EquivariantGB is a package for computing in polynomial rings with an infinite number of variables, but with an action of the infinite symmetric group. Alternatively such a ring can be considered as the limit of a family of rings with symmetric action. A representation of such a ring can be created using the method buildERing.
For example consider the ring R = ℚ[xi,yi | i,j∈ℤ≥0], the coordinate ring of 2 by infinite matrices. The infinite symmetric group acts by permuting columns.
i1 : R = buildERing({symbol x,symbol y},{1,1},QQ,4) o1 = R o1 : PolynomialRing |
i2 : vars R o2 = | x_3 x_2 x_1 x_0 y_3 y_2 y_1 y_0 | 1 8 o2 : Matrix R <--- R |
Here the output ring stores only a truncation of the set of variables, with indices from 0 to 3, but this bound will be adjusted as necessary in the computations.
We now consider ideals of R that are closed under the symmetric group action. For example, let I be the set of vanishing equations of the rank 1 matrices. I is generated by all 2 by 2 minors xiyj - xjyi.