1 | //////////////////////////////////////////////////////////////// |
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2 | version="version fpalgebras.lib 4.1.1.0 Mar_2018 "; // $Id$ |
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3 | category="Noncommutative"; |
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4 | info=" |
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5 | LIBRARY: fpalgebras.lib Generation of various algebras in the letterplace case |
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6 | AUTHORS: Karim Abou Zeid, karim.abou.zeid at rwth-aachen.de |
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7 | @* Grischa Studzinski, grischa.studzinski at rwth-aachen.de |
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8 | |
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9 | Support: Project II.6 in the transregional collaborative research centre |
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10 | SFB-TRR 195 'Symbolic Tools in Mathematics and their Application' of the German DFG |
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11 | |
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12 | OVERVIEW: |
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13 | Generation of various algebras, including group algebras of finitely presented groups in the Letterplace ring |
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14 | |
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15 | PROCEDURES: |
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16 | operatorAlgebra(string a, int d); |
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17 | baumslagSolitar(int n, int m, int d, list #); |
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18 | baumslag(int m, int n, int d); |
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19 | crystallographicGroupP1(int d); |
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20 | crystallographicGroupPM(int d); |
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21 | crystallographicGroupPG(int d); |
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22 | crystallographicGroupP2MM(int d); |
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23 | crystallographicGroupP2(int d); |
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24 | crystallographicGroupP2GG(int d); |
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25 | crystallographicGroupCM(int d); |
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26 | crystallographicGroupC2MM(int d); |
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27 | crystallographicGroupP4(int d); |
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28 | crystallographicGroupP4MM(int d); |
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29 | crystallographicGroupP4GM(int d); |
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30 | crystallographicGroupP3(int d); |
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31 | crystallographicGroupP31M(int d); |
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32 | crystallographicGroupP3M1(int d); |
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33 | crystallographicGroupP6(int d); |
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34 | crystallographicGroupP6MM(int d); |
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35 | dyckGroup1(int n, int d, intvec P); |
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36 | dyckGroup2(int n, int d, intvec P); |
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37 | dyckGroup3(int n, int d, intvec P); |
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38 | fibonacciGroup(int m, int d); |
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39 | tetrahedronGroup(int g, int d); |
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40 | triangularGroup(int g, int d); |
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41 | "; |
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42 | |
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43 | LIB "freegb.lib"; |
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44 | LIB "general.lib"; |
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45 | //////////////////////////////////////////////////////////////////// |
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46 | |
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47 | //////////////////////////////////////////////////////////////////// |
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48 | // Operator Algebras /////////////////////////////////////////////// |
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49 | //////////////////////////////////////////////////////////////////// |
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50 | |
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51 | proc operatorAlgebra(string a, int d) |
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52 | "USAGE: operatorAlgebra(a,d); a a string, d an integer |
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53 | RETURN: ring |
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54 | NOTE: - the ring contains the ideal I, which contains the required relations |
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55 | @* - a gives the name of the algebra |
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56 | @* - d gives the degreebound for the Letterplace ring |
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57 | @* |
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58 | @* a must be one of the following: |
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59 | @* integrodiff3 |
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60 | @* toeplitz |
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61 | @* weyl1 |
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62 | @* usl2 |
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63 | @* usl2h |
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64 | @* shift1inverse |
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65 | @* exterior2 |
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66 | @* quadrowmm |
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67 | @* shift1 |
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68 | @* weyl1inverse |
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69 | @* |
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70 | @* This is a collection of common algebras |
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71 | @* |
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72 | " |
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73 | { |
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74 | if (d < 2) { |
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75 | ERROR("Degbound d is too small. Must be at least 2."); |
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76 | } |
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77 | int baseringdef; |
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78 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
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79 | { |
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80 | def save = basering; |
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81 | baseringdef = 1; |
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82 | } |
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83 | |
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84 | if (a == "integrodiff3") { |
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85 | ring r = 0,(d,I,x),dp; |
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86 | def R = makeLetterplaceRing(d); |
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87 | setring(R); |
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88 | ideal I = d(1)*x(2)-x(1)*d(2)-1, |
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89 | I(1)*x(2)-x(1)*I(2)+I(1)*I(2), |
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90 | d(1)*I(2)-1; |
