Almost-invariant polynomials under dihedral group action Think about the dihedral group $D_4$ acting on the polynomial algebra $\mathbb C[x_1, \cdots, x_4]$ via generating permutations $(x_1\ x_2)$, $(x_3\ x_4)$, and $(x_1\ x_3)(x_2\ x_4)$.  I'd like to look at the polynomials that are invariant under the first two permutations and change sign under the last one; the two obvious examples are $$P_n = x_1^n + x_2^n - x_3^n - x_4^n$$ and $$Q_n = x_1^nx_2^n - x_3^nx_4^n.$$
Normally I'd think about the ring of invariants, but I don't even have a ring here - $P_1^2$ doesn't have the right invariance properties!
My main question is: How can I effectively describe the polynomials that obey these (anti)symmetries?
It seems like I'm going to end up having to consider the $D_4$-invariant polynomials, since if I have a polynomial $s$ that's $D_4$-partially-anti-invariant (as above), and a polynomial $t$ that's $D_4$-invariant, then clearly $st$ will be $D_4$-partially-anti-invariant.  But I've got no clue how to describe the (vector space of) polynomials that are $D_4$-partially-anti-invariant in the first place.
EDIT:  After reading a bit about this and doing some thinking, I can make the question more explicit.  Write $R = \mathbb{C}[x_1, x_2, x_3, x_4]$, let $R^{D_4}$ be the ring of $D_4$-invariant polynomials (under the obvious permutation action), and let $\chi: D_4 \to \mathbb{C}^{\times}$ be the character above.  Then I'd like a description (or a good reference for how to construct such a description) of the semi-invariants $R^{D_4}_{\chi}$ - those polynomials $p$ such that $\sigma(p) = \chi(\sigma)\cdot p$ - as a module over the ring of invariants $R^{D_4}$.
 A: What you want are the $D_4$-invariant elements of $\mathbb{C}[x_1, ...,x_4]$, but with a weird group action, which isn't as simple as permuting the variables. To describe the group action you want, I'll need to describe some other homomorphisms first.
There is a group homomorphism $h: D_4 \mapsto \{1, -1\}$ that can be thought of as follows: an element of $D_4$ can be thought of as a symmetry of a square, and we can ask whether the symmetry swaps the diagonals or keeps them the same. If it swaps them, map to $-1$, otherwise, map to $1$. There is also a group homomorphism $i: D_4 \mapsto S_4$ based on the permutation of the corners, which we'll label, in clockwise order, 1, 3, 2, 4 (this labelling is chosen based on the specific generators you wanted). We can now describe the desired action of $D_4$ on $\mathbb{C}[x_1, ...,x_4]$: an element $u$ of $D_4$ acts by performing the permutation $i(u)$ on the variables, and then multiplying by $h(u)$. Now the polynomials you are interested are exactly the $D_4$-invariant ones, with this group action.
