gh-37393: Tensor, exterior, and symmetric product of semigroup repres…

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This allows us to take tensor,. exterior, and symmetric products of
semigroup representations.
We also provide the following additional changes/improvements to help
build examples and fix bugs:

- Reflection representation of Coxeter groups.
- Latex output of the identity in Artin groups and permutation groups.
- ascii and unicode art of CFMs that do not have a `one_basis()`.

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### ⌛ Dependencies

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URL: #37393
Reported by: Travis Scrimshaw
Reviewer(s): Frédéric Chapoton, Matthias Köppe, Travis Scrimshaw, trevorkarn

src/sage/algebras/clifford_algebra.py

@@ -51,7 +51,7 @@ class CliffordAlgebraIndices(UniqueRepresentation, Parent):
     A facade parent for the indices of Clifford algebra.
     Users should not create instances of this class directly.
     """
-    def __init__(self, Qdim):
+    def __init__(self, Qdim, degree=None):
         r"""
         Initialize ``self``.
 
@@ -67,9 +67,30 @@ def __init__(self, Qdim):
             1111
             sage: type(i)
             <class 'sage.data_structures.bitset.FrozenBitset'>
+
+            sage: idx = CliffordAlgebraIndices(7, 3)
+            sage: idx._nbits
+            7
+            sage: idx._degree
+            3
+            sage: idx._cardinality
+            35
+
+            sage: idx = CliffordAlgebraIndices(7, 0)
+            sage: idx._nbits
+            7
+            sage: idx._degree
+            0
+            sage: idx._cardinality
+            1
         """
         self._nbits = Qdim
-        self._cardinality = 2 ** Qdim
+        if degree is None:
+            self._cardinality = 2 ** Qdim
+        else:
+            from sage.arith.misc import binomial
+            self._cardinality = binomial(Qdim, degree)
+        self._degree = degree
         # the if statement here is in case Qdim is 0.
         category = FiniteEnumeratedSets().Facade()
         Parent.__init__(self, category=category, facade=True)
@@ -92,10 +113,16 @@ def _element_constructor_(self, x):
             00001
             000001
             0000001
+
+            sage: idx = CliffordAlgebraIndices(0)
+            sage: idx([])
+            0
         """
         if isinstance(x, (list, tuple, set, frozenset)):
             if len(x) > self._nbits:
                 raise ValueError(f"{x=} is too long")
+            if not x:
+                return FrozenBitset()
             return FrozenBitset(x)
 
         if isinstance(x, int):
@@ -129,6 +156,12 @@ def cardinality(self):
             True
             sage: len(idx) == 2^7
             True
+
+            sage: idx = CliffordAlgebraIndices(7, 3)
+            sage: idx.cardinality() == binomial(7, 3)
+            True
+            sage: len(idx) == binomial(7, 3)
+            True
         """
         return self._cardinality
 
@@ -149,14 +182,20 @@ def _repr_(self):
             Subsets of {0}
             sage: CliffordAlgebraIndices(2)
             Subsets of {0,1}
+            sage: CliffordAlgebraIndices(5, 3)
+            Subsets of {0,1,...,4} of size 3
         """
+        if self._degree is not None:
+            extra = f" of size {self._degree}"
+        else:
+            extra = ""
         if self._nbits == 0:
-            return "Subsets of {}"
+            return "Subsets of {}" + extra
         if self._nbits == 1:
-            return "Subsets of {0}"
+            return "Subsets of {0}" + extra
         if self._nbits == 2:
-            return "Subsets of {0,1}"
-        return f"Subsets of {{0,1,...,{self._nbits-1}}}"
+            return "Subsets of {0,1}" + extra
+        return f"Subsets of {{0,1,...,{self._nbits-1}}}" + extra
 
     def _latex_(self):
         r"""
@@ -166,21 +205,27 @@ def _latex_(self):
 