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91 | } |
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92 | if (a == "toeplitz") { |
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93 | ring r = 0,(y,x),dp; |
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94 | def R = makeLetterplaceRing(d); |
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95 | setring(R); |
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96 | ideal I = y(1)*x(2)-1; |
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97 | } |
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98 | if (a == "weyl1") { |
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99 | ring r = 0,(d,x),dp; |
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100 | def R = makeLetterplaceRing(d); |
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101 | setring(R); |
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102 | ideal I = d(1)*x(2)-x(1)*d(2)-1; |
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103 | } |
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104 | if (a == "usl2") { |
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105 | ring r = 0,(h,f,e),dp; |
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106 | def R = makeLetterplaceRing(d); |
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107 | setring(R); |
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108 | ideal I = f(1)*e(2)-e(1)*f(2)+h(1), |
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109 | h(1)*e(2)-e(1)*h(2)-2*e(1), |
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110 | h(1)*f(2)-f(1)*h(2)+2*f(1); |
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111 | } |
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112 | if (a == "usl2h") { |
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113 | ring r = 0,(H,h,f,e),dp; |
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114 | def R = makeLetterplaceRing(d); |
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115 | setring(R); |
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116 | ideal I = f(1)*e(2)-e(1)*f(2)+h(1)*H(2), |
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117 | h(1)*e(2)-e(1)*h(2)-2*e(1)*H(2), |
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118 | h(1)*f(2)-f(1)*h(2)+2*f(1)*H(2), |
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119 | f(1)*H(2)-H(1)*f(2), |
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120 | e(1)*H(2)-H(1)*e(2), |
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121 | h(1)*H(2)-H(1)*h(2); |
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122 | } |
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123 | if (a == "shift1inverse") { |
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124 | ring r = 0,(d,x,t),dp; |
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125 | def R = makeLetterplaceRing(d); |
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126 | setring(R); |
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127 | ideal I = d(1)*x(2)-x(1)*d(2)-d(1), |
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128 | t(1)*x(2)-1, |
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129 | x(1)*t(2)-1; |
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130 | } |
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131 | if (a == "exterior2") { |
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132 | ring r = 0,(y,x),dp; |
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133 | def R = makeLetterplaceRing(d); |
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134 | setring(R); |
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135 | ideal I = y(1)*x(2)+x(1)*y(2), |
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136 | x(1)*x(2), |
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137 | y(1)*y(2); |
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138 | } |
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139 | if (a == "quadrowmm") { |
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140 | ring r = 0,(y,x),dp; |
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141 | def R = makeLetterplaceRing(d); |
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142 | setring(R); |
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143 | ideal I = y(1)*x(2)-x(1)*y(2), |
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144 | x(1)*x(2), |
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145 | y(1)*y(2); |
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146 | } |
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147 | if (a == "shift1") { |
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148 | ring r = 0,(s,x),dp; |
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149 | def R = makeLetterplaceRing(d); |
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150 | setring(R); |
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151 | ideal I = s(1)*x(2)-x(1)*s(2)-s(1); |
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152 | } |
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153 | if (a == "weyl1inverse") { |
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154 | ring r = 0,(d,x,t),dp; |
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155 | def R = makeLetterplaceRing(d); |
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156 | setring(R); |
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157 | ideal I = d(1)*x(2)-x(1)*d(2)-1, |
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158 | t(1)*x(2)-1, |
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159 | x(1)*t(2)-1; |
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160 | } |
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161 | |
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162 | if (!defined(I)) { |
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163 | ERROR("Illegal argument for algebra"); |
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164 | } |
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165 | |
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166 | export(I); |
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167 | if (baseringdef == 1) {setring save;} |
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168 | return(R); |
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169 | } |
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170 | |
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171 | //////////////////////////////////////////////////////////////////// |
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172 | // Baumslag //////////////////////////////////////////////////////// |
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173 | // from Grischa Studzinski ///////////////////////////////////////// |
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174 | //////////////////////////////////////////////////////////////////// |
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175 | |
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176 | proc baumslagSolitar(int n, int m, int d, list #) |
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177 | "USAGE: baumslagSolitar(m,n,d[,IsGroup]); n an integer, m an integer, d an integer, IsGroup an optional integer |
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178 | RETURN: ring |
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179 | NOTE: - the ring contains the ideal I, which contains the required relations |
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180 | @* - in the group case: A = a^(-1), B = b^(-1) |
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181 | @* - negativ input is only allowed in the group case! |
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182 | @* - d gives a degreebound and must be >m,n |
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183 | @* |
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184 | @* This is a family |
