In the course of doing some calculations on a project I am working on, I came across the following presentation of a vector space. It is generated by homogenous polynomials of even degree $n$ over a vector space $V\oplus V\oplus V$, satisfying the following conditions.
1. $f(x,y,z)=-f(y,x,z)$
2. $f(x,y,z)=-f(x,z,y)$
3. $f(-x,y,z)=f(x,y,z)$
4. $-f(x,y,z)-f(x+y,-x,z)+f(x,y-x,z)=0$-f(x,y,z)-f(-y,x+y,z)+f(x,x+y,z)=0$Computer calculations indicate that when$V$is$1$-dimensional, the dimensions of this space, starting with$n=2$are$0,0,0,0,1,0,1,1,1,1,2,1,2,$which looks like the dimensions of the spaces of classical cusp forms. I'm curious if anyone can see whether there really is an isomorphism. There is a similar problem for polynomials over$V\oplus V$which I do know how to solve. In this case, the polynomial is supposed to satisfy: 1.$f(x,y)=f(y,x)$2.$f(x,y)=-f(-x,y)$3.$f(x,y)+f(y,-x-y)+f(-x-y,x)=0$Over a general$V$the answer is a bit complicated, but when$\dim V=1$, you do get classical cusp forms. Basically one uses the Eichler-Shimura isomorphism$H^1_{cusp}(SL_2(\mathbb Z),Sym^k(\mathbb C^2))\cong\mathcal S_{k+2}\oplus \overline{\mathcal S_{k+2}}$, where$\mathcal S_{k+2}$denotes the space of cusp forms of weight$k+2$. Examining the group cohomology chain complex, one can derive the isomorphism. It might be enlightening to have a more direct computation of the dimension. Anyway, in summary, I'm looking for a proof that the above presentation of polynomials over$V\oplus V\oplus V$gives classical cusp forms when$\dim V=1$and any further ideas about what happens as$\dim V$increases would be helpful too! It's clear that the answer will decompose as a direct sum of Schur functors$\mathbb S_{\lambda}(V)$where$\lambda$is a partition of$n$with$\leq 3$rows. Presumably the multiplicities will be given by modular form spaces. The$\dim V=1$computation is picking up the partition$(n)$. Edit (7/25/2012): There was a mistake in condition 4 as stated, which I have now corrected. Here are some basis elements, as per Kevin Buzzard's suggestion. These are easy enough to generate. I can post more if necessary. Two variable case ($V=\mathbb R$) 1.$42 x^5 y^5-25 \left(y^3 x^7+y^7 x^3\right)+4 \left(y x^9+y^9