precond(1)

1precond(1)                       User Commands                      precond(1)
2
3
4

NAME

6       precond - precondensation of tensor products
7

SYNOPSIS

9       precond [OPTIONS] <Info> <M> <N>
10

DESCRIPTION

12       This program performs two tasks:
13
14       · It  compares  the  irreducible constituents of M_H and N_H, and finds
15         all pairs (S_i,T_j) of constituents where S_i≅T_j^*.
16
17       · For each pair (S,T) of constituents found in step 1, the program cal‐
18         culates  the  embedding of (S⊗T)e_H into S⊗T as a direct summand, and
19         the corresponding projection of S⊗T onto (S⊗T)e_H.
20
21       If there is no peak word for a constituent, precond will issue a  warn‐
22       ing  but  continue.  However, the P and Q matrices for this constituent
23       are zero.
24

OPTIONS

26       -Q     Quiet, no messages.
27
28       -V     Verbose, more messages.
29
30       -T <MaxTime>
31              Set CPU time limit
32

IMPLEMENTATION DETAILS

34       Step 1, matching of constituents, is implemented in the same way as  in
35       chop(1)  and  cfcomp(1), i.e., by using the standard basis with respect
36       to identifying words.  Step 2 is based on two observations:
37
38       A   V⊗V^*≅Hom_k(V,V), and (S⊗T)e_H≅End_kH(V) as kH-Modules.
39
40       B   There is a natural, H-invariant non-degenerate  scalar  product  on
41           Hom_k(V,V), given by Γ(phi,phi)=Trace(phi∘phi).
42
43       From  (A)  it  is clear that calculating the embedding of (S⊗T)e_H into
44       S⊗T is equivalent to computing a basis of  End_kH(V).   The  latter  is
45       easily  accomplished using the peak word of V.  As a consequence of the
46       second  observation,  there  is  a  natural  one-to-one  correspondence
47       between  H-invariant linear forms on Hom_k(V,V) and End_kH(V), which is
48       used to calculate the projection from Hom_k(V,V) on End_kH(V).
49
50       More details on the algorithm used in Step 2 can be found in M.  Ringe,
51       "Bemerkungen zur Kondensation von Tensorprodukten irreduzibler Moduln",
52       Nov. 1998, unpublished.
53

INPUT FILES

55       M.cfinfo
56              Info file for M, produced by chop(1)/pwkond(1).
57
58       N.cfinfo
59              Info file for N, produced by chop(1)/pwkond(1).
60
61       M<Cf>.std.{1,2,...}
62              Standard generators for each constituent.
63
64       N<Cf>.std.{1,2,...}
65              Standard generators for each constituent.
66

OUTPUT FILES

68       Info.tki
69              Tensor condensation info file.
70
71       Info.q.{1,2,...}
72              Embeddings for each constituent.
73
74       Info.p.{1,2,...}
75              Projections for each constituent.
76