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185 | @* |
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186 | " |
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187 | { |
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188 | int isGroup = 0; |
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189 | if (size(#) > 0) {isGroup = #[1];} |
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190 | |
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191 | if (isGroup != 0) |
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192 | { |
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193 | int baseringdef; |
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194 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
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195 | { |
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196 | def save = basering; |
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197 | baseringdef = 1; |
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198 | } |
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199 | if (m < 0 || n < 0) {ERROR("Exponent can't be negativ in monoid rings!");} |
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200 | if (d < 1 || d < m || d < n) {ERROR("Degree bound must be positiv and greater then m,n!");} |
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201 | int i; |
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202 | ring mr = 0,(a,b),Dp; |
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203 | def Mr = makeLetterplaceRing(d); |
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204 | setring Mr; |
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205 | poly p,q; |
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206 | if (n==0) {p = b(1);} |
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207 | else |
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208 | { |
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209 | p = a(1)*b(2); |
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210 | for (i = 1; i < n; i++) {p = lpMult(a(1),p);} |
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211 | } |
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212 | if (m==0) {q = b(1);} |
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213 | else |
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214 | { |
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215 | q = b(1)*a(2); |
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216 | for (i = 1; i < m; i++) {q = lpMult(q,a(1));} |
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217 | } |
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218 | ideal I = p - q; |
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219 | export(I); |
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220 | if (baseringdef == 1) {setring save;} |
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221 | return(Mr); |
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222 | } |
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223 | else |
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224 | { |
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225 | int baseringdef; |
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226 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
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227 | { |
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228 | def save = basering; |
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229 | baseringdef = 1; |
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230 | } |
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231 | int i; |
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232 | if (d < 1 || d < absValue(m) || d < absValue(n)) {ERROR("Degree bound must be positiv and greater then |m|,|n|!");} |
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233 | ring gr = 0,(a,b,A,B),Dp; |
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234 | def Gr = makeLetterplaceRing(d); |
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235 | setring Gr; |
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236 | poly p,q; |
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237 | if (n==0) {p = b(1);} |
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238 | else |
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239 | {if (n > 0) |
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240 | { |
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241 | p = a(1)*b(2); |
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242 | for (i = 1; i < n; i++) {p = lpMult(a(1),p);} |
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243 | } |
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244 | else |
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245 | { |
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246 | p = A(1)*b(2); |
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247 | for (i = 1; i < -n; i++) {p = lpMult(A(1),p);} |
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248 | } |
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249 | } |
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250 | if (m==0) {q = b(1);} |
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251 | else |
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252 | {if (m > 0) |
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253 | { |
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254 | q = b(1)*a(2); |
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255 | for (i = 1; i < m; i++) {q = lpMult(q,a(1));} |
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256 | } |
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257 | else |
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258 | { |
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259 | q = A(1)*b(2); |
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260 | for (i = 1; i < -m; i++) {q = lpMult(q,A(1));} |
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261 | } |
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262 | } |
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263 | ideal I = p - q, a(1)*A(2) - 1, b(1)*B(2) - 1, a(1)*A(2) - A(1)*a(2), b(1)*B(2) - B(1)*b(2); |
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264 | export(I); |
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265 | if (baseringdef == 1) {setring save;} |
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266 | return(Gr); |
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267 | } |
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268 | } |
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269 | example { |
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270 | "EXAMPLE:"; echo = 2; |
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271 | def R = baumslagSolitar(2,3,4); setring R; |
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272 | I; |
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273 | } |
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274 | |
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275 | proc baumslagGroup(int m, int n, int d) |
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276 | "USAGE: baumslagGroup(m,n,d); m an integer, n an integer, d an integer |
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277 | RETURN: ring |
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278 | NOTE: - the ring contains the ideal I, which contains the required relations |
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279 | @* - Baumslag group with the following presentation |
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280 | @* < a, b | a^m = b^n = 1 > |
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281 | @* -d gives the degreebound for the Letterplace ring |
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282 | @* |
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283 | @* This is a family |
