Products of Irreducible Representations
A common group theoretical task is the direct product or symmetric direct product of irreducible representations. The composition of a product is obtained as
$ pm irr-rep-product --point-group Oh --irr-reps T2g T2g
# T2gxT2g = A1g+Eg+T1g+T2g
The composition of a symmetric product is obtained as
$ pm irr-rep-symmetric-product --point-group Oh --irr-reps T2g T2g
# [T2gxT2g] = A1g+Eg+T2g
An arbitrary number of irreps can be handled
$ pm irr-rep-product --point-group Oh --irr-reps T2g T2g Eg Eg A2g
# T2gxT2gxEgxEgxA2g = 2A1g+2A2g+4Eg+4T1g+4T2g
Another important task is constructing the irreducible representation vectors of the product representation in terms of the rows of the original irreducible representations.
$ pm irr-rep-product --point-group Oh --irr-reps Eg Eg --print-irr-vectors
# EgxEg = A1g+A2g+Eg
#
# irr Eg.0xEg.0 Eg.0xEg.1 Eg.1xEg.0 Eg.1xEg.1
# A1g 0.7071 0.0 0.0 0.7071
# A2g 0.0 0.7071 -0.7071 0.0
# Eg.0 0.0 0.7071 0.7071 0.0
# Eg.1 0.7071 0.0 0.0 -0.7071
The same can be done for the symmetric product
$ pm irr-rep-symmetric-product --point-group Oh --irr-reps Eg Eg --print-irr-vectors
# [EgxEg] = A1g+Eg
#
# irr Eg.0xEg.0 Eg.0xEg.1 Eg.1xEg.1
# A1g 0.7071 0.0 0.7071
# Eg.0 0.0 1.0 0.0
# Eg.1 0.7071 0.0 -0.7071
It might also be useful to print out the transpose of the results, which can be done with the output-transpose flag:
$ pm irr-rep-symmetric-product --point-group Oh --irr-reps Eg Eg --print-irr-vectors --output-transpose
# [EgxEg] = A1g+Eg
#
# A1g Eg.0 Eg.1
# Eg.0xEg.0 0.7071 0.0000 0.7071
# Eg.0xEg.1 0.0000 1.0000 0.0000
# Eg.1xEg.1 0.7071 0.0000 -0.7071
In many cases, one is only concerned with the identity representation, and the flag only-print-identity will only print identity representations and will prune out any basis vectors that are not contained in the product:
$ pm irr-rep-symmetric-product --point-group Oh --irr-reps Eg Eg --print-irr-vectors --only-print-identity
# [EgxEg] = A1g+Eg
#
# Eg.0xEg.0 Eg.1xEg.1
# A1g 0.7071 0.7071
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