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Let $V_d$ be the vector space of homogeneous polynomials of degree $d$ in three variables $x, y,$ and $z$, and let the symmetric group $S_3$ act on $V_d$ by permuting the variables. Find the irreducible subrepresentations of $V_d$ in case $d = 1, 2,$ and $3.$

I am confused about what this question is actually asking. In the case where $d=1$ we have the space of polynomials of the form $ax+by+cz$. The group $S_3$ will act by permuting the variables. Considering the polynomials as vectors in the $x,y,z$ basis we can write them as $(a,b,c)$ and then the six permutations of $S_3$ map to linear transformations permuting the basis. So what does it mean to find the irreducible subrepresentations? I though an irreducible subrepresentation was something you have on the group, i.e. one of the three irreducible subrepresentations of $S_3$ is the trivial representation on a one dimensional vector space where all of $S_3$ is sent to the identity. Why am I being asked to find the irreducible representations the vector space (rather than the group)?


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I don't understand what you are asking, really. You seen yo be confused about what representations are... May I suggest you reread your notes/textbook? – Mariano Suárez-Alvarez Mar 18 '12 at 1:44
Do you understand what the question (the first paragraph) is asking (it's verbatim from my prof)? I can't seem to parse it into a question that makes sense. – Steven-Owen Mar 18 '12 at 1:55
@jake Unrelated: Please do not add subject at the title of your question. They are what the tags are for. – user21436 Mar 18 '12 at 1:59
up vote 3 down vote accepted

I will do the example for $d=1$, because you seem to be confused about what kind of work you have to do. When you know what you have to do perhaps you will be able to do the rest on your own, so I'll stick with the $d=1$ case which is the most easy one to give you an idea.

You have $6$ elements in $S_3$, namely $()$, $(12)$, $(13)$, $(23)$, $(123)$ and $(132)$. You are given a representation of $S_3$ over the vector space $V_1$ of homogeneous polynomials of degree $1$, which is a vector space (over $\mathbb C$? You didn't specify the ground field of the vector space ; this is a very important fact when treating representation theory!) A basis of $V_d$ is given by $\{x_1, x_2, x_3\}$ (I prefer this notation, you will see why in a second) because given this, we can say that if $\pi$ is a permutation of $\{1,2,3\}$, then $$ \pi \cdot p(x_1, x_2, x_3) = p(x_{\pi(1)},x_{\pi(2)}, x_{\pi(3)}) $$ or in other words, you "apply the permutation to the indices".

Now we know some trivial submodules of $V_1$ : for instance, the submodule generated by $x_1 + x_2 + x_3$. By Maschke's theorem, there exists a complement to this submodule which is also a submodule, so that they are in a direct sum ; in other words, you can find a basis of $V_d$ of the form $\{x_1 + x_2 + x_3, y_2, y_3\}$ such that $\langle y_2, y_3 \rangle$ is a submodule of $V_1$. In matrix terms, this means that over this new basis, your representation has this form. $$ \rho(\pi) = \begin{bmatrix} * & 0 & 0 \\ 0 & * & * \\ 0 & * & * \end{bmatrix} $$ Now it remains to show if the last block is irreducible (representations of degree $1$ cannot be reduced).

It is very hard to precisely give you what your teacher wants you to answer, because there are many tools to do this ; character theory, module theory, "doing things by hand" (which is computationally speaking very hard, mostly for $d=3$) but if you tell me more I guess I could try to help you more.

Hope that helps,

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The first paragraph (i.e. the question your professor is asking) makes complete sense. The fact that it doesn't make sense to you seems to be because you are confusing irreducible representations of a group with irreducible subrepresentations of a given representation of the group. You should review the concept of subrepresentation. (It is surely in your notes and textbook, and is likely discussed in many online resources as well.)

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