End VChar.
Section Isometries.
Variables (gT : finGroupType) (L G : {group gT}) (S : seq 'CF(L)).
Implicit Type nu : {additive 'CF(L) -> 'CF(G)}.

(* Specifications: *)
Lemma Zisometry_of_cfnorm (tauS : seq 'CF(G)) :
pairwise_orthogonal S -> pairwise_orthogonal tauS ->
map cfnorm tauS = map cfnorm S -> {subset tauS <= 'Z[irr G]} ->
{tau : {linear 'CF(L) -> 'CF(G)} | map tau S = tauS
& {in 'Z[S], isometry tau, to 'Z[irr G]}}.
Lemma Zisometry_of_iso f :
free S -> {in S, isometry f, to 'Z[irr G]} ->
{tau : {linear 'CF(L) -> 'CF(G)} | {in S, tau =1 f}
& {in 'Z[S], isometry tau, to 'Z[irr G]}}.
Lemma Zisometry_inj A nu :
{in 'Z[S, A] &, isometry nu} -> {in 'Z[S, A] &, injective nu}.