# explicit components of regular representation of $S_4$

Consider (left) regular complex representation of $S_4$. It has two 2-dimensional irreducible components. I need exact form of elements in those components (probably, having one element I may get three more by left and right multiplication). May I find them online, or is there very quick way to get them which I do not see?

## 1 Answer

All $S_n$'s have an irreducible representation of degree $n-1$: take the natural permutation representation of degree $n$, say on the permuted basis $v_1,v_2,\ldots,v_n$, then the subspace spanned by $v_j-v_{j-1},\ j>1$ is in fact a subrepresentation and is irreducible. You can always multiply this representation by signum and get another irreducible representation of the same degree.

• Perhaps I should add that every irreducible representation occurs in the regular representation with multiplicity equalling its degree. So you can look for this. – P Vanchinathan Mar 24 '15 at 0:35
• Ok, but how does it help to get an explicit element of group algebra lying in 2-dimensional irreducible component? – Fedor Petrov Mar 24 '15 at 5:48
• This is by Young diagram. From each diagram corresponding to a partition you can define column stabilizer and row stabilizer subgroup. Take the sum of elements in subgroup and alternating sum in another and take the product in the group algebra: the subrepresentation generated by that element is what you are looking for. – P Vanchinathan Mar 24 '15 at 6:33
• Oh, I see. Million of thanks! – Fedor Petrov Mar 24 '15 at 10:38