Composition Factors

Computes the composition factors of the n-th tensor power of the free associative algebra in terms of coefficients cf[lambda_][mu] indexing the terms in the irreducible decomposition.

This file implements:

• A fast algorithm computing the coefficients cf[lambda_][mu] for partitions lambda_, mu.
• A data-structure Lie capturing the representation theory of the free Lie algebra.
• Visualisations of the composition factors and certain stability phenomena.

There is a white paper describing this algorithm on the arXiv.

Examples

Compute all coefficients for partitions of size up n with cf = CompositionFactors(n).

Given two partitions lambda_ and mu, get at the coefficient cf[lambda_][mu] as follows.

sage: cf = CompositionFactors(6) # computes all coefficients of degree <= 6
sage: lambda_ = Partition([4,2])
sage: mu = Partition([1,1,1])
sage: cf[lambda_][mu]
2

You can see the all coefficients up a chosen degree in with the display method.

sage: cf = CompositionFactors(7)
sage: cf.display()

We also provide a Visualisations class to investigate new stability phenomena. Here we investigate PD-module stability among the coefficients. Concretely, this is the stability that occurs when you add one box to the first row in each partition lambda_ and mu. The method PD_stability plots how the coefficients evolve under this stability.

sage: cf = CompositionFactors(10)
sage: vis = Visualisations(cf)
sage: vis.PD_stability()