Code
Code
Much of my research involves finding formulas for symmetric group representations in terms of symmetric functions. Below is some SageMath code that I've used in the past to calculate the graded Frobenius characteristics of quotients of polynomial rings by Sn-invariant ideals. Examples include coinvariant rings, Garsia-Procesi modules, and more general representations coming from diagonal harmonics. The first file is for ideals in a polynomial ring generated by a single set of variables QQ[x1,...,xn], and the second file is for ideals in QQ[x1,...,xn,y1,...,yn] that are invariant under the diagonal action of the symmetric group.
A fair warning to users: The code does not throw any exceptions, so make sure that you input an ideal that is genuinely homogeneous and Sn-invariant. Note that the init(n) function must be run first before using the other functions. Some of the functions only work for finite-dimensional quotients but could easily be modified for infinite-dimensional ones. The code for quotients in the two sets of variables case is quite slow but may help with small examples.
If you find any bugs or have any ideas on how to speed up the computations, please let me know!