Equivariant matroid relaxation

Ferroni, Nasr, and Vecchi show a remarkable connection between the Kazhdan-Lusztig polynomial of a matroid and the operation of matroid relaxation. Recent work shows this extends to an equivariant version. This code computes the decomposition into irreducible Mathieu representations of the polynomial corresponding to various Steiner systems.

For example, we can compute the equivariant Kazhdan-Lusztig polynomial of the largest known Steiner system S(5, 8, 24):

sage: load('equivariant-matroid-relaxation.sage')
sage: steiner_system_KL_coeff(d, k, n, 1).values()
[735, 15, 0, 0, 0, 15, 0, 0, 63, 7, 3, 0, -1, 7, 3, 0, 0, 0, 0, 0, 0, 0, -2, 1, -1, -1]
sage: steiner_system_KL_coeff(d,k,n,2).values()
[4830, 6, -5, 1, 1, 54, 0, 0, 110, 2, 6, 0, 0, 6, 2, 0, 0, -2, -2, 0, 0, -1, 1, 0, 0, 0]

We can also decompose it:

sage: steiner_system_KL_coeff(d,k,n,2).decompose()
((1,
  Character of Mathieu group of degree 24 and order 244823040 as a permutation group),
 (1,
  Character of Mathieu group of degree 24 and order 244823040 as a permutation group),
 (1,
  Character of Mathieu group of degree 24 and order 244823040 as a permutation group))

For more information about ClassFunctions in SageMath, see the SageMath documentation.

WARNING

We use SageMath functions which are wrappers around GAP functions. GAP does not provide conjugacy classes in any standard order, nor is it consistent from function-call to function-call. (For more, see GAP's documentation). Thus, values obtained via this code will agree with the forthcoming paper containing these results as a set, but will not neccessarily be listed in the same order.