Sn = symmetric group

For each p in Sn associate a +1(p even) or -1(p odd)

a group algebra can be defined on the elements of Sn.

Theorem: The number of standard young tableau corresponding to a particular young diagram = dim of associated irrep denoted by the diagram.

Sn has the general symmetry elements built from the symmetrizer, antisymmetrizer, and young diagrams.

Define a general symmetry as:

Yl,q=Sl,q*Al,q

where l = the particular young diagram, q= the particular numbering of the young diagram, Sl,q= symmetry on the rows, and Al,q=antisymmetry on the columns.

Example:

1 2

3

=(e+12)(e-13)

12

34

= (e+(12))(e+(34))(e-(13))(e-(24))

source

jl@crush.caltech.edu index

projection_operator

symmetric_group

antisymmetrizer

group_division

young_diagram

symmetrizer

An