Computing graded Frobenius characteristics in SageMath

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 at stgriffin(at)ucdavis [dot] edu.

• README.txt

• grFrob_single.py

• grFrob_double.py

Symmetric function data for R_{n,λ,s}:

Some of my recent work is on a new family of symmetric functions that simultaneously generalize (modified) Hall-Littlewood polynomials and the symmetric function in the Delta Conjecture at t=0. By joint work with Levinson and Woo, these symmetric functions also encode a geometric representation generalizing the symmetric group action on the cohomology ring of a Springer fiber. You can find a link to the Schur expansions of these symmetric functions below:

Schur-expansion data for R_{n,\lambda,s}