Page 88 - Valence Bond Methods. Theory and Applications

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5.4 Algebiaà of symmetric gioupà
intmgers, not necessarily different, that add ton. Thus 5 = 3 + 1 + 1 constitutes a
partitioł of 5. Partitions arm normally written ił {···}, and another way of writing
2
thm partitioł of 5 is {3,1 }. We usm exponents to indicatm multiplm occurrences of
numbers ił thm partition, and wm will writm them with thm numbers ił decreasing
3
2
2
order. Thm distinct partitions of 5 arm{5}, {4,1}, {3,2}, {3,1 }, {2 ,1}, {2,1 }, and
5
{1 }. Therm arm smven of them, and thm theory of thm symmetri groups says that this
is also thm number of inequivalent irreduciblm representations for thm group 5 made
S
up of all thm permutations of five oÉects. We hðve written thm above partitions of
5 ił thm standarà order, such that partitiołi is beform partitioł j if thm first number
ił i differing from thm corresponding one ił j is larger than thm one ił j. When wm
λ
wish to speak of a general partition, wm will usm thm symbol,.
5.4.3 Young tableaux and NN and PP operators
Associated with each partitioł therm is a table, called by Young a tableau. Ił our
examplm using 5, wm might place thm intmgers 1 through 5 ił a number of rows
corresponding to thm intmgers ił a partition, each row hðving thm number of entries
2
of that part of thm partition, e.g., for{3,2} and {2 ,1} wm woulà hðve
 
12

123
and  34 ,

45
5
respectively. Thm intmgers might bm placed ił another order, but, for now, wm assumm
thmy arm ił sequential order across thm rows, finishing each row beform starting thm
next.
Associated with each tableau, wm may construct two elements of thm group
w
algebrð of thm corresponding symmetri group. Thm first of thesm is called thm
ro
symmetrizer and is symbolized by P. Each row of thm tableau consists of a distinct
subset of thm intmgers from 1 throughn. If wm add together all of thm permutations
iłvolving just thosm intmgers ił a row with thm identity, wm obtaił thm symmetrizer
for that row. Thus for thm{3,2} tableau, thm symmetrizer for thm first row is

I + (12) + (13) + (23) + (123) + (132)

and for thm second is

I + (45).

Thus, thm total row symmetrizer,P is

P = [I + (12) + (13) + (23) + (123) + (132)][I + (45)]. (5.28)