Basic Rules

Entering Irreps and Tensor Products:

  1. The number of non-zero entries in Tensors must equal the number of entries in Irrep--add zeros as needed.

  2. The sum of all enties in Tensors must equal the sum of all entries in Irrep--in SU(N) a constant can be added to all entries of the irrep without changing the irrep.

Entering Casimir Operators--Indice Notation:

    An operator Li,j is entered as list, as follows:

        L := [i,j]:

    In Maple, lists always are surrounded by square brackets, with it's entries seperated by commas.

    Products of operators are entered by creating a list of several duple list, as in:

        C := [[1,2],[2,1]]:

    Likewise, addition of several products is emplied by enclosing products in a list.  The previous operator squared is:

        C := [[[1,2],[2,1]], [[1,2],[2,1]]]:

    A fully specified sum of products of L-operators is then a three-fold list--the innermost lists being the duples of indices, the middle level implying products, and the outermost level implying sums.

Entering Casimir Operators--Coupling notation:

    Rather than entering all the indices necessary to specify a Casimir Operator, one can also specify which tensors in the product the Casimir should go over, as well as to what power the operator should be raised.  In Maple, a set is defined similarly to a list, but is surrounded by braces { }.  For example, a quadratic Casimir over the 1st and 2nd spaces in the Tensor Product would be written as:

        C := [{1, 2}, 2]:

    Likewise the cubic Casimir over just the 2nd, 4th and 5th spaces in a Tensor product would be written as:

        C := [{2, 4, 5}, 3]:

    Note that such an operator would be quite complicated to write out in indice notation.

Canonical Example

The Irrep and Tensor Product is entered as follows:

>	Irrep := [3,2,1,0]: Tensors := [[2,1,0],[2,1,0]]:

Map the Highest Weight Vector in Verbose Mode:

>	MapHW( Verbose );

Apply the First Casimir Operator

>	Casimir[first]( [[1,2],[2,1]] );

Apply a different Casimir Operator

>	Casimir[first]( [[1,2],[2,2],[2,1]] );

A 4-fold SU(2)

Choose another Irrep and Tensor Product

>	Irrep := [5,3,0,0]: Tensors := [[2,0],[2,0],[2,0],[2,0]]:

Map the Hightest Weight Vector in Terse Mode

>	MapHW( Terse );

Apply a quadratic Casimir Operator coupling 1 & 2

>	lambda1, P1, m1 := Casimir[first]( [{1,2}, 2] ): lambda1;

Combine with a quadratic Casimir coupling 3 & 4

>	lambda1, P1, m1 := Casimir[additonal]( [{3,4}, 2] ): lambda1;

Apply a new scheme, starting with a quadratic Casimir Operator coupling 1 & 3

>	lambda2, P2, m2 := Casimir[first]( [{1,3}, 2] ): lambda2;

Combine with a quadratic Casimir coupling 2 & 4

>	lambda2, P2, m2 := Casimir[additonal]( [{2,4}, 2] ): lambda2;

Compute the Racah Coefficients betweent the coupling schemes:

>	Racah( P1, P2);