gh-40743: add type annotations "-> str" in algebras/
    
very boring ticket, adding "-> str" annotations in many places in the
`algebras` folder

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #40743
Reported by: Frédéric Chapoton
Reviewer(s):

Author: Release Manager

src/sage/algebras/affine_nil_temperley_lieb.py

@@ -113,7 +113,7 @@ def one_basis(self):
         """
         return self.weyl_group().one()
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         EXAMPLES::
 
@@ -242,7 +242,7 @@ def has_no_braid_relation(self, w, i) -> bool:
                 return j not in wi.descents()
         return self.has_no_braid_relation(w * s[w.first_descent()], i)
 
-    def _repr_term(self, t, short_display=True):
+    def _repr_term(self, t, short_display=True) -> str:
         """
         EXAMPLES::
 

src/sage/algebras/askey_wilson.py

@@ -311,7 +311,7 @@ def exp(l, e):
         var_names = ['A', 'B', 'C', '\\alpha', '\\beta', '\\gamma']
         return ''.join(exp(l, e) for l, e in zip(var_names, t))
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of ``self``.
 

src/sage/algebras/associated_graded.py

@@ -196,7 +196,7 @@ def __init__(self, A, category=None):
         phi = self.module_morphism(diagonal=lambda x: base_one, codomain=A)
         self._A.register_conversion(phi)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return a string representation of ``self``.
 
@@ -213,7 +213,7 @@ def _repr_(self):
             return "Graded Algebra of {}".format(self._A)
         return "Graded Module of {}".format(self._A)
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return a latex representation of ``self``.
 

src/sage/algebras/cellular_basis.py

@@ -207,7 +207,7 @@ def __init__(self, A, to_algebra=None, from_algebra=None, **kwargs):
         to_cellular.register_as_coercion()
         from_cellular.register_as_coercion()
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         String representation of ``self``.
 
@@ -219,7 +219,7 @@ def _repr_(self):
         """
         return "Cellular basis of {}".format(self._algebra)
 
-    def _latex_term(self, x):
+    def _latex_term(self, x) -> str:
         r"""
         Return a latex representation of the term indexed by ``x``.
 

src/sage/algebras/clifford_algebra.py

@@ -167,7 +167,7 @@ def cardinality(self):
 
     __len__ = cardinality
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of ``self``.
 
@@ -197,7 +197,7 @@ def _repr_(self):
             return "Subsets of {0,1}" + extra
         return f"Subsets of {{0,1,...,{self._nbits-1}}}" + extra
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return a latex representation of ``self``.
 
@@ -552,7 +552,7 @@ def __init__(self, Q, names, category=None):
                                          sorting_key=tuple)
         self._assign_names(names)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of ``self``.
 
@@ -568,7 +568,7 @@ def _repr_(self):
         """
         return "The Clifford algebra of the {}".format(self._quadratic_form)
 
-    def _repr_term(self, m):
+    def _repr_term(self, m) -> str:
         """
         Return a string representation of the basis element indexed by ``m``.
 
@@ -594,7 +594,7 @@ def _repr_term(self, m):
             term += self.variable_names()[i]
         return term
 
-    def _latex_term(self, m):
+    def _latex_term(self, m) -> str:
         r"""
         Return a `\LaTeX` representation of the basis element indexed
         by ``m``.
@@ -1480,7 +1480,7 @@ def __init__(self, R, names):
         cat = HopfAlgebrasWithBasis(R).FiniteDimensional().Supercommutative().Supercocommutative()
         CliffordAlgebra.__init__(self, QuadraticForm(R, len(names)), names, category=cat)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of ``self``.
 
@@ -1491,7 +1491,7 @@ def _repr_(self):
         """
         return "The exterior algebra of rank {} over {}".format(self.ngens(), self.base_ring())
 
-    def _repr_term(self, m):
+    def _repr_term(self, m) -> str:
         """
         Return a string representation of the basis element indexed by
         ``m``.
@@ -1547,7 +1547,7 @@ def _unicode_art_term(self, m):
         wedge = unicodedata.lookup('LOGICAL AND')
         return unicode_art(*[self.variable_names()[i] for i in m], sep=wedge)
 
-    def _latex_term(self, m):
+    def _latex_term(self, m) -> str:
         r"""
         Return a `\LaTeX` representation of the basis element indexed
         by ``m``.
@@ -2295,7 +2295,7 @@ class ExteriorAlgebraBoundary(ExteriorAlgebraDifferential):
 
     - :wikipedia:`Exterior_algebra#Lie_algebra_homology`
     """
-    def _repr_type(self):
+    def _repr_type(self) -> str:
         """
         TESTS::
 
@@ -2566,7 +2566,7 @@ def __init__(self, E, s_coeff):
                 self._cos_coeff[m] = self._cos_coeff.get(m, zero) + c * k
         ExteriorAlgebraDifferential.__init__(self, E, s_coeff)
 
