Casimir
This module contains procedures related to Generalized Casimir Opperators.

Calling Sequence

Casimir[first] ( operator );

Casimir[additional] ( operator );

Casimir[Couple] ( spaces, degree );

Casimir[EV] ( degree, irrep );

Parameters

operator	Either an appropriate list to specify the ranges of the L-operators, or spaces, degree as defined below.
spaces	A set of integers specifying which spaces to couple
degree	An integer greater than 1, specifies the degree of the opperator--quadratic, cubic, etc.
irrep	A monotonically decreasing list of positive integers, specifying an irrep.

Description

• Casimir[first] applies a Generalized Casimir Operator to the polynomials created the last time MapHW was run. Attemting to apply a Casimir Operator before mapping the irrep results in an error.
   
• Casimir[additional] applies a Generalized Casimir Operator and attempts to simultainiusly diagnolize it with previous operators.
   
• Both Casimir[first] and Casimir[additional] return a matrix of eigenvalues and a matrix of eigenvectors. Each column of the eigenvalues matrix corresponds to a different operator, while each row are the labels/values corresponding to a particular eigenvector. The eigenvector matrix returned will need to be stored to a global variable for later use. This matrix will store all the needed information to represent the specific coupling scheme for both the Generalized Racah and Clebsch-Gordan coefficients.
• As the names would suggest, Casimir[first] always assumes the user is specifying the first operator in a set of commuting operators, and resets the internal memory of previous operators. Casimir[additional], however, always assumes the specified operator is part of a set of commuting operators and simultainiously diagnolizes the opperator with all other operators applied, back to the last time Casimir[first] was computed.
   
• The operators can be specified as a product of left operators, sums of such products, or as a coupling scheme.
   
• A product of operators is specified by a list of lists: the inner list containing the pairs of indices for the operators. The details of the requirements on these indices (in order for them to represent a Generalized Casimir Operator) is found in the publication. Sums of products are representing a list of such products. For example,

  Operator	Entered As
  L1,2L2,1	[[1,2],[2,1]]
  L1,3L3,1	[[1,3],[3,1]]
  L1,2L2,1 + L1,3L3,1	[ [[1,2],[2,1]], [[1,3],[3,1]] ]

  
   
• Instead of explicitly defining the indices, the operator can be specified by a list of two items: the first is a set containing the indices of the spaces the operator should cover, and the second item in the list is the degree of the operator. The procedure then automatically uses Casimir[Couple] to determine the actual L-operators. For instance, the quadratic operator over the set of spaces {1,3,4} is specified as [ {1,3,4}, 2 ].
   
• The user can also directly view the related L-operators by running the Casimir[Couple] procedure themselves. Passing the results of this procedure to Casimir[first] or Casimir[additional] procedures is equivilant to just specifying the spaces and degree directly. This is helpful, since just a few of the terms in the operator can have equal (or greater) ability to break the degeneracy than the full operator specified by all the terms. The actual eigenvalues will of course be different, but this smaller operator can carry the full implication ("breaking power") of the coupling scheme.
   
• Further details about chosing operators to represent coupling schemes and break degeneracy can be found in Additional Documentation .
   
• Casimir[EV] uses a well known formula to compute the eigenvalue of the (non-generalized) Casmir operator of specified degree over the specified irrep. This procedure is needing in computing Clebsh-Gordan Coefficients, specifically to relate eigenvectors to specific Gelfand-Zetlin Tableau.

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