# Determine the 2x4 window representation of the product of two Weyl
# tensors. Tests @asym, @acanonicalorder and also
# @riemann_index_regroup.

# The \tableau{2 2 2 2} rep of W^2:

W_{m n p q}::WeylTensor.

WW:=      W_{r1 r2 s1 s2} W_{r3 r4 s3 s4};
WWmanip:= (5/9)*@(WW);

@young_project!(%){2,2,2,2}{0,2,1,3,4,6,5,7}:
# @sym!(%){_{r1},_{s1}}:
# @sym!(%){_{r2},_{s2}}:
# @sym!(%){_{r3},_{s3}}:
# @sym!(%){_{r4},_{s4}}:
# 
# @asym!(%){_{r1},_{r2},_{r3},_{r4}}:
# @asym!(%){_{s1},_{s2},_{s3},_{s4}}:

@acanonicalorder!(%){_{r1},_{r2},_{r3},_{r4}}:
@acanonicalorder!(%){_{s1},_{s2},_{s3},_{s4}}:
@indexsort!(%):
@collect_terms!(%);

@riemann_index_regroup!(%){r1,r2,r3,r4}{s1,s3,s3,s4};
@collect_terms!(%);

tst:= @(WW)-@(WWmanip);
@collect_terms!(%);
@assert(tst);

