proposition: Lemma coord_map_homsym (b : #|{:'P_d}|.-tuple HSFR) j (f : HSFR) : basis_of fullv b -> basis_of fullv (map_tuple map_homsym b) -> coord (map_tuple map_homsym b) j (map_homsym f) = mor (coord b j f).
proof: Proof. move=> Hbasis Hmap_basis. have toSf : f \in span b by rewrite (span_basis Hbasis) // memvf. rewrite {1}(coord_span toSf) raddf_sum /=. rewrite (eq_bigr (fun i : 'I_#|{:'P_d}| => (mor (coord b i f)) *: (map_tuple map_homsym b)`_i )). by rewrite coord_sum_free //; apply: (basis_free Hmap_basis). move=> i _; rewrite linearZ /=. congr (_ *: _); apply esym; apply nth_map. by rewrite size_tuple ltn_ord. Qed.
imports: HB(structures) mathcomp(all_ssreflect) mathcomp(ssralg matrix vector ssrnum algC archimedean) mathcomp(fingroup perm) mathcomp(ssrcomplements freeg mpoly) tools sorted ordtype permuted partition permcent antisym Schur_mpoly Schur_altdef sympoly