             sage: from sage.algebras.clifford_algebra import CliffordAlgebraIndices
             sage: latex(CliffordAlgebraIndices(7))
-            \mathcal{P}({0,1,\ldots,6})
+            \mathcal{P}(\{0,1,\ldots,6\})
             sage: latex(CliffordAlgebraIndices(0))
             \mathcal{P}(\emptyset)
             sage: latex(CliffordAlgebraIndices(1))
-            \mathcal{P}({0})
+            \mathcal{P}(\{0\})
             sage: latex(CliffordAlgebraIndices(2))
-            \mathcal{P}({0,1})
+            \mathcal{P}(\{0,1\})
+            sage: latex(CliffordAlgebraIndices(2, 1))
+            \mathcal{P}(\{0,1\}, 1)
         """
+        if self._degree is not None:
+            extra = f", {self._degree}"
+        else:
+            extra = ""
         if self._nbits == 0:
-            return "\\mathcal{P}(\\emptyset)"
+            return f"\\mathcal{{P}}(\\emptyset{extra})"
         if self._nbits == 1:
-            return "\\mathcal{P}({0})"
+            return f"\\mathcal{{P}}(\\{{0\\}}{extra})"
         if self._nbits == 2:
-            return "\\mathcal{P}({0,1})"
-        return f"\\mathcal{{P}}({{0,1,\\ldots,{self._nbits-1}}})"
+            return f"\\mathcal{{P}}(\\{{0,1\\}}{extra})"
+        return f"\\mathcal{{P}}(\\{{0,1,\\ldots,{self._nbits-1}\\}}{extra})"
 
     def __iter__(self):
         r"""
@@ -200,9 +245,25 @@ def __iter__(self):
             101
             011
             111
+
+            sage: idx = CliffordAlgebraIndices(5, 3)
+            sage: list(idx)
+            [111, 1101, 11001, 1011, 10101, 10011, 0111, 01101, 01011, 00111]
+
+            sage: idx = CliffordAlgebraIndices(7, 0)
+            sage: list(idx)
+            [0]
         """
         import itertools
         n = self._nbits
+        if self._degree is not None:
+            if self._degree == 0:  # special corner case
+                yield FrozenBitset()
+                return
+            for C in itertools.combinations(range(n), self._degree):
+                yield FrozenBitset(C)
+            return
+
         yield FrozenBitset()
         k = 1
         while k <= n:
@@ -226,11 +287,35 @@ def __contains__(self, elt):
             True
             sage: FrozenBitset('000001') in idx
             False
+
+            sage: idx = CliffordAlgebraIndices(6, 3)
+            sage: FrozenBitset('01011') in idx
+            True
+            sage: FrozenBitset('00011') in idx
+            False
+            sage: int(7) in idx
+            True
+            sage: int(8) in idx
+            False
+
+            sage: idx = CliffordAlgebraIndices(7, 0)
+            sage: FrozenBitset() in idx
+            True
+            sage: FrozenBitset('01') in idx
+            False
+            sage: int(0) in idx
+            True
+            sage: int(5) in idx
+            False
         """
         if isinstance(elt, int):
+            if self._degree is not None and sum(ZZ(elt).bits()) != self._degree:
+                return False
             return elt < self._cardinality and elt >= 0
         if not isinstance(elt, FrozenBitset):
             return False
+        if self._degree is not None and len(elt) != self._degree:
+            return False
         return elt.capacity() <= self._nbits
 
     def _an_element_(self):
@@ -252,14 +337,26 @@ def _an_element_(self):
             sage: idx = CliffordAlgebraIndices(3)
             sage: idx._an_element_()
             11
+            sage: idx = CliffordAlgebraIndices(5, 3)
+            sage: idx._an_element_()
+            111
+            sage: idx = CliffordAlgebraIndices(7, 0)
+            sage: idx._an_element_()
+            0
         """
         if not self._nbits:
             return FrozenBitset()
 
+        if self._degree is not None:
+            if self._degree == 0:  # special corner case
+                return FrozenBitset()
+            return FrozenBitset(range(self._degree))
+
         from sage.combinat.subset import SubsetsSorted
         X = SubsetsSorted(range(self._nbits))
         return FrozenBitset(X.an_element())
 
+
 class CliffordAlgebra(CombinatorialFreeModule):
     r"""
     The Clifford algebra of a quadratic form.