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284 | @* |
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285 | " |
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286 | { |
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287 | if (m < 0 || n < 0 ) {ERROR("m,n must be non-negativ integers!");} |
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288 | if (d < 1 || d < m || d < n) {ERROR("degreebound must be positiv and larger than n and m!");} |
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289 | int i; |
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290 | ring r = 0,(a,b),dp; |
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291 | def R = makeLetterplaceRing(d); |
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292 | setring R; |
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293 | poly p,q; |
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294 | p = 1; q = 1; |
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295 | for (i = 1; i <= m; i++){p = lpMult(p,a(1));} |
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296 | for (i = 1; i <= n; i++){q = lpMult(q,b(1));} |
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297 | ideal I = p-1,q-1; |
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298 | export(I); |
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299 | return(R); |
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300 | } |
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301 | example { |
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302 | "EXAMPLE:"; echo = 2; |
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303 | def R = baumslag(2,3,4); setring R; |
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304 | I; |
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305 | } |
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306 | |
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307 | //////////////////////////////////////////////////////////////////// |
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308 | // Crystallographic Groups ////////////////////////////////////////// |
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309 | // from Grischa Studzinski ///////////////////////////////////////// |
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310 | //////////////////////////////////////////////////////////////////// |
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311 | |
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312 | proc crystallographicGroupP1(int d) |
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313 | "USAGE: crystallographicGroupP1(d); d an integer |
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314 | RETURN: ring |
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315 | NOTE: - the ring contains the ideal I, which contains the required relations |
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316 | @* - p1 group with the following presentation |
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317 | @* < x, y | [x, y] = 1 > |
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318 | @* -d gives the degreebound for the Letterplace ring |
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319 | " |
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320 | { |
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321 | if (d < 2){ERROR("Degreebound is to small for choosen example!");} |
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322 | |
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323 | int baseringdef; |
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324 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
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325 | { |
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326 | def save = basering; |
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327 | baseringdef = 1; |
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328 | } |
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329 | ring r = 2,(x,y,X,Y),dp; |
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330 | def R = makeLetterplaceRing(d); |
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331 | setring R; |
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332 | ideal I = x(1)*y(2)-y(1)*x(2)-1, X(1)*x(2)-1, x(1)*X(2)-1, y(1)*Y(2)-1, Y(1)*y(2)-1; |
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333 | I = simplify(I,2); |
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334 | export(I); |
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335 | if (baseringdef == 1) {setring save;} |
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336 | return(R); |
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337 | } |
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338 | example { |
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339 | "EXAMPLE:"; echo = 2; |
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340 | def R = crystallographicGroupP1(5); setring R; |
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341 | I; |
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342 | } |
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343 | |
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344 | // old? there is already another crystallographicGroupP2 proc |
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345 | /* proc crystallographicGroupP2(int d) */ |
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346 | /* " */ |
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347 | /* p2 group with the following presentation */ |
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348 | /* < x, y, r | [x, y] = r^2 = 1, r^-1*x*r = x^-1, r^-1*y*r = y^-1 > */ |
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349 | /* Note: r = r^-1 */ |
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350 | /* " */ |
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351 | /* { */ |
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352 | /* if (d < 3){ERROR("Degreebound is to small for choosen example!");} */ |
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353 | |
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354 | /* int baseringdef; */ |
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355 | /* if (defined(basering)) // if a basering is defined, it should be saved for later use */ |
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356 | /* { */ |
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357 | /* def save = basering; */ |
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358 | /* baseringdef = 1; */ |
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359 | /* } */ |
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360 | /* ring r = 2,(x,y,r,X,Y),dp; */ |
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361 | /* def R = makeLetterplaceRing(d); */ |
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362 | /* setring R; */ |
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363 | /* ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2), r(1)*r(2)-1, r(1)*x(2)*r(3)-X(1), r(1)*y(2)*r(3)-Y(1),x(1)*X(2)-1, */ |
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364 | /* X(1)*x(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; */ |
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365 | /* I = simplify(I,2); */ |
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366 | /* export(I); */ |
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367 | /* if (baseringdef == 1) {setring save;} */ |
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368 | /* return(R); */ |
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369 | /* } */ |
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370 | |
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371 | proc crystallographicGroupPM(int d) |
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372 | "USAGE: crystallographicGroupPM(d); d an integer |
