Mathematica 5.0 for Linux Copyright 1988-2003 Wolfram Research, Inc. -- Terminal graphics initialized -- In[1]:= In[2]:= In[2]:= In[2]:= In[2]:= Starting gapread.m for the graph C4 In[3]:= In[3]:= In[3]:= In[3]:= Starting defineGramSchmidt.m In[4]:= In[4]:= In[4]:= In[4]:= In[5]:= In[5]:= In[5]:= In[5]:= Starting readChars.m allIrrepsToRelevant = {{1}, {1, 2}, {1, 2}, {1, 2}, {1, 2}, {1, 2}, {1, 2}, > {1, 2}, {1, 2, 3, 0}, {1, 2, 0, 3}, {1, 2, 0, 3}, {1, 2, 3, 4}, > {1, 2, 3, 4}, {1, 0, 0, 2, 3}, {1, 0, 0, 2, 3}, > {0, 0, 0, 1, 2, 0, 0, 0, 0, 3}} In[6]:= In[6]:= In[7]:= In[7]:= In[7]:= In[7]:= Starting readBifs.m There are 16 stabilizers = symmetries = isotropy subgroups bifInfoOfStabs = {{}} {{}, {{1, 2}}} {{}, {{1, 2}}} {{}, {{1, 2}}} {{}, {{1, 2}}} {{}, {{1, 2}}} {{}, {{1, 2}}} {{}, {{1, 2}}} {{}, {{3, 2}}, {{2, 2}}} {{}, {{2, 2}}, {{4, 2}}} {{}, {{3, 2}}, {{4, 2}}} {{}, {{4, 2}}, {{5, 2}}, {{8, 2}}} {{}, {{4, 2}}, {{6, 2}}, {{7, 2}}} {{}, {{9, 2}}, {{2, 2}, {5, 2}}} {{}, {{9, 2}}, {{2, 2}, {7, 2}}} {{{15, 2}}, {{14, 2}}, {{10, 2}, {12, 2}}} In[8]:= In[8]:= In[8]:= In[8]:= In[9]:= In[9]:= In[9]:= In[9]:= Starting defineFIC.m The symmetry group of the graph is the stabilizer with label = 14 In other words, autGLabel = 14 The stabilizers of the eigenfunctions used to produce the basis are: oneMaximalStab = {15, 14, 12} FIC 1 is made up of 1 eigenspaces of dimension 1 FIC 2 is made up of 1 eigenspaces of dimension 1 FIC 3 is made up of 1 eigenspaces of dimension 2 Now find one eigenvector within each FIC with each maximal stabilizer. One eigenvector in FIC 1 with symmetry 15 is {-0.5, 0.5, -0.5, 0.5} One eigenvector in FIC 2 with symmetry 14 is {-0.5, -0.5, -0.5, -0.5} One eigenvector in FIC 3 with symmetry 10 is {0.707107, 0., -0.707107, 0.} One eigenvector in FIC 3 with symmetry 12 is {0.5, 0.5, -0.5, -0.5} The FIC Basis is orthonormal, as it should be. FICcopiesDim = j Copies Dim 1 1 1 2 1 1 3 1 2 In[10]:= In[10]:= In[10]:= In[10]:= In[10]:= In[10]:= Beginning writeIrreps.m In[11]:= In[11]:= Beginning writeInfo.m 0 : 0 = stabilizer : stabilizer class {1, 1, 2} = isotypic component dimensions. 1. 0 0 0 0 1. 0 0 0 0 1. 0 0 0 0 1. 1 : 1 = stabilizer : stabilizer class {1, 1, 2} = isotypic component dimensions. 1. 0 0 0 0 1. 0 0 0 0 1. 0 0 0 0 1. 2 : 2 = stabilizer : stabilizer class {1, 1, 2} = isotypic component dimensions. 0 1. 0 0 1. 0 0 0 0 0 1. 0 0 0 0 1. 3 : 3 = stabilizer : stabilizer class {1, 1, 1, 1} = isotypic component dimensions. 0 0 0 1. 0 0 1. 0 0 1. 0 0 1. 0 0 0 4 : 3 = stabilizer : stabilizer class {1, 1, 1, 1} = isotypic component dimensions. 0 0 1. 0 0 0 0 1. 0 1. 0 0 1. 0 0 0 5 : 4 = stabilizer : stabilizer class {1, 2, 1} = isotypic component dimensions. 0 0 0.707107 -0.707107 1. 0 0 0 0 1. 0 0 0 0 0.707107 0.707107 6 : 4 = stabilizer : stabilizer class {1, 2, 1} = isotypic component dimensions. 0 0 0.707107 0.707107 1. 0 0 0 0 1. 