great orthogonality theorem

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Theorem:

Consider two unitary, irreducible matrix representations \Gamma^{(i)}(G)\, and \Gamma^{(j)}(G)\, of a group G\,. Then

\sum_{g} \Gamma^{(i)*}(g)_{\alpha\beta} \Gamma^{(j)}(g)_{\gamma\delta}  = \frac{[G]}{l_{i}} \delta^{ij}    \delta_{\alpha\gamma} \delta_{\beta\delta} \,,

where [G]\, is the order of the group G\,, and l_i\, is the dimension of \Gamma^{(i)}(G)\,.

Proof:

Define the matrix M\,:

M \equiv \sum_{g\in G} \Gamma^{(i)}(g) X \Gamma^{(j)\dagger}(g)\,,

where X\, is some unspecified matrix. Then

\Gamma^{(i)}(h) M\,  = \Gamma^{(i)}(h) \sum_{g} \Gamma^{(i)}(g) X \Gamma^{(j)\dagger}(g)\,,
 = \sum_{g} \Gamma^{(i)}(hg) X \Gamma^{(j)\dagger}(g)\,,
 = \sum_{hg} \Gamma^{(i)}(hg) X \Gamma^{(j)\dagger}(h^{-1}hg)\,,
 = \sum_{g} \Gamma^{(i)}(g) X \Gamma^{(j)\dagger}(h^{-1}g)\,, (relabeling hg \to g\,)
 = \sum_{g} \Gamma^{(i)}(g) X \Gamma^{(j)\dagger}(g) \Gamma^{(j)}(h)\,,
 = M \Gamma^{(j)}(h)\,.

By the converse of Schur's lemma, either i=j\,, or M = 0\,. If i = j\,, then by Schur's lemma M = m \mathbf{1}\,, where we find m\, by taking the trace.

\sum_{g} \Gamma^{(i)}(g) X \Gamma^{(j)\dagger}(g) = m \delta^{ij} \mathbf{1}\,,
[G] \operatorname{tr} X = m l_{(i)}\,. So
\sum_{g} \Gamma^{(i)}(g) X \Gamma^{(j)\dagger}(g) = \frac{[G]}{l_{i}} \delta^{ij} \operatorname{tr}X \mathbf{1}\,,

or, with index notation

\sum_{g} \Gamma^{(i)}(g)_{\alpha\beta} X_{\beta\gamma} \Gamma^{(j)\dagger}(g)_{\gamma\delta} = \frac{[G]}{l_{i}}\delta_{\beta\gamma} X_{\beta\gamma} \delta^{ij} \delta_{\alpha\delta}\,,

but, since X_{\beta\gamma}\, is arbitrary,

\sum_{g} \Gamma^{(i)}(g)_{\alpha\beta} \Gamma^{(j)\dagger}(g)_{\gamma\delta} = \frac{[G]}{l_{i}} \delta^{ij}\delta_{\beta\gamma}   \delta_{\alpha\delta}\,,

or, relabeling indices,

\sum_{g} \Gamma^{(i)*}(g)_{\alpha\beta} \Gamma^{(j)}(g)_{\gamma\delta}  = \frac{[G]}{l_{i}} \delta^{ij}    \delta_{\alpha\gamma} \delta_{\beta\delta} \,.



Consequences

Choosing \Gamma^{(j)}(G)\, to be the identity representation, we get

\sum_{g} \Gamma^{(i)}(g)_{\alpha\beta} = 0\,.

Also, the great orthogonality theorem implies an orthogonality between characters.

References

[1] [2] [3] [4]

  1. M. Tinkham (1992). Group Theory and Quantum Mechanics. Dover Publications. ISBN 978-0486432472. 
  2. M. Hamermesh (1989). Group Theory and its Applications to Physical Problems. Dover publications. ISBN 978-0486661810. 
  3. W. Miller, Jr. (1972). Symmetry Groups and their Applications. Academic Press. ISBN 978-0124974609. 
  4. J. F. Cornwell (1997). Group Theory in Physics (Three volumes), Volume 1. Academic Press. ISBN 978-0121898007. 
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