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373 | RETURN: ring |
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374 | NOTE: - the ring contains the ideal I, which contains the required relations |
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375 | @* - pm group with the following presentation |
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376 | @* < x, y, m | [x, y] = m^2 = 1, m^-1*x*m = x, m^-1*y*m = y^-1 > |
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377 | @* - d gives the degreebound for the Letterplace ring |
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378 | " |
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379 | { |
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380 | if (d < 3){ERROR("Degreebound is to small for choosen example!");} |
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381 | |
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382 | int baseringdef; |
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383 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
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384 | { |
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385 | def save = basering; |
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386 | baseringdef = 1; |
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387 | } |
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388 | ring r = 2,(x,y,m,X,Y),dp; |
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389 | def R = makeLetterplaceRing(d); |
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390 | setring R; |
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391 | ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-m(1)*m(2), m(1)*m(2)-1, m(1)*x(2)*m(3)-x(1), m(1)*y(2)*m(3)-Y(1),x(1)*X(2)-1, |
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392 | X(1)*x(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; |
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393 | I = simplify(I,2); |
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394 | export(I); |
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395 | if (baseringdef == 1) {setring save;} |
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396 | return(R); |
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397 | } |
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398 | example { |
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399 | "EXAMPLE:"; echo = 2; |
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400 | def R = crystallographicGroupPM(5); setring R; |
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401 | I; |
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402 | } |
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403 | |
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404 | proc crystallographicGroupPG(int d) |
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405 | "USAGE: crystallographicGroupPG(d); d an integer |
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406 | RETURN: ring |
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407 | NOTE: - the ring contains the ideal I, which contains the required relations |
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408 | @* - pg group with the following presentation |
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409 | @* < x, y, t | [x, y] = 1, t^2 = x, t^-1*y*t = y^-1 > |
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410 | @* - d gives the degreebound for the Letterplace ring |
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411 | " |
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412 | { |
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413 | if (d < 3){ERROR("Degreebound is to small for choosen example!");} |
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414 | |
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415 | int baseringdef; |
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416 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
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417 | { |
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418 | def save = basering; |
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419 | baseringdef = 1; |
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420 | } |
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421 | ring r = 2,(x,y,t,X,Y,T),dp; |
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422 | def R = makeLetterplaceRing(d); |
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423 | setring R; |
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424 | ideal I = x(1)*y(2)-y(1)*x(2)-1, t(1)*t(2) - x(1), T(1)*y(2)*t(3)-Y(1), X(1)*x(2)-1, x(1)*X(2)-1, |
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425 | Y(1)*y(2)-1, y(1)*Y(2)-1, t(1)*T(2)-1, T(1)*t(2)-1; |
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426 | I = simplify(I,2); |
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427 | export(I); |
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428 | if (baseringdef == 1) {setring save;} |
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429 | return(R); |
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430 | } |
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431 | example { |
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432 | "EXAMPLE:"; echo = 2; |
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433 | def R = crystallographicGroupPG(5); setring R; |
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434 | I; |
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435 | } |
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436 | |
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437 | |
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438 | proc crystallographicGroupP2MM(int d) |
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439 | "USAGE: crystallographicGroupP2MM(d); d an integer |
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440 | RETURN: ring |
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441 | NOTE: - the ring contains the ideal I, which contains the required relations |
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442 | @* - p2mm group with the following presentation |
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443 | @* < x, y, p, q | [x, y] = [p, q] = p^2 = q^2 = 1, p^-1*x*p = x, q^-1*x*q = x^-1, p^-1*y*p = y^-1, q^-1*y*q = y > |
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444 | @* - d gives the degreebound for the Letterplace ring |
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445 | " |
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446 | { |
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447 | if (d < 3){ERROR("Degreebound is to small for choosen example!");} |
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448 | |
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449 | int baseringdef; |
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450 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
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451 | { |
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452 | def save = basering; |
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453 | baseringdef = 1; |
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454 | } |
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455 | ring r = 2,(x,y,p,q,X,Y),dp; |
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456 | def R = makeLetterplaceRing(d); |
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457 | setring R; |
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458 | ideal I = x(1)*y(2)-y(1)*x(2)-1, p(1)*q(2)-q(1)*p(2)-1, p(1)*p(2) - 1, q(1)*q(2) - 1, p(1)*y(2)*p(3)-Y(1), p(1)*x(2)*p(3)-x(1), |
---|
459 | q(1)*y(2)*q(3)-y(1), q(1)*x(2)*q(3)-X(1), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1, x(1)*y(2)-y(1)*x(2)- p(1)*p(2), |
---|
460 | x(1)*y(2)-y(1)*x(2)- q(1)*q(2), p(1)*p(2)-q(1)*q(2); |
