# Find the dimension of the set of all polynomials of degree at most 3, as well as the subspace consisting of symmetric and skew-symmetric polynomials.

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.

• 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. Jan 6, 2022 at 4:00
• You're missing the "anti-" in your statement of part 3 of the problem. Jan 6, 2022 at 4:00