Add centralizer examples

examples/centralizer.g

@@ -0,0 +1,56 @@
+#! @BeginChunk Example_CentralizerBlocksOfRepresentation
+#! @BeginExample
+G := DihedralGroup(8);;
+irreps := IrreducibleRepresentations(G);;
+# rho is the sum of two isomorphic degree 1 irreps, and a degree
+# 2 irrep.
+rho := DirectSumOfRepresentations([irreps[4], irreps[4], irreps[5]]);;
+# Compute a basis for the centralizer (in blocks)
+cent_basis_blocks := CentralizerBlocksOfRepresentation(rho);;
+# Verify that the dimension is the sum of the multiplicities squared,
+# in this case 2^2 + 1 = 5.
+Length(cent_basis_blocks) = 5;
+#! true
+#! @EndExample
+#! @EndChunk
+
+#! @BeginChunk Example_CentralizerOfRepresentation
+#! @BeginExample
+# This is the actual basis for the centralizer.
+cent_basis := CentralizerOfRepresentation(rho);;
+# All matrices in the span should commute with the image of rho.
+ForAll(G, g -> ForAll(cent_basis, M -> Image(rho, g)*M = M*Image(rho,g)));
+#! true
+#! @EndExample
+#! @EndChunk
+
+#! @BeginChunk Example_ClassSumCentralizer
+#! @BeginExample
+# Now we have a basis for the centralizer, we can sum a conjugacy class
+# of G.
+class := List(ConjugacyClasses(G)[3]);;
+# We can do the computation naively, with no centralizer basis given:
+sum1 := ClassSumCentralizer(rho, class, fail);;
+# Before summing with th centralizer basis given, we need to
+# orthonormalize it. It's already orthogonal, but not normal:
+orth_basis := OrthonormalBasis@RepnDecomp(cent_basis);;
+IsOrthonormalSet(orth_basis, InnerProduct@RepnDecomp);
+#! true
+# And with the centralizer given, should be more efficient in certain
+# cases (small degree, low multiplicities, but very large group)
+sum2 := ClassSumCentralizer(rho, class, orth_basis);;
+# Should be the same:
+sum1 = sum2;
+#! true
+#! @EndExample
+#! @EndChunk
+
+
+#! @BeginChunk Example_ClassSumCentralizerNC
+#! @BeginExample
+# The very same as the above, but with no checks on orthonormality.
+sum3 := ClassSumCentralizerNC(rho, class, orth_basis);;
+sum1 = sum3;
+#! true
+#! @EndExample
+#! @EndChunk

examples/mymethod.g

@@ -5,8 +5,8 @@ irreps := IrreducibleRepresentations(G);;
 rho := DirectSumOfRepresentations([irreps[4], irreps[5]]);;
 # Jumble rho up a bit so it's not so easy for the library.
 A := [ [ 3, -3, 2, -4, 0, 0 ], [ 4, 0, 1, -5, 1, 0 ], [ -3, -1, -2, 4, -1, -2 ],
-       [ 4, -4, -1, 5, -3, -1 ], [ 3, -2, 1, 0, 0, 0 ], [ 4, 2, 4, -1, -2, 1 ] ];
-rho := ComposeHomFunction(rho, B -> A^-1 * B * A);
+       [ 4, -4, -1, 5, -3, -1 ], [ 3, -2, 1, 0, 0, 0 ], [ 4, 2, 4, -1, -2, 1 ] ];;
+rho := ComposeHomFunction(rho, B -> A^-1 * B * A);;
 # We've constructed rho from two degree 3 irreps, so there are a few
 # things we can check for correctness:
 decomposition := REPN_ComputeUsingMyMethod(rho);;

lib/centralizer.gd

@@ -2,53 +2,63 @@
 
 #! @Section Finding a basis for the centralizer
 
+#! @Description Let $G$ have irreducible representations $\rho_i$ with
+#! multiplicities $m_i$. The centralizer has dimension $\sum_i m_i^2$
+#! as a $\mathbb{C}$-vector space. This function gives the minimal
+#! number of generators required.
+#!
+#! <P/>
+#! @InsertChunk Example_CentralizerBlocksOfRepresentation
+#! <P/>
 #! @Arguments rho
-
 #! @Returns List of vector space generators for the centralizer ring
 #! of $\rho(G)$, written in the basis given by <Ref
 #! Func="BlockDiagonalBasisOfRepresentation" />.  The matrices are
 #! given as a list of blocks.
-
-#! @Description Let $G$ have irreducible representations $\rho_i$ with
-#! multiplicities $m_i$. The centralizer has dimension $\sum_i m_i^2$
-#! as a $\mathbb{C}$-vector space. This function gives the minimal
-#! number of generators required.
 DeclareGlobalFunction( "CentralizerBlocksOfRepresentation" );
 
-#! @Returns List of vector space generators for the centralizer ring
-#! of $\rho(G)$.
-
 #! @Description This gives the same result as <Ref
 #! Func="CentralizerBlocksOfRepresentation" />, but with the matrices
 #! given in their entirety: not as lists of blocks, but as full
 #! matrices.
+#!
+#! <P/>
+#! @InsertChunk Example_CentralizerOfRepresentation
+#! <P/>
+#! @Returns List of vector space generators for the centralizer ring
+#! of $\rho(G)$.
 DeclareGlobalFunction( "CentralizerOfRepresentation" );
 
 #! @Section Using the centralizer for computations
 
-#! @Arguments rho, class, cent_basis
-
-#! @Returns $\sum_{s \in t^G} \rho(s)$, where $t$ is a representative
-#! of the conjugacy class <A>class</A> of $G$.
-
 #! @Description We require that <A>rho</A> is unitary. Uses the given
 #! orthonormal basis (with respect to the inner product $\langle A, B
 #! \rangle = \mbox{Trace}(AB^*)$) for the centralizer ring of
 #! <A>rho</A> to calculate the sum of the conjugacy class <A>class</A>
 #! quickly, i.e. without summing over the class.
-
+#!
 #! NOTE: Orthonormality of <A>cent_basis</A> and unitarity of
 #! <A>rho</A> are checked. See <Ref Func="ClassSumCentralizerNC" />
 #! for a version of this function without checks. The checks are not
 #! very expensive, so it is recommended you use the function with
 #! checks.
-DeclareGlobalFunction( "ClassSumCentralizer" );
-
+#!
+#! <P/>
+#! @InsertChunk Example_ClassSumCentralizer
+#! <P/>
 #! @Arguments rho, class, cent_basis
+#! @Returns $\sum_{s \in t^G} \rho(s)$, where $t$ is a representative
+#! of the conjugacy class <A>class</A> of $G$.
+DeclareGlobalFunction( "ClassSumCentralizer" );
 
 #! @Description The same as <Ref Func="ClassSumCentralizer" />, but
 #! does not check the basis for orthonormality or the representation
 #! for unitarity.
+#!
+#! <P/>
+#! @InsertChunk Example_ClassSumCentralizerNC
+#! <P/>
+#! @Arguments rho, class, cent_basis
 DeclareGlobalFunction( "ClassSumCentralizerNC" );
 
 DeclareGlobalFunction( "SizesToBlocks" );

lib/centralizer.gi

@@ -130,7 +130,7 @@ InstallGlobalFunction( ClassSumCentralizer, function(rho, class, cent_basis)
         Error("<rho> is not unitary!");
     fi;
 
-    if not IsOrthonormalSet(cent_basis, InnerProduct@) then
+    if cent_basis <> fail and not IsOrthonormalSet(cent_basis, InnerProduct@) then
         Error("<cent_basis> is not an orthonormal set!");
     fi;