-    def _repr_type(self):
+    def _repr_type(self) -> str:
         """
         TESTS::
 

src/sage/algebras/clifford_algebra_element.pyx

@@ -36,7 +36,7 @@ cdef class CliffordAlgebraElement(IndexedFreeModuleElement):
         sage: elt = ((x^3-z)*x + y)^2
         sage: TestSuite(elt).run()
     """
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return a string representation of ``self``.
 

src/sage/algebras/commutative_dga.py

@@ -321,7 +321,7 @@ def _call_(self, x):
                 idx += 1
         return res
 
-    def _repr_defn(self):
+    def _repr_defn(self) -> str:
         r"""
         Return a string showing where ``self`` sends each generator.
 
@@ -338,7 +338,7 @@ def _repr_defn(self):
         """
         return '\n'.join(f"{i} --> {self(i)}" for i in self.domain().gens())
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of ``self``.
 
@@ -1051,7 +1051,7 @@ def __init__(self, base, R=None, I=None, names=None, degrees=None, category=None
         category = Algebras(R.base_ring()).Graded().or_subcategory(category)
         QuotientRing_nc.__init__(self, R, I, names, category=category)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Print representation.
 
@@ -1787,7 +1787,7 @@ def __init__(self, base, degrees, names=None, R=None, I=None, category=None):
         self._degrees_multi = degrees
         self._grading_rank = len(list(degrees[0]))
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Print representation.
 
@@ -2198,7 +2198,7 @@ def _base_repr(self):
         """
         return GCAlgebra._repr_(self).replace('Graded Commutative', 'Commutative Differential Graded')
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         EXAMPLES::
 
@@ -3911,7 +3911,7 @@ def is_graded(self, total=False):
                        and x.degree(total=total) == y.degree(total=total))
                    for (x, y) in zip(self.domain().gens(), self.im_gens()))
 
-    def _repr_type(self):
+    def _repr_type(self) -> str:
         """
         EXAMPLES::
 
@@ -3929,7 +3929,7 @@ def _repr_type(self):
             return "Commutative Differential Graded Algebra"
         return "Graded Commutative Algebra"
 
-    def _repr_defn(self):
+    def _repr_defn(self) -> str:
         """
         EXAMPLES::
 
@@ -4130,7 +4130,7 @@ def __hash__(self):
         """
         return hash(self._x)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         EXAMPLES::
 
@@ -4140,7 +4140,7 @@ def _repr_(self):
         """
         return '[{}]'.format(self._x)
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         EXAMPLES::
 

src/sage/algebras/down_up_algebra.py

@@ -215,7 +215,7 @@ def __init__(self, alpha, beta, gamma, base_ring):
         CombinatorialFreeModule.__init__(self, base_ring, indices, category=cat, sorting_reverse=True)
         self._assign_names(['d', 'u'])
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of ``self``.
 
@@ -228,7 +228,7 @@ def _repr_(self):
         return "Down-Up algebra with parameters ({}, {}, {}) over {}".format(
             self._alpha, self._beta, self._gamma, self.base_ring())
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return a latex representation of ``self``.
 
@@ -241,7 +241,7 @@ def _latex_(self):
         """
         return "\\mathcal{DU}(%s,%s,%s)" % (self._alpha, self._beta, self._gamma)
 
-    def _repr_term(self, m):
+    def _repr_term(self, m) -> str:
         r"""
         Return a string representation of the basis element indexed by ``m``.
 
@@ -273,7 +273,7 @@ def _repr_term(self, m):
                 ret += f"{s}^{m[i]}"
         return ret
 
-    def _latex_term(self, m):
+    def _latex_term(self, m) -> str:
         r"""
         Return a latex representation for the basis element indexed by ``m``.
 
@@ -622,7 +622,7 @@ def _la_iter():
         CombinatorialFreeModule.__init__(self, R, NonNegativeIntegers(),
                                          prefix='v', category=cat)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of ``self``.
 
@@ -634,7 +634,7 @@ def _repr_(self):
         """
         return f"Verma module of weight {self._weights[0]} of {self._DU}"
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return a latex representation of ``self``.
 

src/sage/algebras/finite_dimensional_algebras/finite_dimensional_algebra.py

@@ -266,7 +266,7 @@ def __init__(self, k, table, names='e', category=None):
         # No further validity checks necessary!
         Parent.__init__(self, base=k, names=names, category=category)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return a string representation of ``self``.
 

src/sage/algebras/finite_dimensional_algebras/finite_dimensional_algebra_element.pyx

@@ -283,7 +283,7 @@ cdef class FiniteDimensionalAlgebraElement(AlgebraElement):
         return sum([self._vector[0, i] * A.left_table()[i] for
                     i in range(A.degree())])
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return the string representation of ``self``.