Young’s projection operators
Associated to a Young tableau with boxes, we have two elements
of the group ring
of the permutation group
on symbols. To
construct the operators, we first construct the antisymmetrizing
operators corresponding to the columns and the symmetrizing
operators corresponding to the rows. Then one operator corresponding
to the tableau consists of the product of the symmetrizing operators
corresponding to the rows multiplied by the product of the
antisymmetrizing operators corresponding to the columns and the
other consists of the product of the antisymmetrizing operators
corresponding to the columns multiplied by the symmetrizing operators
corresponding to the rows.
How this works may be illustrated with a simple example. Consider the tableau
Corresponding to the first row, we have the symmetrization operator
Corresponding to the second row, we have the symmetrization operator
Multiplying these two symmetrization operators (the order does not
matter because they involve permutations of different elements)
produces
Corresponding to the first column, we have the antisymmetrization operator
Corresponding to the second column, we have the antisymmetrization operator
Multiplying these two antisymmetrization operators (the order does not matter because they involve permutations of different elements) produces
To obtain one Young projector, we multiply the product of the symmetrization operators by the product of the antisymmetrization operators.
To obtain the other projector, we multiply in the other order.
Title | Young’s projection operators |
---|---|
Canonical name | YoungsProjectionOperators |
Date of creation | 2013-03-22 16:48:25 |
Last modified on | 2013-03-22 16:48:25 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 16 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 20C30 |
Classification | msc 11P99 |
Classification | msc 05A17 |