sagemathgh-36980: Specht modules for Ariki-Koike algebras

    
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We provide an implementation of Specht modules for the Ariki-Koike
algebra $H_{r,n}(q,u)$ over a field (more generally, a commutative ring
with $q^k u_i - u_j$ for all applicable $i, j,k$ are invertible). To
help ease the construction, we add an additional optional argument to
the AK algebra constructor to have the base ring convert to the fraction
field.

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URL: sagemath#36980
Reported by: Travis Scrimshaw
Reviewer(s): Matthias Köppe, Travis Scrimshaw

src/doc/en/reference/algebras/index.rst

@@ -79,6 +79,7 @@ Hecke algebras
    :maxdepth: 1
 
    sage/algebras/hecke_algebras/ariki_koike_algebra
+   sage/algebras/hecke_algebras/ariki_koike_specht_modules
    sage/algebras/iwahori_hecke_algebra
    sage/algebras/nil_coxeter_algebra
    sage/algebras/yokonuma_hecke_algebra

src/doc/en/reference/references/index.rst

@@ -2315,6 +2315,9 @@ REFERENCES:
 .. [DPVAR2000] \J. Daemen, M. Peeters, G. Van Assche, and V. Rijmen,
                *Nessie proposal: NOEKEON*; in First Open NESSIE Workshop, (2000).
 
+.. [DR2001] \J. Du and H. Rui. *Specht modules for Ariki–Koike algebras*,
+            Comm. Alg., **29** (2001), 4710-4719.
+
 .. [DR2002] Joan Daemen, Vincent Rijmen. *The Design of
             Rijndael*. Springer-Verlag Berlin Heidelberg, 2002.
 
@@ -4621,13 +4624,20 @@ REFERENCES:
              Providence, RI, 1999. xiv+188 pp. ISBN: 0-8218-1926-7
              :mathscinet:`MR1711316`
 
-.. [Mat2002] Jiří Matousek, "Lectures on Discrete Geometry", Springer,
-             2002
+.. [Mathas2002] Andrew Mathas.
+                *The representation theory of the Ariki-Koike and
+                cyclotomic* `q`-*Schur algebras*.
+                Representation Theory of Algebraic Groups and Quantum Groups,
+                Adv. Stud. Pure Math., vol. 40, Math. Soc. Japan, Tokyo (2004),
+                pp. 261-320. :arxiv:`math/0204025`.
 
 .. [Mathas2004] Andrew Mathas.
                 *Matrix units and generic degrees for the Ariki-Koike algebras*.
                 J. Algebra. **281** (2004) pp. 695-730.
 
+.. [Mat2002] Jiří Matousek, "Lectures on Discrete Geometry", Springer,
+             2002
+
 .. [Mas1995] Mason, Geoffrey. *The quantum double of a finite group and its role
              in conformal field theory*. Groups '93 Galway/St. Andrews, Vol. 2,
              405-417, London Math. Soc. Lecture Note Ser., 212, Cambridge, 1995.

src/sage/algebras/hecke_algebras/ariki_koike_algebra.py

@@ -192,6 +192,8 @@ class ArikiKoikeAlgebra(Parent, UniqueRepresentation):
       default is the generators of `\ZZ[u_1, \ldots, u_r]`
     - ``R`` -- (optional) a commutative ring containing ``q`` and ``u``;
       the default is the parent of `q` and `u_1, \ldots, u_r`
+    - ``use_fraction_field`` -- (default: ``False``) whether to use the
+      fraction field or not
 
     EXAMPLES:
 
@@ -268,7 +270,7 @@ class ArikiKoikeAlgebra(Parent, UniqueRepresentation):
         True
     """
     @staticmethod
-    def __classcall_private__(cls, r, n, q=None, u=None, R=None):
+    def __classcall_private__(cls, r, n, q=None, u=None, R=None, use_fraction_field=False):
         r"""
         Standardize input to ensure a unique representation.
 
