dft for symmetric group algebra when p|n!

src/sage/combinat/symmetric_group_algebra.py

@@ -1909,12 +1909,14 @@
         """
         if form == "seminormal":
             return self._dft_seminormal(mult=mult)
+        if form == "modular":
+            return self._dft_modular()
         else:
             raise ValueError("invalid form (= %s)" % form)
 
     def _dft_seminormal(self, mult='l2r'):
         """
-        Return the seminormal form of the discrete Fourier for ``self``.
+        Return the seminormal form of the discrete Fourier transform for ``self``.
 
         INPUT:
 
@@ -1941,6 +1943,42 @@
         snb = self.seminormal_basis(mult=mult)
         return matrix([vector(b) for b in snb]).inverse().transpose()
 
+    def _dft_modular(self):
+        """
+        Return the discrete Foruier transform when the characterisc divides the order of the group.
+        The usual .dft() throws ZeroDivisionError when p|n.
+
+        EXAMPLES::
+
+            sage: GF2S3 = SymmetricGroupAlgebra(GF(2),3)
+            sage: GF2S3._dft_modular()
+            [1 0 0 0 1 0]
+            [0 1 0 0 0 1]
+            [0 0 1 0 0 1]
+            [0 0 0 1 1 0]
+            [1 0 0 1 1 0]
+            [0 1 1 0 0 1]
+        """
+        #create a spanning set for the block corresponding to an idempotent
+        spanning_set = lambda idem: [self(sigma)*idem for sigma in self.group()]
+        #compute central primitive orthogonal idempotents
+        idempotents = self.central_orthogonal_idempotents()
+        #project v onto each block U_i = F_p[S_n]*e_i via \pi_i: v |--> v*e_i
+        blocks = [self.submodule(spanning_set(idem)) for idem in idempotents]
+        #helper function to flatten a list
+        flatten = lambda l: [item for sublist in l for item in sublist]
+        #compute the list of basis vectors lifed to the SGA from each block
+        block_decomposition_basis = flatten([[u.lift() for u in block.basis()] for block in blocks])
+        #the elements of the symmetric group are ordered, giving the map from the standard basis
+        sym_group_list = list(self.group())
+        change_of_basis_matrix = []
+        for b in block_decomposition_basis:
+            coord_vector = [0]*len(sym_group_list)
+            for pair in list(b):
+                coord_vector[sym_group_list.index(pair[0])] = pair[1]
+            change_of_basis_matrix.append(coord_vector)
+        return matrix(self.base_ring(),change_of_basis_matrix).transpose()
+
     def epsilon_ik(self, itab, ktab, star=0, mult='l2r'):
         r"""
         Return the seminormal basis element of ``self`` corresponding to the