gh-39403: conversion of symmetric functions to and from magma
    
as this can be useful

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.
    
URL: #39403
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/combinat/sf/elementary.py

@@ -98,6 +98,24 @@ def P(i):
         T = self.tensor_square()
         return T.sum_of_monomials( (P(j), P(i-j)) for j in range(i+1) )
 
+    def _magma_init_(self, magma):
+        """
+        Used in converting this ring to the corresponding ring in MAGMA.
+
+        EXAMPLES::
+
+            sage: # optional - magma
+            sage: E = SymmetricFunctions(QQ).e()
+            sage: t = 4*E[3,2]+9
+            sage: mt = magma(t); mt
+            9 + 4*$.[3,2]
+            sage: mt.sage()
+            9*e[] + 4*e[3, 2]
+        """
+        B = magma(self.base_ring())
+        Bref = B._ref()
+        return f"SymmetricFunctionAlgebraElementary({Bref})"
+
     class Element(classical.SymmetricFunctionAlgebra_classical.Element):
         def omega(self):
             r"""

src/sage/combinat/sf/homogeneous.py

@@ -125,6 +125,24 @@ def P(i):
         T = self.tensor_square()
         return T.sum_of_monomials( (P(j), P(i-j)) for j in range(i+1) )
 
+    def _magma_init_(self, magma):
+        """
+        Used in converting this ring to the corresponding ring in MAGMA.
+
+        EXAMPLES::
+
+            sage: # optional - magma
+            sage: H = SymmetricFunctions(QQ).h()
+            sage: t = 4*H[3,2]+9
+            sage: mt = magma(t); mt
+            9 + 4*$.[3,2]
+            sage: mt.sage()
+            9*h[] + 4*h[3, 2]
+        """
+        B = magma(self.base_ring())
+        Bref = B._ref()
+        return f"SymmetricFunctionAlgebraHomogeneous({Bref})"
+
     class Element(classical.SymmetricFunctionAlgebra_classical.Element):
         def omega(self):
             r"""

src/sage/combinat/sf/monomial.py

@@ -276,6 +276,24 @@ def antipode_by_coercion(self, element):
         s = self.realization_of().schur()
         return self(s.antipode(s(element)))
 
+    def _magma_init_(self, magma):
+        """
+        Used in converting this ring to the corresponding ring in MAGMA.
+
+        EXAMPLES::
+
+            sage: # optional - magma
+            sage: M = SymmetricFunctions(QQ).m()
+            sage: t = 4*M[3,2]+9
+            sage: mt = magma(t); mt
+            9 + 4*$.[3,2]
+            sage: mt.sage()
+            9*m[] + 4*m[3, 2]
+        """
+        B = magma(self.base_ring())
+        Bref = B._ref()
+        return f"SymmetricFunctionAlgebraMonomial({Bref})"
+
     class Element(classical.SymmetricFunctionAlgebra_classical.Element):
         def expand(self, n, alphabet='x'):
             """

src/sage/combinat/sf/powersum.py

@@ -216,6 +216,24 @@ def eval_at_permutation_roots_on_generators(self, k, rho):
         """
         return self.base_ring().sum(d*list(rho).count(d) for d in divisors(k))
 
+    def _magma_init_(self, magma):
+        """
+        Used in converting this ring to the corresponding ring in MAGMA.
+
+        EXAMPLES::
+
+            sage: # optional - magma
+            sage: P = SymmetricFunctions(QQ).p()
+            sage: t = 4*P[3,2]+9
+            sage: mt = magma(t); mt
+            9 + 4*$.[3,2]
+            sage: mt.sage()
+            9*p[] + 4*p[3, 2]
+        """
+        B = magma(self.base_ring())
+        Bref = B._ref()
+        return f"SymmetricFunctionAlgebraPower({Bref})"
+
     class Element(classical.SymmetricFunctionAlgebra_classical.Element):
         def omega(self):
             r"""

src/sage/combinat/sf/schur.py

@@ -224,6 +224,24 @@ def _repeated_bernstein_creation_operator_on_basis(self, la, nu):
             return (-1)**m * self([a+b for (a,b) in zip(ga, range(-r,0))])
         return self.zero()
 
+    def _magma_init_(self, magma):
+        """
+        Used in converting this ring to the corresponding ring in MAGMA.
+
+        EXAMPLES::
+
+            sage: # optional - magma
+            sage: S = SymmetricFunctions(QQ).s()
+            sage: t = 4*S[3,2]+9
+            sage: mt = magma(t); mt
+            9 + 4*$.[3,2]
+            sage: mt.sage()
+            9*s[] + 4*s[3, 2]
+        """
+        B = magma(self.base_ring())
+        Bref = B._ref()
+        return f"SymmetricFunctionAlgebraSchur({Bref})"
+
     class Element(classical.SymmetricFunctionAlgebra_classical.Element):
         def __pow__(self, n):
             """

src/sage/ext_data/magma/sage/basic.m

@@ -245,6 +245,33 @@ intrinsic Sage(I::RngOrdIdl) -> MonStgElt, BoolElt
 return Sprintf("%o.ideal(%o)", Sage(K),Sage(seq)), true;
 end intrinsic;
 
+/* Symmetric functions */
+
+intrinsic Sage(X::AlgSym) -> MonStgElt, BoolElt
+{}
+if HasSchurBasis(X) then
+  return Sprintf("SymmetricFunctions(%o).s()", Sage(BaseRing(X))), false;
+elif HasHomogeneousBasis(X) then
+  return Sprintf("SymmetricFunctions(%o).h()", Sage(BaseRing(X))), false;
+elif HasElementaryBasis(X) then
+  return Sprintf("SymmetricFunctions(%o).e()", Sage(BaseRing(X))), false;
+elif HasPowerSumBasis(X) then
+  return Sprintf("SymmetricFunctions(%o).p()", Sage(BaseRing(X))), false;
+elif HasMonomialBasis(X) then
+  return Sprintf("SymmetricFunctions(%o).m()", Sage(BaseRing(X))), false;
+end if;
+end intrinsic;
+
+intrinsic Sage(X::AlgSymElt) -> MonStgElt, BoolElt
+{}
+PA := Parent(X);
+SF := Sage(PA);
+BR := Sage(BaseRing(PA));
+parts, coeffs := Support(X);
+dict := (&* [ Sprintf("Partition(%o):%o(%o),", Sage(parts[i]), BR, Sage(coeffs[i])) : i in [1..#parts] ]);
+return Sprintf("%o._from_dict({%o})", SF, dict), false;
+end intrinsic;
+
 /* Elliptic curves */
 
 intrinsic Sage(X::CrvEll) -> MonStgElt, BoolElt

src/sage/modules/with_basis/indexed_element.pyx

@@ -1020,6 +1020,16 @@ cdef class IndexedFreeModuleElement(ModuleElement):
         x_inv = B(x) ** -1
         return type(self)(F, scal(x_inv, D))
 
+    def _magma_init_(self, magma):
+        r"""
+        Convert ``self`` to Magma.
+        """
+        # Get a reference to Magma version of parent.
+        R = magma(self.parent()).name()
+        # use dict {key: coefficient}.
+        return '+'.join(f"({c._magma_init_(magma)})*{R}.{m._magma_init_(magma)}"
+                        for m, c in self.monomial_coefficients().items())
+
 
 def _unpickle_element(C, d):
     """