0 0 0 0 0.707107 -0.707107 7 : 5 = stabilizer : stabilizer class {2, 1, 1} = isotypic component dimensions. 1. 0 0 0 0 1. 0 0 0 0 0.707107 -0.707107 0 0 0.707107 0.707107 8 : 6 = stabilizer : stabilizer class {2, 2} = isotypic component dimensions. 1. 0 0 0 0 0 1. 0 0 1. 0 0 0 0 0 1. 9 : 6 = stabilizer : stabilizer class {2, 2} = isotypic component dimensions. 1. 0 0 0 0 0 0 1. 0 1. 0 0 0 0 1. 0 10 : 7 = stabilizer : stabilizer class {2, 2} = isotypic component dimensions. 0 1. 0 0 0 0 0 1. 1. 0 0 0 0 0 1. 0 11 : 7 = stabilizer : stabilizer class {2, 2} = isotypic component dimensions. 0 1. 0 0 0 0 1. 0 1. 0 0 0 0 0 0 1. 12 : 8 = stabilizer : stabilizer class {2, 2} = isotypic component dimensions. 0 0 1. 0 0 0 0 1. 1. 0 0 0 0 1. 0 0 13 : 9 = stabilizer : stabilizer class {3, 1} = isotypic component dimensions. 1. 0 0 0 0 1. 0 0 0 0 0.707107 -0.707107 0 0 0.707107 0.707107 14 : 9 = stabilizer : stabilizer class {3, 1} = isotypic component dimensions. 1. 0 0 0 0 1. 0 0 0 0 0.707107 0.707107 0 0 0.707107 -0.707107 15 : 10 = stabilizer : stabilizer class {4} = isotypic component dimensions. 1. 0 0 0 0 1. 0 0 0 0 1. 0 0 0 0 1. isoCompDims = {{4}, {3, 1}, {3, 1}, {2, 2}, {2, 2}, {2, 2}, {2, 2}, {2, 2}, > {2, 1, 1}, {1, 2, 1}, {1, 2, 1}, {1, 1, 1, 1}, {1, 1, 1, 1}, {1, 1, 2}, > {1, 1, 2}, {1, 1, 2}} {{{1., 0, 0, 0}, {0, 1., 0, 0}, {0, 0, 1., 0}, newIsoCompBasis = > {0, 0, 0, 1.}}} {{{1., 0, 0, 0}, {0, 1., 0, 0}, > {0, 0, 0.707107, 0.707107}}, > {{0, 0, 0.707107, -0.707107}}} {{{1., 0, 0, 0}, {0, 1., 0, 0}, > {0, 0, 0.707107, -0.707107}}, > {{0, 0, 0.707107, 0.707107}}} {{{0, 0, 1., 0}, {0, 0, 0, 1.}}, > {{1., 0, 0, 0}, {0, 1., 0, 0}}} {{{0, 1., 0, 0}, {0, 0, 1., 0}}, > {{1., 0, 0, 0}, {0, 0, 0, 1.}}} {{{0, 1., 0, 0}, {0, 0, 0, 1.}}, > {{1., 0, 0, 0}, {0, 0, 1., 0}}} {{{1., 0, 0, 0}, {0, 0, 0, 1.}}, > {{0, 1., 0, 0}, {0, 0, 1., 0}}} {{{1., 0, 0, 0}, {0, 0, 1., 0}}, > {{0, 1., 0, 0}, {0, 0, 0, 1.}}} {{{1., 0, 0, 0}, {0, 1., 0, 0}}, > {{0, 0, 0.707107, -0.707107}}, > {{0, 0, 0.707107, 0.707107}}} {{{0, 0, 0.707107, 0.707107}}, > {{1., 0, 0, 0}, {0, 1., 0, 0}}, > {{0, 0, 0.707107, -0.707107}}} {{{0, 0, 0.707107, -0.707107}}, > {{1., 0, 0, 0}, {0, 1., 0, 0}}, > {{0, 0, 0.707107, 0.707107}}} {{{0, 0, 1., 0}}, {{0, 0, 0, 1.}}, {{0, 1., 0, 0}}, > {{1., 0, 0, 0}}} {{{0, 0, 0, 1.}}, {{0, 0, 1., 0}}, {{0, 1., 0, 0}}, > {{1., 0, 0, 0}}} {{{0, 1., 0, 0}}, {{1., 0, 0, 0}}, > {{0, 0, 1., 0}, {0, 0, 0, 1.}}} {{{1., 0, 0, 0}}, {{0, 1., 0, 0}}, > {{0, 0, 1., 0}, {0, 0, 0, 1.}}} {{{1., 0, 0, 0}}, {{0, 1., 0, 0}}, > {{0, 0, 1., 0}, {0, 0, 0, 1.}}} bifInfoOfStabs = {{}} {{}, {{1, 2}}} {{}, {{1, 2}}} {{}, {{1, 2}}} {{}, {{1, 2}}} {{}, {{1, 2}}} {{}, {{1, 2}}} {{}, {{1, 2}}} {{}, {{3, 2}}, {{2, 2}}} {{}, {{2, 2}}, {{4, 2}}} {{}, {{3, 2}}, {{4, 2}}} {{}, {{4, 2}}, {{5, 2}}, {{8, 2}}} {{}, {{4, 2}}, {{6, 2}}, {{7, 2}}} {{}, {{9, 2}}, {{2, 2}, {5, 2}}} {{}, {{9, 2}}, {{2, 2}, {7, 2}}} {{{15, 2}}, {{14, 2}}, {{10, 2}, {12, 2}}} 0 = stabilizer, 3 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {1, 2, 1, 1} Basis for MaximalRed = 1. 