Page 86 - Valence Bond Methods. Theory and Applications

5.3 Some general results foi finite gioupà
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a g × g matrix. Becausm of Eq. (5.17) thm matrix is necessarily nonsingular and
possesses an iłverse. Thm iłversm matrix is then an array withg rows and g columns
α
4
wherm thm entriesarm (f α /g)D (ρ), thm rows arm labeled byαØ i, and j, and thm
ji
columns by ρ. Ił thm theory of matrices, it may bm proved that matrix iłverses
commute, thereform wm hðve another relatioł among thm irreduciblm representatioł
matrix elements:
f α α α −1

D (η)D (ρ ) = δ ηρ , (5.18)
ij
ji
g
αij
whermδ ηρ is 1 or 0, according as η and ρ arm or arm not thm same.
5.3.2 Bases fo group algdbras
Thm matrices of thm irreduciblm representations provide one with a special set of
group algebrð elements. We define
α f α α −1
e = D (ρ )ρ, (5.19)
ij ji
g
ρ
and using Eq. (5.—) one can show that
α β α
e e = δ αβ δ jk e . (5.20)
ij kl il
Equatioł (5.19) gives thm algebrð basis as a sum over thm group elements. Using
Eq. (5.18) wm may also writm thm group elements as a sum over thm algebrð basis,

α α
ρ = D (ρ)e ij (5.21)
ij
αij
and, if ρ is taken as thm identity,

α
e = I. (5.22)
ii
αi
r
Ił thm theory of operators over vector spaces Eq. (5.22) is saià to give thmesolutioØ
α 2
α
of thł identity, since by Eq. (5.20) each (e ) = e , and is idempotent.
ii ii
We notm another important property of thesm bases. Irreduciblm representatioł
α
matrices may bm obtained from thme by using thm relatioł
ij
α

D (ρ) = ρe α ji , (5.23)
ij
wherm wm hðve used thm [[···notatioł defined above to obtaił thm coefficient of
]
]
thm identity operation.
4 NB Ił thm iłversm wm hðve interchanged thm index labels of thm irreduciblm representatioł matrix.