MeatAxe: precond - Precondensation of Tensor Products"

MeatAxe 2.4

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precond - Precondensation of Tensor Products"

## Command Line

    precond Options Info M N

Options
  Standard options, see Standard Command Line Options

Info
  Name of the tensor condensation data file.

M
  Name of first module (left factor).

N
  Name of second module (right factor).

## Input Files

M.cfinfo, N.cfinfo
  Constituent information.

MCf.std.1, MCf.std.2, ..., NCf.std.1, NCf.std.2, ...
  Standard generators of the condensation subgroup H for each constituent.

## Output Files

Info.tki
  Tensor condensation info file.

Info.q.1, Info.q.2, ...
  Embeddings for each constituent.

Info.p.1, Info.p.2, ...
  Projections for each constituent.

## Description

This program performs two tasks:

- It compares the irreducible constituents of MH and NH, and finds all pairs (Si,Tj) of constituents where Si≅Tj⋆.
- For each pair (S,T) of constituents found in step 1, the program calculates the embedding of (S⊗T)eH into S⊗T as an direct summand, and the corresponding projection of S⊗T onto (S⊗T)eH. If there is no peak word for a constituent, precond will issue a warning but continue. However, the P and Q matrices for this constituent are zero.

## Implementation Details

Step 1, matching of constituents, is implemented in the same way as in chop and cfcomp, i.e., by using the standard basis with respect to identifying words. Step 2 is based on two observations:

- (A): V⊗V*≅Homk(V,V)$, and (S⊗T)e_H≅EndkH(V) as $kH$-Modules.
- (B): There is a natural, H-invariant non-degenerate scalar product on Homk(V,V), given by Γ(φ,ψ)=Trace(φ∘ψ).

From (A) it is clear that calculating the embedding of (S⊗T)eH into S⊗T is equivalent to computing a basis of EndkH(V). The latter is easily accomplished using the peak word of V. As a consequence of the second observation, there is a natural one-to-one correspondence between H-invariant linear forms on Hom_k(V,V) and EndkH(V), which is used to calculate the projection from Homk(V,V) on EndkH(V).

More details on the algorithm used in Step 2 can be found in [Ri98].