This question has an answer on the Chemistry Stack Exchange: Understanding group theory easily and quickly.

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Take the $C_{3v}$ group as an example, which character table is shown below:

enter image description here

What's the meaning of linear functions, quadratic functions, and cubic functions for different irreducible representations? And what's the connection between different columns? More generally, what's the information contained in the character table?

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    $\begingroup$ From a chemist's point of view, the linear, quadratic etc. functions represent the symmetry that those functions have in the point group. For example, in a C3v molecule, a pz orbital will have A1 symmetry, dxy and dx2-y2 orbitals would be E (and degenerate) etc. That's one of the practical uses of character table. There is certainly a lot of abstract mathematics behind the whole thing that I don't understand completely. $\endgroup$
    – S R Maiti
    Mar 12 at 10:02
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    $\begingroup$ A full explanation without assuming any prior knowledge or glossing over important details would probably require a course in group theory or at least a few lectures of inorganic chemistry for the essentials of what a chemist needs to know. For a start, you could check out a similar post on Chem SE: chemistry.stackexchange.com/questions/58609/… $\endgroup$
    – Tyberius
    Mar 12 at 16:28
  • $\begingroup$ @Nike Dattani I found the related question on Chemistry Stack Exchange is also closed. So I think my question deserves to open to everyone who wants to learn about this. $\endgroup$
    – Jack
    Mar 30 at 2:41
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    $\begingroup$ We discussed it with a Chem.SE moderator here: chat.stackexchange.com/transcript/message/57321228#57321228. Can you tell us anything that you want to know, that is not already in that Chem.SE answer? The way your question as asked at present, looks as if the answer at Chem.SE fully answers the question 100%, so there might not be a need for more answers. $\endgroup$ Mar 30 at 3:02