---|
461 | I = simplify(I,2); |
---|
462 | export(I); |
---|
463 | if (baseringdef == 1) {setring save;} |
---|
464 | return(R); |
---|
465 | } |
---|
466 | example { |
---|
467 | "EXAMPLE:"; echo = 2; |
---|
468 | def R = crystallographicGroupP2MM(5); setring R; |
---|
469 | I; |
---|
470 | } |
---|
471 | |
---|
472 | proc crystallographicGroupP2(int d) |
---|
473 | "USAGE: crystallographicGroupP2(d); d an integer |
---|
474 | RETURN: ring |
---|
475 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
476 | @* - p2 group with the following presentation |
---|
477 | @* < x, y, m, t | [x, y] = t^2 = 1, m^2 = y, t^-1*x*t = x, m^-1*x*m = x^-1, t^-1*y*t = y^-1, t^-1*m*t = m^-1 > |
---|
478 | @* - d gives the degreebound for the Letterplace ring |
---|
479 | " |
---|
480 | { |
---|
481 | if (d < 3){ERROR("Degreebound is to small for choosen example!");} |
---|
482 | |
---|
483 | int baseringdef; |
---|
484 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
485 | { |
---|
486 | def save = basering; |
---|
487 | baseringdef = 1; |
---|
488 | } |
---|
489 | ring r = 2,(x,y,m,t,X,Y,M),dp; |
---|
490 | def R = makeLetterplaceRing(d); |
---|
491 | setring R; |
---|
492 | ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-t(1)*t(2), m(1)*m(2)-y(1), t(1)*t(2) - 1, t(1)*x(2)*t(3)-x(1), |
---|
493 | M(1)*x(2)*m(3)-X(1), t(1)*y(2)*t(3)-Y(1), t(1)*m(2)*t(3)-M(1), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1, |
---|
494 | m(1)*M(2)-1, M(1)*m(2)-1; |
---|
495 | I = simplify(I,2); |
---|
496 | export(I); |
---|
497 | if (baseringdef == 1) {setring save;} |
---|
498 | return(R); |
---|
499 | } |
---|
500 | example { |
---|
501 | "EXAMPLE:"; echo = 2; |
---|
502 | def R = crystallographicGroupP2(5); setring R; |
---|
503 | I; |
---|
504 | } |
---|
505 | |
---|
506 | proc crystallographicGroupP2GG(int d) |
---|
507 | "USAGE: crystallographicGroupP2GG(d); d an integer |
---|
508 | RETURN: ring |
---|
509 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
510 | @* - p2gg group with the following presentation |
---|
511 | @* < x, y, u, v | [x, y] = (u*v)^2 = 1, u^2 = x, v^2 = y, v^-1*x*v = x^-1, u^-1*y*u = y^-1 > |
---|
512 | @* - d gives the degreebound for the Letterplace ring |
---|
513 | " |
---|
514 | { |
---|
515 | if (d < 4){ERROR("Degreebound is to small for choosen example!");} |
---|
516 | |
---|
517 | int baseringdef; |
---|
518 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
519 | { |
---|
520 | def save = basering; |
---|
521 | baseringdef = 1; |
---|
522 | } |
---|
523 | ring r = 2,(x,y,u,v,X,Y,u,v),dp; |
---|
524 | def R = makeLetterplaceRing(d); |
---|
525 | setring R; |
---|
526 | ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-u(1)*v(2)*u(3)*v(4), u(1)*v(2)*u(3)*v(4)-1, u(1)*u(2)-x(1), v(1)*v(2) - y, |
---|
527 | V(1)*x(2)*v(3)-X(1), U(1)*y(2)*u(3)-Y(1), |
---|
528 | X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1, u(1)*U(2)-1, U(1)*u(2)-1, v(1)*V(2)-1, V(1)*v(2)-1; |
---|
529 | I = simplify(I,2); |
---|
530 | export(I); |
---|
531 | if (baseringdef == 1) {setring save;} |
---|
532 | return(R); |
---|
533 | } |
---|
534 | example { |
---|
535 | "EXAMPLE:"; echo = 2; |
---|
536 | def R = crystallographicGroupP2GG(5); setring R; |
---|
537 | I; |
---|
538 | } |
---|
539 | |
---|
540 | proc crystallographicGroupCM(int d) |
---|
541 | "USAGE: crystallographicGroupCM(d); d an integer |
---|
542 | RETURN: ring |
---|
543 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
544 | @* - cm group with the following presentation |
---|
545 | @* < x, y, t | [x, y] = t^2 = 1, t^-1*x*t = x*y, t^-1*y*t = y^-1 > |
---|
546 | @* - d gives the degreebound for the Letterplace ring |
---|
547 | " |
---|
548 | { |
---|
549 | if (d < 3){ERROR("Degreebound is to small for choosen example!");} |
---|
550 | |
---|
551 | int baseringdef; |
---|
552 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
553 | { |
---|
554 | def save = basering; |
---|
555 | baseringdef = 1; |
---|
556 | } |
---|
557 | ring r = 2,(x,y,t,X,Y),dp; |
---|
558 | def R = makeLetterplaceRing(d); |
---|
559 | setring R; |
---|
560 | ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-t(1)*t(2), t(1)*t(2)-1, |
---|
561 | t(1)*x(2)*t(3)-x(1)*y(2), t(1)*y(2)*t(3)-Y(1), |
---|
562 | X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; |
---|
563 | I = simplify(I,2); |
---|
564 | export(I); |
---|
565 | if (baseringdef == 1) {setring save;} |
---|
566 | return(R); |
---|
567 | } |
---|
568 | example { |
---|
569 | "EXAMPLE:"; echo = 2; |
---|
570 | def R = crystallographicGroupCM(5); setring R; |
---|
571 | I; |
---|
572 | } |
---|
573 | |
---|
574 | proc crystallographicGroupC2MM(int d) |
---|
575 | "USAGE: crystallographicGroupC2MM(d); d an integer |
---|
576 | RETURN: ring |
---|
577 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
578 | @* - c2mm group with the following presentation |
---|
579 | @* < x, y, m, r | [x, y] = m^2 = r^2 = 1, m^-1*y*m = y^-1, m^-1*x*m = x*y, r^-1*y*r = y^-1, r^-1*x*r = x^-1, m^-1*r*m = r^-1 > |
---|
580 | @* - d gives the degreebound for the Letterplace ring |
---|
581 | " |
---|
582 | { |
---|
583 | if (d < 3){ERROR("Degreebound is to small for choosen example!");} |
---|
584 | |
---|
585 | int baseringdef; |
---|
586 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
587 | { |
---|
588 | def save = basering; |
---|
589 | baseringdef = 1; |
---|
590 | } |
---|
591 | ring r = 2,(x,y,m,r,X,Y),dp; |
---|
592 | def R = makeLetterplaceRing(d); |
---|
593 | setring R; |
---|
594 | ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-m(1)*m(2), x(1)*y(2)-y(1)*x(2)-r(1)*r(2), m(1)*m(2)-1, r(1)*r(2)-1, |
---|
595 | m(1)*m(2)-r(1)*r(2), m(1)*y(2)*m(3)-Y(1), m(1)*x(2)*m(3)-x(1)*y(2), (1)*y(2)*r(3)-Y(1), r(1)*x(2)*r(3)-X(1), m(1)*r(2)*m(3)-r(1), |
---|
596 | X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; |
---|
597 | I = simplify(I,2); |
---|
598 | export(I); |
---|
599 | if (baseringdef == 1) {setring save;} |
---|
600 | return(R); |
---|
601 | } |
---|
602 | example { |
---|
603 | "EXAMPLE:"; echo = 2; |
---|
604 | def R = crystallographicGroupC2MM(5); setring R; |
---|
605 | I; |
---|
606 | } |
---|
607 | |
---|
608 | proc crystallographicGroupP4(int d) |
---|
609 | "USAGE: crystallographicGroupP4(d); d an integer |
---|
610 | RETURN: ring |
---|
611 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
612 | @* - p4 group with the following presentation |
---|
613 | @* < x, y, r | [x, y] = r^4 = 1, r^-1*x*r = x^-1, r^-1*x*r = y > |
---|
614 | @* - d gives the degreebound for the Letterplace ring |
---|
615 | " |
---|
616 | { |
---|
617 | if (d < 5){ERROR("Degreebound is to small for choosen example!");} |
---|
618 | |
---|
619 | int baseringdef; |
---|
620 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
621 | { |
---|
622 | def save = basering; |
---|
623 | baseringdef = 1; |
---|
624 | } |
---|
625 | ring r = 2,(x,y,r,X,Y),dp; |
---|
626 | def R = makeLetterplaceRing(d); |
---|
627 | setring R; |
---|
628 | ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4), r(1)*r(2)*r(3)*r(4)-1, |
---|
629 | r(1)*r(2)*r(3)*x(4)*r(5)-X(1), r(1)*r(2)*r(3)*x(4)*r(5)-y(1), |
---|
630 | X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; |
---|
631 | I = simplify(I,2); |
---|
632 | export(I); |
---|
633 | if (baseringdef == 1) {setring save;} |
---|
634 | return(R); |
---|
635 | } |
---|
636 | example { |
---|
637 | "EXAMPLE:"; echo = 2; |
---|
638 | def R = crystallographicGroupP4(5); setring R; |
---|
639 | I; |
---|
640 | } |
---|
641 | |
---|
642 | proc crystallographicGroupP4MM(int d) |
---|
643 | "USAGE: crystallographicGroupP4MM(d); d an integer |
---|
644 | RETURN: ring |
---|
645 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
646 | @* - p4mm group with the following presentation |
---|
647 | @* < x, y, r, m | [x, y] = r^4 = m^2 = 1, r^-1*y*r = x^-1, r^-1*x*r = y, m^-1*x*m = y, m^-1*r*m = r^-1 > |
---|
648 | @* - d gives the degreebound for the Letterplace ring |
---|
