From e2d2ea8265cfb915e55a20e89ea731eed2f16b11 Mon Sep 17 00:00:00 2001
From: Matthias Koeppe <mkoeppe@math.ucdavis.edu>
Date: Sat, 27 Aug 2022 10:59:51 -0700
Subject: [PATCH] src/sage/tensor/modules/finite_rank_free_module.py:
 isomorphism_with_fixed_basis with codomain=symmetric matrices

---
 .../tensor/modules/finite_rank_free_module.py | 32 +++++++++++++++++++
 1 file changed, 32 insertions(+)

diff --git a/src/sage/tensor/modules/finite_rank_free_module.py b/src/sage/tensor/modules/finite_rank_free_module.py
index 9da331cf3b3..2a84cee63a9 100644
--- a/src/sage/tensor/modules/finite_rank_free_module.py
+++ b/src/sage/tensor/modules/finite_rank_free_module.py
@@ -846,6 +846,36 @@ def isomorphism_with_fixed_basis(self, basis=None, codomain=None):
             [2 4 5]
             [3 5 6]
 
+        Same but explicitly in the subspace of symmetric bilinear forms::
+
+            sage: Sym2Vdual = V.dual_symmetric_power(2); Sym2Vdual
+            Free module of type-(0,2) tensors
+             with Fully symmetric 2-indices components w.r.t. (1, 2, 3)
+             on the 3-dimensional vector space over the Rational Field
+            sage: Sym2Vdual.is_submodule(T02)
+            True
+            sage: Sym2Vdual.rank()
+            6
+            sage: e_Sym2Vdual = Sym2Vdual.basis("e"); e_Sym2Vdual
+            Standard basis on the Free module of type-(0,2) tensors
+             with Fully symmetric 2-indices components w.r.t. (1, 2, 3)
+             on the 3-dimensional vector space over the Rational Field
+             induced by Basis (e_1,e_2,e_3) on the 3-dimensional vector space over the Rational Field
+            sage: W_basis = [phi_e_T02(b) for b in e_Sym2Vdual]; W_basis
+            [
+            [1 0 0]  [0 1 0]  [0 0 1]  [0 0 0]  [0 0 0]  [0 0 0]
+            [0 0 0]  [1 0 0]  [0 0 0]  [0 1 0]  [0 0 1]  [0 0 0]
+            [0 0 0], [0 0 0], [1 0 0], [0 0 0], [0 1 0], [0 0 1]
+            ]
+            sage: W = MatrixSpace(QQ, 3).submodule(W_basis); W
+            Free module generated by {0, 1, 2, 3, 4, 5} over Rational Field
+            sage: phi_e_Sym2Vdual = Sym2Vdual.isomorphism_with_fixed_basis(e_Sym2Vdual, codomain=W); phi_e_Sym2Vdual
+            Generic morphism:
+            From: Free module of type-(0,2) tensors
+             with Fully symmetric 2-indices components w.r.t. (1, 2, 3)
+             on the 3-dimensional vector space over the Rational Field
+            To:   Free module generated by {0, 1, 2, 3, 4, 5} over Rational Field
+
         Sending tensors to elements of the tensor square of :class:`CombinatorialFreeModule`::
 
             sage: T20 = V.tensor_module(2, 0); T20
@@ -909,6 +939,8 @@ def isomorphism_with_fixed_basis(self, basis=None, codomain=None):
         else:
             # assume that the keys of the codomain should be used
             key_pairs = zip(codomain_basis.keys(), basis.keys())
+        # Need them several times, can't keep as generators
+        key_pairs = tuple(key_pairs)
 
         def _isomorphism(x):
             r"""