# What is the product of two irreducible representations on the character table?

Let's say the direct product of $\ce{A1}$ and $\ce{B1}$ on $\ce{C}_{2v}$ character table, what is the direct product of the two for water molecule? or for any other instance in general, what does the direct product of irreducible representations mean, and how to evaluate it?

• Homework questions should show the work or thinking that you have already done in an initial attempt to answer the question. For help asking a good homework question, see: How do I ask homework questions on Chemistry Stack Exchange? So what do you think? What is your current thought process? – Geoff Hutchison Sep 21 '14 at 19:08
• Thanks, I think direct product of A1 and B1 will be B1 itself. – Kinformationist Sep 21 '14 at 22:10
• Indeed, the direct product of the totally symmetric representation (here $A_1$) will always be the other representation, much like 1 times X is X. – Geoff Hutchison Sep 22 '14 at 0:38

I'll start with this clue. Any function can be expressed by the irreducible representations (or a linear combination of them). So you're doing a direct product of $A_1$ and $B_1$ under $C_{2v}$ symmetry.