649 | " |
---|
650 | { |
---|
651 | if (d < 5){ERROR("Degreebound is to small for choosen example!");} |
---|
652 | |
---|
653 | int baseringdef; |
---|
654 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
655 | { |
---|
656 | def save = basering; |
---|
657 | baseringdef = 1; |
---|
658 | } |
---|
659 | ring r = 2,(x,y,r,m,X,Y),dp; |
---|
660 | def R = makeLetterplaceRing(d); |
---|
661 | setring R; |
---|
662 | ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4), r(1)*r(2)*r(3)*r(4)-1, |
---|
663 | r(1)*r(2)*r(3)*x(4)*r(5)-X(1), r(1)*r(2)*r(3)*x(4)*r(5)-y(1), |
---|
664 | X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; |
---|
665 | I = simplify(I,2); |
---|
666 | export(I); |
---|
667 | if (baseringdef == 1) {setring save;} |
---|
668 | return(R); |
---|
669 | } |
---|
670 | example { |
---|
671 | "EXAMPLE:"; echo = 2; |
---|
672 | def R = crystallographicGroupP4MM(5); setring R; |
---|
673 | I; |
---|
674 | } |
---|
675 | |
---|
676 | proc crystallographicGroupP4GM(int d) |
---|
677 | "USAGE: crystallographicGroupP4GM(d); d an integer |
---|
678 | RETURN: ring |
---|
679 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
680 | @* - p4gm group with the following presentation |
---|
681 | @* < x, y, r, t | [x, y] = r^4 = t^2 = 1, r^-1*y*r = x^-1, r^-1*x*r = y, t^-1*x*t = y, t^-1*r*t = x^-1*r^-1> |
---|
682 | @* - d gives the degreebound for the Letterplace ring |
---|
683 | " |
---|
684 | { |
---|
685 | if (d < 5){ERROR("Degreebound is to small for choosen example!");} |
---|
686 | |
---|
687 | int baseringdef; |
---|
688 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
689 | { |
---|
690 | def save = basering; |
---|
691 | baseringdef = 1; |
---|
692 | } |
---|
693 | ring r = 2,(x,y,r,t,X,Y),dp; |
---|
694 | def R = makeLetterplaceRing(d); |
---|
695 | setring R; |
---|
696 | ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4), r(1)*r(2)*r(3)*r(4)-1, x(1)*y(2)-y(1)*x(2)-t(1)*t(2), |
---|
697 | t(1)*t(2)-1, r(1)*r(2)*r(3)*r(4)-t(1)*t(2), r(1)*r(2)*r(3)*y(4)*r(5)-X(1), r(1)*r(2)*r(3)*x(4)*r(5)-y(1), |
---|
698 | t(1)*r(2)*t(3)-X(1)*r(2)*r(3)*r(4), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; |
---|
699 | I = simplify(I,2); |
---|
700 | export(I); |
---|
701 | if (baseringdef == 1) {setring save;} |
---|
702 | return(R); |
---|
703 | } |
---|
704 | example { |
---|
705 | "EXAMPLE:"; echo = 2; |
---|
706 | def R = crystallographicGroupP4GM(5); setring R; |
---|
707 | I; |
---|
708 | } |
---|
709 | |
---|
710 | proc crystallographicGroupP3(int d) |
---|
711 | "USAGE: crystallographicGroupP3(d); d an integer |
---|
712 | RETURN: ring |
---|
713 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
714 | @* - p3 group with the following presentation |
---|
715 | @* < x, y, r | [x, y] = r^3 = 1, r^-1*x*r = x^-1*y, r^-1*y*r = x^-1> |
---|
716 | @* - d gives the degreebound for the Letterplace ring |
---|
717 | " |
---|
718 | { |
---|
719 | if (d < 4){ERROR("Degreebound is to small for choosen example!");} |
---|
720 | |
---|
721 | int baseringdef; |
---|
722 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
723 | { |
---|
724 | def save = basering; |
---|
725 | baseringdef = 1; |
---|
726 | } |
---|
727 | ring r = 2,(x,y,r,X,Y),dp; |
---|
728 | def R = makeLetterplaceRing(d); |
---|
729 | setring R; |
---|
730 | ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3), r(1)*r(2)*r(3)-1, |
---|
731 | r(1)*r(2)*x(3)*r(4)-X(1)*y(2), r(1)*r(2)*y(3)*r(4)-X(1), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; |
---|
732 | I = simplify(I,2); |
---|
733 | export(I); |
---|
734 | if (baseringdef == 1) {setring save;} |
---|
735 | return(R); |
---|
736 | } |
---|
737 | example { |
---|
738 | "EXAMPLE:"; echo = 2; |
---|
739 | def R = crystallographicGroupP3(5); setring R; |
---|
740 | I; |
---|
741 | } |
---|
742 | |
---|
743 | proc crystallographicGroupP31M(int d) |
---|
744 | "USAGE: crystallographicGroupP31M(d); d an integer |
---|
745 | RETURN: ring |
---|
746 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
747 | @* - p31m group with the following presentation |
---|
748 | @* < x, y, r, t | [x, y] = r^2 = t^2 = (t*r)^3 = 1, r^-1*x*r = x, t^-1*y*t = y, t^-1*x*t = x^-1*y, r^-1*y*r = x*y^-1 > |
---|
749 | @* - d gives the degreebound for the Letterplace ring |
---|
750 | " |
---|
751 | { |
---|
752 | if (d < 6){ERROR("Degreebound is to small for choosen example!");} |
---|
753 | |
---|
754 | int baseringdef; |
---|
755 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
756 | { |
---|
757 | def save = basering; |
---|
758 | baseringdef = 1; |
---|
759 | } |
---|
760 | ring r = 2,(x,y,r,t,X,Y),dp; |
---|
761 | def R = makeLetterplaceRing(d); |
---|
762 | setring R; |
---|
763 | ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2), x(1)*y(2)-y(1)*x(2)-t(1)*t(2), r(1)*r(2)-1, t(1)*t(2)-1, |
---|
764 | t(1)*r(2)*t(3)*r(4)*t(5)*r(6)-1, r(1)*r(2)-t(1)*t(2), x(1)*y(2)-y(1)*x(2)-t(1)*r(2)*t(3)*r(4)*t(5)*r(6), |
---|
765 | t(1)*r(2)*t(3)*r(4)*t(5)*r(6)-r(1)*r(2), t(1)*r(2)*t(3)*r(4)*t(5)*r(6)-t(1)*t(2), |
---|
766 | r(1)*x(2)*r(3)-x(1), t(1)*y(2)*t(3)-y(1), t(1)*x(2)*t(3)-X(1)*y(2), r(1)*y(2)*r(3)-x(1)*Y(2), |
---|
767 | X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; |
---|
768 | I = simplify(I,2); |
---|
769 | export(I); |
---|
770 | if (baseringdef == 1) {setring save;} |
---|
771 | return(R); |
---|
772 | } |
---|
773 | example { |
---|
774 | "EXAMPLE:"; echo = 2; |
---|
775 | def R = crystallographicGroupP31M(5); setring R; |
---|
776 | I; |
---|
777 | } |
---|
778 | |
---|
779 | proc crystallographicGroupP3M1(int d) |
---|
780 | "USAGE: crystallographicGroupP3M1(d); d an integer |
---|
781 | RETURN: ring |
---|
782 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
783 | @* - p3m1 group with the following presentation |
---|
784 | @* < x, y, r, m | [x, y] = r^3 = m^2 = 1, m^-1*r*m = r^2, r^-1*x*r = x^-1*y, r^-1*y*r = x^-1, m^-1*x*m = x^-1, m^-1*y*m = x^-1*y > |
---|
785 | @* - d gives the degreebound for the Letterplace ring |
---|
786 | " |
---|
787 | { |
---|
788 | if (d < 4){ERROR("Degreebound is to small for choosen example!");} |
---|
789 | |
---|
790 | int baseringdef; |
---|
791 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
792 | { |
---|
793 | def save = basering; |
---|
794 | baseringdef = 1; |
---|
795 | } |
---|
796 | ring r = 2,(x,y,r,m,X,Y),dp; |
---|
797 | def R = makeLetterplaceRing(d); |
---|
798 | setring R; |
---|
799 | ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3), x(1)*y(2)-y(1)*x(2)-m(1)*m(2), r(1)*r(2)*r(3)-1, m(1)*m(2)-1, |
---|
800 | r(1)*r(2)*r(3)-m(1)*m(2), m(1)*r(2)*m(3)-r(1)*r(2), r(1)*r(2)*x(3)*r(4)-X(1)*y(2), r(1)*r(2)*y(3)*r(4)-X(1),m(1)*x(2)*m(3)-X(1), |
---|
801 | m(1)*y(2)*m(3)-X(1)*y(2), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; |
---|
802 | I = simplify(I,2); |
---|
803 | export(I); |
---|
804 | if (baseringdef == 1) {setring save;} |
---|
805 | return(R); |
---|
806 | } |
---|
807 | example { |
---|
808 | "EXAMPLE:"; echo = 2; |
---|
809 | def R = crystallographicGroupP3M1(5); setring R; |
---|
810 | I; |
---|
811 | } |
---|
812 | |
---|
813 | proc crystallographicGroupP6(int d) |
---|
814 | "USAGE: crystallographicGroupP6(d); d an integer |
---|
815 | RETURN: ring |
---|
816 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
817 | @* - p6 group with the following presentation |
---|
818 | @* < x, y, r | [x, y] = r^6 = 1, r^-1*x*r = y, r^-1*y*r = x^-1*y> |
---|
819 | @* - d gives the degreebound for the Letterplace ring |
---|
820 | " |
---|
821 | { |
---|
822 | if (d < 7){ERROR("Degreebound is to small for choosen example!");} |
---|
823 | |
---|
824 | int baseringdef; |
---|
825 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
826 | { |
---|
827 | def save = basering; |
---|
828 | baseringdef = 1; |
---|
829 | } |
---|
830 | ring r = 2,(x,y,r,X,Y),dp; |
---|
831 | def R = makeLetterplaceRing(d); |
---|
832 | setring R; |
---|
833 | ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4)*r(5)*r(6), r(1)*r(2)*r(3)*r(4)*r(5)*r(6)-1, |