src/sage/categories/coxeter_groups.py

@@ -952,14 +952,52 @@ def sign_representation(self, base_ring=None, side="twosided"):
                 Sign representation of
                  Weyl Group of type ['A', 1, 1] (as a matrix group acting on the root space)
                  over Integer Ring
-
             """
             if base_ring is None:
                 from sage.rings.integer_ring import ZZ
                 base_ring = ZZ
             from sage.modules.with_basis.representation import SignRepresentationCoxeterGroup
             return SignRepresentationCoxeterGroup(self, base_ring)
 
+        def reflection_representation(self, base_ring=None, side="left"):
+            r"""
+            Return the reflection representation of ``self``.
+
+            This is also the canonical faithful representation of a
+            Coxeter group.
+
+            INPUT:
+
+            - ``base_ring`` -- (optional) the base ring; the default is
+              the base ring of :meth:`canonical_representation`
+            - ``side`` -- ignored
+
+            EXAMPLES::
+
+                sage: W = CoxeterGroup(['D', 4])
+                sage: W.reflection_representation()
+                Reflection representation of Finite Coxeter group over
+                 Integer Ring with Coxeter matrix:
+                [1 3 2 2]
+                [3 1 3 3]
+                [2 3 1 2]
+                [2 3 2 1]
+
+                sage: W = CoxeterGroup(['I', 13])
+                sage: W.reflection_representation()
+                Reflection representation of Finite Coxeter group over
+                 Universal Cyclotomic Field with Coxeter matrix:
+                [ 1 13]
+                [13  1]
+
+                sage: W = WeylGroup(["B", 3, 1])
+                sage: W.reflection_representation(QQ)
+                Reflection representation of Weyl Group of type ['B', 3, 1]
+                 (as a matrix group acting on the root space)
+            """
+            from sage.modules.with_basis.representation import ReflectionRepresentation
+            return ReflectionRepresentation(self, base_ring)
+
         def demazure_product(self, Q):
             r"""
             Return the Demazure product of the list ``Q`` in ``self``.
@@ -1169,6 +1207,11 @@ def canonical_representation(self):
             r"""
             Return the canonical faithful representation of ``self``.
 
+            .. SEEALSO::
+
+                To obtain the underlying module with the action, use
+                :meth:`reflection_representation`.
+
             EXAMPLES::
 
                 sage: W = WeylGroup("A3")                                               # needs sage.combinat sage.groups

src/sage/groups/artin.py

@@ -57,7 +57,7 @@ def _latex_(self):
 
         TESTS::
 
-            sage: A = ArtinGroup(['B',3])                                               # needs sage.rings.number_field
+            sage: A = ArtinGroup(['B', 3])                                              # needs sage.rings.number_field
             sage: b = A([1, 2, 3, -1, 2, -3])                                           # needs sage.rings.number_field
             sage: b._latex_()                                                           # needs sage.rings.number_field
             '\\sigma_{1}\\sigma_{2}\\sigma_{3}\\sigma_{1}^{-1}\\sigma_{2}\\sigma_{3}^{-1}'
@@ -66,9 +66,14 @@ def _latex_(self):
             sage: b = B([1, 2, 3, -1, 2, -3])
             sage: b._latex_()
             '\\sigma_{1}\\sigma_{2}\\sigma_{3}\\sigma_{1}^{-1}\\sigma_{2}\\sigma_{3}^{-1}'
+            sage: B.one()._latex_()
+            '1'
         """
+        word = self.Tietze()
+        if not word:
+            return '1'
         return ''.join(r"\sigma_{%s}^{-1}" % (-i) if i < 0 else r"\sigma_{%s}" % i
-                       for i in self.Tietze())
+                       for i in word)
 
     def exponent_sum(self):
         """

src/sage/groups/perm_gps/permgroup_element.pyx

@@ -948,10 +948,15 @@ cdef class PermutationGroupElement(MultiplicativeGroupElement):
             sage: S = SymmetricGroup(['a', 'b'])
             sage: latex(S.gens())
             \left((\text{\texttt{a}},\text{\texttt{b}})\right)
+            sage: latex(S.one())
+            1
         """
+        cycle_tuples = self.cycle_tuples()
+        if not cycle_tuples:
+            return '1'
         from sage.misc.latex import latex
         return "".join(("(" + ",".join(latex(x) for x in cycle) + ")")
-                       for cycle in self.cycle_tuples())
+                       for cycle in cycle_tuples)
 
     def __getitem__(self, i):
         r"""