0 0 0 bifComponent 2 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {2, 2, 1, 1} Basis for MaximalRed = 0 1. 0 0 bifComponent 3 2 = dim E, 2 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {4, 2, 1, 1} Basis for MaximalRed = 0 0 1. 0 {maximal, |D|, dimMaximalE, dimMaximalRed} = {6, 2, 1, 1} Basis for MaximalRed = 0 0 0.707107 0.707107 1 = stabilizer, 2 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {7, 2, 1, 1} Basis for MaximalRed = 0 1. 0 0 bifComponent 2 2 = dim E, 2 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {9, 2, 1, 1} Basis for MaximalRed = 0 0 0 1. {maximal, |D|, dimMaximalE, dimMaximalRed} = {14, 2, 1, 1} Basis for MaximalRed = 0 0 0.707107 0.707107 2 = stabilizer, 2 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {7, 2, 1, 1} Basis for MaximalRed = 1. 0 0 0 bifComponent 2 2 = dim E, 2 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {11, 2, 1, 1} Basis for MaximalRed = 0 0 1. 0 {maximal, |D|, dimMaximalE, dimMaximalRed} = {14, 2, 1, 1} Basis for MaximalRed = 0 0 0.707107 0.707107 3 = stabilizer, 3 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {12, 2, 1, 1} Basis for MaximalRed = 0 0 1. 0 bifComponent 2 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {10, 2, 1, 1} Basis for MaximalRed = 0 1. 0 0 bifComponent 3 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {9, 2, 1, 1} Basis for MaximalRed = 1. 0 0 0 4 = stabilizer, 3 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {12, 2, 1, 1} Basis for MaximalRed = 0 0 0 1. bifComponent 2 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {11, 2, 1, 1} Basis for MaximalRed = 0 1. 0 0 bifComponent 3 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {8, 2, 1, 1} Basis for MaximalRed = 1. 0 0 0 5 = stabilizer, 2 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {13, 2, 1, 2} Basis for MaximalRed = 1. 0 0 0 0 1. 0 0 bifComponent 2 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {12, 2, 1, 1} Basis for MaximalRed = 0 0 0.707107 0.707107 6 = stabilizer, 2 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {14, 2, 1, 2} Basis for MaximalRed = 1. 0 0 0 0 1. 0 0 bifComponent 2 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {12, 2, 1, 1} Basis for MaximalRed = 0 0 0.707107 -0.707107 7 = stabilizer, 2 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {13, 2, 1, 1} Basis for MaximalRed = 0 0 0.707107 -0.707107 bifComponent 2 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {14, 2, 1, 1} Basis for MaximalRed = 0 0 0.707107 0.707107 8 = stabilizer, 1 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {15, 2, 1, 2} Basis for MaximalRed = 0 1. 0 0 0 0 0 1. 9 = stabilizer, 1 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {15, 2, 1, 2} Basis for MaximalRed = 0 1. 0 0 0 0 1. 