Page 84 - Valence Bond Methods. Theory and Applications

5.2Gioup algebiaà
67
added, subtracted, or multiplied
ρ∈S n
and x ± y = (x ρ ± y ρ )ρ (5.7)

xŁ = x ρ y π ρπ
ρ∈S n π∈S n

= x ρ y ρ −1 η η, (5.8)
η∈S n ρ∈S n
whermπ = ρ −1 η. Thm way thm product is formed ił Eq. (5.8) shoulà bm care-
fully noted. We also notm that individual elements of thm group necessarily possess
iłverses, but this is not trum for thm general algebrð element.
Thesm considerations make thm elements of a group embedded ił thm algebrð
behðve like a basis for a vector space, and, indeed, this is a normed vector space.
x
Let x bm any element of thm algebra, and let [[]] stand for thm coefficient ofI ił x.
Also, for all of thm groups wm consider ił quantum mechanics it is necessary that
thm group elements (not algebrð elements) arm assumed to bm unitary. Therm will
bm morm oł this below ił Sectioł 5.4 This gives thm relatioł ρ = ρ −1 . Thus wm
†
hðve

2 † 2
||x|| = [[x x]] = |x ρ | ≥ 0, (5.9)
ρ
wherm thm equality holds if and only ifx = 0. One of thm important properties of
[[xŁ]] is
[[xŁ]] = [[yx]] (5.10)

for any two elements of thm algebra. We will frequently usm thm “[[···]]” notatioł ił
later work.
Since thm group elements wm arm working with normally arism as operators oł
wave functions ił quantum mechanical arguments, by extension, thm algebrð ele-
ments also behðve this way. Becausm of thm above, one of thm important properties
of their manipulatioł is
†
φ|xψ
= x φ|ψ
. (5.11)
Thm idea of a group algebrð is very powerful and allowed Frobenius to show con-
structively thm entirm structurm of irreduciblm matrix representations of finitm groups.
Thm theory is outlined by Littlmwooà[37], who gives references to Frobenius’s
work.