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I am looking to find the invariants of products of fields under SU(5) and other possible gauge groups (but lets take SU(5) as an example). Take, for example, two matter fields in the5*and10and two Higgses in5and5*(called [itex]H_{5}[/itex] and [itex]\bar{H}_{5*}[/itex]).

Then the term

5*10[itex]\bar{H}_{5*}[/itex]

is invariant under SU(5), as expected.

So long as we simply multiply the fields, I suppose it is easy to find the possible combinations simply by counting the number of [itex]U[/itex]'s and [itex] U^{\dagger}[/itex]'s (for [itex]U\in[/itex]SU(5)) that will enter the term upon transforming it and checking that the number balances (in the abovementioned example,5*will be multiplied by one [itex] U^{\dagger}[/itex], as will the Higgs. The10will be multiplied by two [itex]U[/itex]'s and since [itex]U U U^{\dagger}U^{\dagger}[/itex]=1, the whole term is invariant).

So far, so good.

But checking the literature, one notices that there seems to be another invariant term (to be found in the superpotential of SUSY SU(5)) which is not of the aforementioned form. It looks like this:

[itex]\epsilon_{XYZWK}[/itex]10[itex]^{XY}[/itex]10[itex]^{ZW}[/itex][itex]H_{5}^{K}[/itex]

Firstly, I have tried checking that this term is indeed invariant under SU(5) but so far havent managed to show it... how can one see that its invariant?

And secondly, how can one find terms like this? If I have some group G and want to construct all invariant terms under G, how would I find out that a term like above even exists?

Thank you!

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# Invariants under group actions

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