@@ -311,11 +313,12 @@ def __classcall_private__(cls, r, n, q=None, u=None, R=None):
                     R = cm.common_parent(q.parent(), *[val.parent() for val in u])
             elif q is None:
                 q = 'q'
-            u = [R(val) for val in u]
         if R not in Rings().Commutative():
             raise TypeError("base ring must be a commutative ring")
+        if use_fraction_field:
+            R = R.fraction_field()
         q = R(q)
-        u = tuple(u)
+        u = tuple([R(val) for val in u])
         return super().__classcall__(cls, r, n, q, u, R)
 
     def __init__(self, r, n, q, u, R):
@@ -414,6 +417,21 @@ def a_realization(self):
         """
         return self.LT()
 
+    def specht_module(self, la):
+        r"""
+        Return the Specht module of ``self`` corresponding to the shape ``la``.
+
+        EXAMPLES::
+
+            sage: AK = algebras.ArikiKoike(4, 6)
+            sage: AK.specht_module([[2], [], [1,1,1], [1]])
+            Specht module of shape ([2], [], [1, 1, 1], [1]) for
+             Ariki-Koike algebra of rank 4 and order 6 with q=q and u=(u0, u1, u2, u3)
+             over ... over Integer Ring
+        """
+        from sage.algebras.hecke_algebras.ariki_koike_specht_modules import SpechtModule
+        return SpechtModule(self, la)
+
     class _BasesCategory(Category_realization_of_parent):
         r"""
         The category of bases of a Ariki-Koike algebra.
@@ -566,6 +584,24 @@ def some_elements(self):
                     elts += [self.L(2)**(self._r//2)]
                 return elts
 
+            def specht_module(self, la):
+                r"""
+                Return the Specht module of ``self`` corresponding
+                to the shape ``la``.
+
+                EXAMPLES::
+
+                    sage: AK = algebras.ArikiKoike(4, 3)
+                    sage: LT = AK.LT()
+                    sage: S1 = LT.specht_module([[1], [], [1,1], []])
+                    sage: T = AK.T()
+                    sage: S2 = T.specht_module([[1], [], [1,1], []])
+                    sage: S1 is S2
+                    True
+                """
+                from sage.algebras.hecke_algebras.ariki_koike_specht_modules import SpechtModule
+                return SpechtModule(self.realization_of(), la)
+
     # -----------------------------------------------------
     # Basis classes
     # -----------------------------------------------------
@@ -1186,6 +1222,18 @@ def __init__(self, algebra):
             _Basis.__init__(self, algebra, prefix='T')
             self._assign_names(['T%s' % i for i in range(self._n)])
 
+        def _basis_to_word(self, t):
+            """
+            Return the basis element indexed by ``m`` to a word.
+            """
+            redword = []
+            for i, k in enumerate(t[0]):
+                if not k:
+                    continue
+                redword.extend(list(range(i, 0, -1)) + [0]*k)
+            redword.extend(t[1].reduced_word())
+            return redword
+
         def _repr_term(self, t):
             r"""
             Return a string representation of the basis element indexed by ``m``.
@@ -1196,13 +1244,8 @@ def _repr_term(self, t):
                 sage: T._repr_term( ((1,0,2), Permutation([3,2,1])) )
                 'T[0,2,1,0,0,2,1,2]'
             """
-            redword = []
-            for i,k in enumerate(t[0]):
-                if k == 0:
-                    continue
-                redword += list(reversed(range(1,i+1))) + [0]*k
-            redword += t[1].reduced_word()
-            if len(redword) == 0:
+            redword = self._basis_to_word(t)
+            if not redword:
                 return "1"
             return (self._print_options['prefix']
                     + '[%s]' % ','.join('%d' % i for i in redword))
@@ -1215,15 +1258,10 @@ def _latex_term(self, t):
 
                 sage: T = algebras.ArikiKoike(4, 3).T()
                 sage: T._latex_term( ((1,0,2), Permutation([3,2,1])) )
-                'T_{0}T_{1}T_{0}T_{0}T_{2}T_{1}T_{2}'
+                'T_{0}T_{2}T_{1}T_{0}T_{0}T_{2}T_{1}T_{2}'
             """
-            redword = []
-            for i,k in enumerate(t[0]):
-                if k == 0:
-                    continue
-                redword += list(reversed(range(1,i))) + [0]*k
-            redword += t[1].reduced_word()
-            if len(redword) == 0:
+            redword = self._basis_to_word(t)
+            if not redword:
                 return "1"
             return ''.join("%s_{%d}" % (self._print_options['prefix'], i)
                            for i in redword)
@@ -1529,7 +1567,7 @@ def product_on_basis(self, m1, m2):
 
             # Compute t1 * T * sprod
             def compute(T, sprod):
-                if not T: # T=1, so just do t1 * sprod, each of which is in order
+                if not T:  # T=1, so just do t1 * sprod, each of which is in order
                     return self._from_dict({(t1, s): sprod[s] for s in sprod},
                                            remove_zeros=False, coerce=False)