MeatAxe  2.4 Programs for working with modular representations
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

• 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].

MeatAxe 2.4 documentation, generated on Wed Jan 7 2015 08:38:36