(h)=h.

• The set of all nonzero 2 h* for which L 0 is denoted by and the elements of are called the roots of sl(3,C) with respect to h.

• The decomposition

sl(3,C)=h©(©2L)

is called a Cartan decomposition or root space decomposition of sl}(3,C).

Since

we can write

h*=C[L1, L2, L3]/(L1+L2+L3=0)

where

We find the Cartan decomposition of adjoint representation.

Irreducible representation of sl(3,C)Irreducible representation of sl(3,C)• Let V be a finite dimensional irreducible representatio

n of sl(3,C). • First, we decompose V into eigenspaces of h. For the a

djoint representation was called a root, for other representation is called a weight of V. A diagram of all weight with multiplicity is called a weight diagram. Then one can find a vector v2 V such that Xij(v)=0 which is called a highest weight vector of V and the corresponding weight is called highest weight of V. Then V is generated by the images of v under successive applications of three operatorsYij.

• Since sl(2,C) is immersed if we restrict actions of h to { Xij, Yij, Hij}, we can see it reflects along three line <Hij, L>=0.

• Thus given the highest weight, we can find the hexagonal (possibly triangular) region that all weights of V appear. The following figure shows how to find the region from the given highest weight .

• The next step will be how we deal with the interior of the region.

• In general, we find it by theorems by Freudenthal and Kostant.

• For sl(3,C), first the multiplicity of the boundary is 1. Next, we can shrink the boundary by one time actions of Xij, Yij, if the boundary was hexagonal, the multiplicity of weight for the resulting polygon goes up by one, otherwise it stays the same. We continues the process until we find all multiplicity of weights.

• Let 1 = L1, 2= - L3.

• For finite dimensional representation, the highest weight must be in the region which can be expresses in non-negative integral linear combinations of 1 and 2.

• This region is called the Weyl chamber of sl(3,C). • The following is a weight diagram of an irreducible re

presentation of sl(3,C) of highest weight 41+22, denoted by V42

or V(4,2). The number in the weight d

iagram indicates the multiplicity of the weight.

1) A given weight in the Weyl chamber, there exists an irreducible representation of sl(3,C) with highest weight . Moreover can be written as a linear combination of 1 and b2, a1+b2 where a, b are nonnegative integers. The irreducible representation is denoted by V(a,b) =Va1+b2

.

2) Any finite dimensional representation of sl(3,C) is of the form Va1+b2

.

Tensor RuleTensor Rule• First we can use all weight diagram, for example we

look at the defining representation of sl(3,C), V(1,0) which has three eigenvalues L1, L2 and L3. Also one can find easily that its dual representation V(0,1) has three eigenvalues -L1, -L2 and -L3.

• One can find that V(1,0) V(0,1) had the following weight diagram which decompose into the adjoint representation and the trivial representation.

V(1,0)­ V(2,1)

V(1,0) ­ V(2,1) V(3,1) ©

• V(1,0) ­ V(2,1) V(3,1) ©V(1,2) ©­

• Therefore we find that

V(1,0) V(2,1) V(3,1) © V(1,2) © V(2,0)

There is another way using the Brauer theorem

• Honeycombs, V(1,0) is written by (0,1,1), V(2,1) by (0,2,3). Then we find all possible honeycombs.

• V(3,1) V(2,0) V(1,2)

Invariant spaceInvariant space

• First we find few dimensions of invariants spaces– Dim (Inv (V(1,0)­3)) = 1,

– Dim (Inv (V(0,1)­3)) = 1,

– Dim (Inv (V ­ V*)) = 1,

– Dim (Inv (V(1,0)­2 ­ V(0,1)­2 )) = 2.

• Let x, y, z be eigenvectors of V(1, 0) and let w = x ­ y ­ z – y ­ x ­ z + y ­ z ­ x – x ­ z ­ y + z ­ x ­ y – z ­ y ­ x. Then w 2 Inv (V(1, 0)­3))

• Webs for U_q(sl(3,C)) are generated by the Webs for U_q(sl(3,C)) are generated by the following trivalent verticesfollowing trivalent vertices

with relationswith relations