# precond man page

precond — precondensation of tensor products

## Synopsis

**precond** [Options] *<Info> <M> <N>*

## 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 a 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.

## Options

**-Q**- Quiet, no messages.
**-V**- Verbose, more messages.
**-T***<MaxTime>*- Set CPU time limit

## Implementation Details

Step 1, matching of constituents, is implemented in the same way as in chop(1) and cfcomp(1), 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(phi,phi)=Trace(phi∘phi).

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in Step 2 can be found in M. Ringe, "Bemerkungen zur Kondensation von Tensorprodukten irreduzibler Moduln", Nov. 1998, unpublished.

## Input Files

*M*.cfinfo- Info file for
*M*, produced by chop(1)/pwkond(1). *N*.cfinfo- Info file for
*N*, produced by chop(1)/pwkond(1). *M<Cf>*.std.{1,2,...}- Standard generators for each constituent.
*N<Cf>*.std.{1,2,...}- Standard generators for each constituent.

## Output Files

*Info*.tki- Tensor condensation info file.
*Info*.q.{1,2,...}- Embeddings for each constituent.
*Info*.p.{1,2,...}- Projections for each constituent.