MeatAxe: The Lattice Programs

MeatAxe 2.4

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The Lattice Programs

These programs are used to investigate the structure of matrix representations over finite fields.

The first step is always to find the irreducible constituents of the representation with chop, and to find the corresponding peak words with pwkond. Here are some examples of what you can do with the programs:

- Calculate the complete submodule lattice of a module (using chop, pwkond, mkcycl, mkinc, mkdotl, mksub, genmod).
- Find out if a given irreducible module occurs as a constituent of a module (cfcomp - Compare Irreducible Constituents).
- Draw the submodule lattice in graphical form (mkgraph).
- Calculate a module's socle and radical series (chop, pwkond, soc, rad).
- Calculate all homomorphisms between two modules, or the endomorphism ring of a module (chop, pwkond, MKHOM).
- Decompose a module into direct summands (chop, pwkond, mkhom, decomp).

## Condensation Programs

These programs are used to condense representations. zkd performs a fixed-point condensation of permutation representations. It can be used after the orbits have been calculated with zmo. zuk uncondenses vectors.

precond and tcond are used to condense tensor products or matrix representations, when the restriction to the condensation subgroup is semisimple. The algorithm assumes that the irreducible constituents of the restriction, and corresponding peak words are known, so you must run chop and pwkond before.