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Let $V$ be the set of all polynomials $f(x,y)$ in $x,y$ with real coefficients, of degree at most 3. That is, each term of the polynomial will have the form $x^iy^j$ with $i,j \in \mathbb{N}$ and $i+j \leq 3$.

  1. What is the dimension of V?
  2. Let $V_+$ be the subspace of symmetric functions in $V$, i.e. $V_+$ is the set of all $f(x,y) \in V$ such that $f(y,x) = f(x,y)$. What is the dimension of $V_+$?
  3. Let $V_-$ be the subspace of skew-symmetric functions in $V$, i.e. $V_-$ is the set of all $f(x,y) \in V$ such that $f(y,x) = -f(x,y)$. What is the dimension of $V_-$?

This is what I've done so far:

  1. To find the dimension of $V$ we must find a basis of $V$ say $\mathcal{B}$. We can do this by enumerating all of the valid monomials of a polynomial in $V$. They are, $$ \mathcal{B} =\{x^3,y^3,x^2y,y^2x,x^2,y^2,xy,x,y,1\}.$$ Thus the dimension of $V$ is 10.

  2. This is where I am not sure what I am doing is correct. We must find a basis for $V_+$. One polynomial that surely works is any constant function, so we can include $1$ in our basis for $V_+$. Now, consider the following, $$ x^3 + y^3 \rightarrow y^3 + x^3\\ x^2y + y^2 x \rightarrow y^2x + x^2 y\\ x^2 + y^2 \rightarrow y^2 x^2\\ xy \rightarrow yx\\ x+y \rightarrow y+x.$$ We see that for each of these polynomials, $f(y,x) = f(x,y)$. So our basis is just $1$ and all of the binomials listed above, which implies the dimension of $V_+$ is 6.

  3. Again, I'm unsure if this is correct. My basic idea was to try what I did above, but instead of adding basis vectors of $V$, subtracting. I came up with, $$x^3 - y^3 \rightarrow -y^3 + x^3\\ x^2y-y^2x \rightarrow -y^2x+x^2y\\ x^2-y^2 \rightarrow -y^2 + x^2\\ x-y \rightarrow -y + x\\ $$ Thus the dimension of $V_-$ is 4.

Any feedback would be welcome.

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    $\begingroup$ In 2, you appear to be trying to find elements stabilized by the action $x \leftrightarrow y$. You might be happier thinking about orbit-stabilizer and Burnside's lemma in the context of your 10-element basis. For 2 and 3: Note that every polynomial has a unique representation as the sum of a symmetric part and an antisymmetric part, so you know the sum of these two dimensions is 10. $\endgroup$ Jan 6, 2022 at 4:00
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    $\begingroup$ You're missing the "anti-" in your statement of part 3 of the problem. $\endgroup$ Jan 6, 2022 at 4:00
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    $\begingroup$ This link might help you: math.stackexchange.com/questions/2524343/… $\endgroup$ Jan 6, 2022 at 4:01

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