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 M_H and N_H, and finds all pairs (S_i,T_j) of constituents where S_i≅T_j^⋆.
For each pair (S,T) of constituents found in step 1, the program calculates the embedding of (S⊗T)e_H into S⊗T as an direct summand, and the corresponding projection of S⊗T onto (S⊗T)e_H. 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^*≅Hom_k(V,V)$, and (S⊗T)e_H≅End_kH(V) as $kH$-Modules.
(B): There is a natural, H-invariant non-degenerate scalar product on Hom_k(V,V), given by Γ(φ,ψ)=Trace(φ∘ψ).

From (A) it is clear that calculating the embedding of (S⊗T)e_H into S⊗T is equivalent to computing a basis of End_kH(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 End_kH(V), which is used to calculate the projection from Hom_k(V,V) on End_kH(V).

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