change variable labeling for default grassmann pluecker ideal (Relate…

…d to issue #4018) (#4074)

* change variable labeling for default grassmann pluecker ideal

* Update src/Rings/mpoly-ideals.jl

---------

Co-authored-by: Lars Göttgens <lars.goettgens@rwth-aachen.de>
Co-authored-by: Lars Göttgens <lars.goettgens@gmail.com>

src/Rings/mpoly-ideals.jl

@@ -2063,23 +2063,25 @@ small_generating_set(I::MPolyIdeal{<:MPolyDecRingElem}; algorithm::Symbol=:simpl
 ################################################################################
 #returns Pluecker ideal in ring with standard grading
 function grassmann_pluecker_ideal(subspace_dimension::Int, ambient_dimension::Int)
-  I = convert(MPolyIdeal{QQMPolyRingElem},
-              Polymake.ideal.pluecker_ideal(subspace_dimension, ambient_dimension))
-  base, _ = grade(base_ring(I))
-  o = degrevlex(base)
-
-  return ideal(IdealGens(base,base.(gens(I)), o;
-                   keep_ordering=true,
-                   isReduced=true,
-                   isGB=true))
+  I = convert(MPolyIdeal{QQMPolyRingElem}, 
+	      Polymake.ideal.pluecker_ideal(subspace_dimension, ambient_dimension))
+  base = base_ring(I)
+  base2, x = graded_polynomial_ring(coefficient_ring(base), :x=> sort(subsets(ambient_dimension, subspace_dimension)))
+  h = hom(base, base2, x)
+  o = degrevlex(base2)
+  return ideal(IdealGens(base2, h.(gens(I)), o;
+                keep_ordering=true,
+                isReduced=true,
+                isGB=true))
 end
 
 @doc raw"""
     grassmann_pluecker_ideal([ring::MPolyRing,] subspace_dimension::Int, ambient_dimension::Int)
 
-Given a (possibly graded) ring, an ambient dimension, and a subspace dimension, return the ideal in the ring
+Given a (possibly graded) ring, an ambient dimension $n$ and a subspace dimension $d$, return the ideal in the ring
 generated by the Plücker relations. If the input ring is not graded, return the ideal in the ring with the standard grading.
-If the ring is not specified return the ideal in a multivariate polynomial ring over the rationals with the standard grading.
+If the ring is not specified return the ideal in a multivariate polynomial ring over the rationals with variables indexed by
+elements of ${[n]\choose d}$ with the standard grading.
 
 The Grassmann-Plücker ideal is the homogeneous ideal generated by the relations defined by 
 the Plücker Embedding of the Grassmannian. That is given Gr$(k, n)$ the Moduli 
@@ -2091,7 +2093,7 @@ are given by all $d \times d$ minors of a $d \times n$ matrix. For the algorithm
 ```jldoctest
 julia> grassmann_pluecker_ideal(2, 4)
 Ideal generated by
-  x[1]*x[6] - x[2]*x[5] + x[3]*x[4]
+  x[[1, 2]]*x[[3, 4]] - x[[1, 3]]*x[[2, 4]] + x[[1, 4]]*x[[2, 3]]
 
 julia> R, x = polynomial_ring(residue_ring(ZZ, 7)[1], "x" => (1:2, 1:3))
 (Multivariate polynomial ring in 6 variables over ZZ/(7), zzModMPolyRingElem[x[1, 1] x[1, 2] x[1, 3]; x[2, 1] x[2, 2] x[2, 3]])