0 10 = stabilizer, 1 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {15, 2, 1, 2} Basis for MaximalRed = 1. 0 0 0 0 0 1. 0 11 = stabilizer, 1 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {15, 2, 1, 2} Basis for MaximalRed = 1. 0 0 0 0 0 0 1. 12 = stabilizer, 1 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {15, 2, 1, 2} Basis for MaximalRed = 1. 0 0 0 0 1. 0 0 13 = stabilizer, 1 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {15, 2, 1, 1} Basis for MaximalRed = 0 0 0.707107 0.707107 14 = stabilizer, 1 = numBifComponents bifComponent 1 1 = dim E, 1 = numMaximals {maximal, |D|, dimMaximalE, dimMaximalRed} = {15, 2, 1, 1} Basis for MaximalRed = 0 0 0.707107 -0.707107 15 = stabilizer, 0 = numBifComponents In[12]:= In[12]:= In[12]:= In[12]:= In[13]:= In[13]:= In[14]:= In[14]:= In[15]:= In[15]:= Starting defineLap.m The graph has m = 4 vertices, n = 4 edges The name of the graph is C4 The symmetry group of the graph is the stabilizer with label = 14 In other words, autGLabel = 14 The stabilizers of the eigenfunctions used to produce the basis are: oneMaximalStab = {15, 14, 12} FICcopiesDim = 1 1 1 2 1 1 3 1 2 irrep of Gamma0 label = FIC label = 1 dimFICj = 1 negLapRestricted = 4. evalsRestricted[[1]] = 4. evecsRestricted[[1]] = {1.} evecjp1 = {-0.5, 0.5, -0.5, 0.5} -0.5 0.5 -0.5 0.5 Isotypic component 1 is made up of 1 eigenspaces of dimension 1 irrep of Gamma0 label = FIC label = 2 dimFICj = 1 negLapRestricted = 0 evalsRestricted[[1]] = 0 evecsRestricted[[1]] = {1} evecjp1 = {-0.5, -0.5, -0.5, -0.5} -0.5 -0.5 -0.5 -0.5 Isotypic component 2 is made up of 1 eigenspaces of dimension 1 irrep of Gamma0 label = FIC label = 3 dimFICj = 2 negLapRestricted = 2. evalsRestricted[[1]] = 2. evecsRestricted[[1]] = {1.} evecjp1 = {0.5, 0.5, -0.5, -0.5} 0.5 0.5 -0.5 -0.5 0.5 -0.5 -0.5 0.5 Isotypic component 3 is made up of 1 eigenspaces of dimension 2 Now find eigenvector(s) within each isotypic component with maximal\ > stabilizer. irrep of Gamma0 label = FIC label = 1 dimFICj = 1 numMaximals = 1, and maximals = {15} The eigenvector(s) in the FIC with label 1 with stabilizer label 15 are -0.5 0.5 -0.5 0.5 irrep of Gamma0 label = FIC label = 2 dimFICj = 1 numMaximals = 1, and maximals = {14} The eigenvector(s) in the FIC with label 2 with stabilizer label 14 are -0.5 -0.5 -0.5 -0.5 irrep of Gamma0 label = FIC label = 3 dimFICj = 2 numMaximals = 2, and maximals = {10, 12} The eigenvector(s) in the FIC with label 3 with stabilizer label 10 are 0.707107 0. -0.707107 0. The eigenvector(s) in the FIC with label 3 with stabilizer label 12 are 0.5 0.5 -0.5 -0.5 The FIC Basis is orthonormal, as it should be. FICInfo = eval j p q 4. 1 1 1 0 2 1 1 2. 3 1 1 2. 3 1 2 reducedFICInfo = eval j q 4. 1 1 0 2 1 2. 3 1 2. 3 2 FICcopiesDim = j Copies Dim 1 1 1 2 1 1 3 1 2 The array FICcopiesDim is written to the file basisInfo.txt The name of the graph is C4 In[16]:= In[16]:=