---|
834 | r(1)*r(2)*r(3)*r(4)*r(5)*x(6)*r(7)-y(1), r(1)*r(2)*r(3)*r(4)*r(5)*y(6)*r(7)-X(1)*y(2), |
---|
835 | X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; |
---|
836 | I = simplify(I,2); |
---|
837 | export(I); |
---|
838 | if (baseringdef == 1) {setring save;} |
---|
839 | return(R); |
---|
840 | } |
---|
841 | example { |
---|
842 | "EXAMPLE:"; echo = 2; |
---|
843 | def R = crystallographicGroupP6(5); setring R; |
---|
844 | I; |
---|
845 | } |
---|
846 | |
---|
847 | proc crystallographicGroupP6MM(int d) |
---|
848 | "USAGE: crystallographicGroupP6MM(d); d an integer |
---|
849 | RETURN: ring |
---|
850 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
851 | @* - p6mm group with the following presentation |
---|
852 | @* < x, y, r, m | [x, y] = r^6 = m^2 = 1, r^-1*y*r = x^-1*y, r^-1*x*r = y, m^-1*x*m = x^-1, m^-1*y*m = x^-1*y, m^-1*r*m = r^-1*y> |
---|
853 | @* - d gives the degreebound for the Letterplace ring |
---|
854 | " |
---|
855 | { |
---|
856 | if (d < 7){ERROR("Degreebound is to small for choosen example!");} |
---|
857 | |
---|
858 | int baseringdef; |
---|
859 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
860 | { |
---|
861 | def save = basering; |
---|
862 | baseringdef = 1; |
---|
863 | } |
---|
864 | ring r = 2,(x,y,r,m,X,Y),dp; |
---|
865 | def R = makeLetterplaceRing(d); |
---|
866 | setring R; |
---|
867 | ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4)*r(5)*r(6), r(1)*r(2)*r(3)*r(4)*r(5)*r(6)-1, |
---|
868 | x(1)*y(2)-y(1)*x(2)-m(1)*m(2), r(1)*r(2)*r(3)*r(4)*r(5)*r(6)-m(1)*m(2), m(1)*m(2)-1, m(1)*x(2)*m(3)-X(1), m(1)*y(2)*m(3)-X(1)*y(2), |
---|
869 | r(1)*r(2)*r(3)*r(4)*r(5)*x(6)*r(7)-y(1), r(1)*r(2)*r(3)*r(4)*r(5)*y(6)*r(7)-X(1)*y(2), M(1)*r(2)*m(3)- r(1)*r(2)*r(3)*r(4)*r(5)*y(6) |
---|
870 | X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1, y(1)*Y(2)-1; |
---|
871 | I = simplify(I,2); |
---|
872 | export(I); |
---|
873 | if (baseringdef == 1) {setring save;} |
---|
874 | return(R); |
---|
875 | } |
---|
876 | example { |
---|
877 | "EXAMPLE:"; echo = 2; |
---|
878 | def R = crystallographicGroupP6MM(5); setring R; |
---|
879 | I; |
---|
880 | } |
---|
881 | |
---|
882 | //////////////////////////////////////////////////////////////////// |
---|
883 | // Dyck Group ////////////////////////////////////////////////////// |
---|
884 | // from Grischa Studzinski ///////////////////////////////////////// |
---|
885 | //////////////////////////////////////////////////////////////////// |
---|
886 | |
---|
887 | proc dyckGroup1(int n, int d, intvec P) |
---|
888 | "USAGE: dyckGroup1(n,d,P); n an integer, d an integer, P an intvec |
---|
889 | RETURN: ring |
---|
890 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
891 | @* - The Dyck group with the following presentation |
---|
892 | @* < x_1, x_2, ... , x_n | (x_1)^p1 = (x_2)^p2 = ... = (x_n)^pn = x_1 * x_2 * ... * x_n = 1 > |
---|
893 | @* - negative exponents are allowed |
---|
894 | @* - representation in the form x_i^p_i - x_(i+1)^p_(i+1) |
---|
895 | @* - d gives the degreebound for the Letterplace ring |
---|
896 | @* |
---|
897 | @* This is a family |
---|
898 | @* |
---|
899 | " |
---|
900 | { |
---|
901 | int baseringdef,i,j; |
---|
902 | if (n < 1) {ERROR("There must be at least one variable!");} |
---|
903 | if (d < n) {ERROR("Degreebound is to small!");} |
---|
904 | for (i = 1; i <= size(P); i++) {if (d < absValue(P[i])){ERROR("Degreebound is to small!");}} |
---|
905 | |
---|
906 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
907 | { |
---|
908 | def save = basering; |
---|
909 | baseringdef = 1; |
---|
910 | } |
---|
911 | ring r = 2,(x(1..n),Y(1..n)),dp; |
---|
912 | def R = makeLetterplaceRing(d); |
---|
913 | setring R; |
---|
914 | ideal I; poly p,q; |
---|
915 | p = 1; q = 1; |
---|
916 | for (i = 1; i<= n; i++) {p = lpMult(p,var(i));} |
---|
917 | I = p-1; |
---|
918 | for (i = n; i > 0; i--) |
---|
919 | { |
---|
920 | if (P[i] >= 0) {for (j = 1; j <= P[i]; j++){q = lpMult(q,var(i));}} |
---|
921 | else {for (j = 1; j <= -P[i]; j++){q = lpMult(q,var(i+n));}} |
---|
922 | I = p - q,I; |
---|
923 | p = q; q = 1; |
---|
924 | } |
---|
925 | |
---|
926 | I = simplify(I,2); |
---|
927 | export(I); |
---|
928 | if (baseringdef == 1) {setring save;} |
---|
929 | return(R); |
---|
930 | } |
---|
931 | example { |
---|
932 | "EXAMPLE:"; echo = 2; |
---|
933 | intvec P = 1,2,3; |
---|
934 | def R = dyckGroup1(3,5,P); setring R; |
---|
935 | I; |
---|
936 | } |
---|
937 | |
---|
938 | |
---|
939 | proc dyckGroup2(int n, int d, intvec P) |
---|
940 | "USAGE: dyckGroup2(n,d,P); n an integer, d an integer, P an intvec |
---|
941 | RETURN: ring |
---|
942 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
943 | @* - The Dyck group with the following presentation |
---|
944 | @* < x_1, x_2, ... , x_n | (x_1)^p1 = (x_2)^p2 = ... = (x_n)^pn = x_1 * x_2 * ... * x_n = 1 > |
---|
945 | @* - negative exponents are allowed |
---|
946 | @* - representation in the form x_i^p_i - 1 |
---|
947 | @* - d gives the degreebound for the Letterplace ring |
---|
948 | @* |
---|
949 | @* This is a family |
---|
950 | @* |
---|
951 | " |
---|
952 | { |
---|
953 | int baseringdef,i,j; |
---|
954 | if (n < 1) {ERROR("There must be at least one variable!");} |
---|
955 | if (d < n) {ERROR("Degreebound is to small!");} |
---|
956 | for (i = 1; i <= size(P); i++) {if (d < absValue(P[i])){ERROR("Degreebound is to small!");}} |
---|
957 | |
---|
958 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
959 | { |
---|
960 | def save = basering; |
---|
961 | baseringdef = 1; |
---|
962 | } |
---|
963 | ring r = 2,(x(1..n),Y(1..n)),dp; |
---|
964 | def R = makeLetterplaceRing(d); |
---|
965 | setring R; |
---|
966 | ideal I; poly p; |
---|
967 | p = 1; |
---|
968 | for (i = 1; i<= n; i++) {p = lpMult(p,var(i));} |
---|
969 | I = p-1; |
---|
970 | for (i = n; i > 0; i--) |
---|
971 | { |
---|
972 | p = 1; |
---|
973 | if (P[i] >= 0) {for (j = 1; j <= P[i]; j++){p = lpMult(p,var(i));}} |
---|
974 | else {for (j = 1; j <= -P[i]; j++){p = lpMult(p,var(i+n));}} |
---|
975 | I = p - 1,I; |
---|
976 | } |
---|
977 | |
---|
978 | I = simplify(I,2); |
---|
979 | export(I); |
---|
980 | if (baseringdef == 1) {setring save;} |
---|
981 | return(R); |
---|
982 | } |
---|
983 | example { |
---|
984 | "EXAMPLE:"; echo = 2; |
---|
985 | intvec P = 1,2,3; |
---|
986 | def R = dyckGroup2(3,5,P); setring R; |
---|
987 | I; |
---|
988 | } |
---|
989 | |
---|
990 | |
---|
991 | |
---|
992 | proc dyckGroup3(int n, int d, intvec P) |
---|
993 | "USAGE: dyckGroup2(n,d,P); n an integer, d an integer, P an intvec |
---|
994 | RETURN: ring |
---|
995 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
996 | @* - The Dyck group with the following presentation |
---|
997 | @* < x_1, x_2, ... , x_n | (x_1)^p1 = (x_2)^p2 = ... = (x_n)^pn = x_1 * x_2 * ... * x_n = 1 > |
---|
998 | @* - only positive exponents are allowed |
---|
999 | @* - no inverse generators needed |
---|
1000 | @* - d gives the degreebound for the Letterplace ring |
---|
1001 | @* |
---|
1002 | @* This is a family |
---|
1003 | @* |
---|
1004 | " |
---|
1005 | { |
---|
1006 | int baseringdef,i,j; |
---|
1007 | if (n < 1) {ERROR("There must be at least one variable!");} |
---|
1008 | if (d < n) {ERROR("Degreebound is to small!");} |
---|
1009 | for (i = 1; i <= size(P); i++) {if (P[i] < 0){ERROR("Exponents must be positive!");}} |
---|
1010 | for (i = 1; i <= size(P); i++) {if (d < P[i]){ERROR("Degreebound is to small!");}} |
---|
1011 | |
---|
1012 | |
---|
1013 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
1014 | { |
---|
1015 | def save = basering; |
---|
1016 | baseringdef = 1; |
---|
1017 | } |
---|
1018 | ring r = 2,x(1..n),dp; |
---|
1019 | def R = makeLetterplaceRing(d); |
---|
1020 | setring R; |
---|
1021 | ideal I; poly p; |
---|
1022 | p = 1; |
---|
1023 | for (i = 1; i<= n; i++) {p = lpMult(p,var(i));} |
---|
1024 | I = p-1; |
---|
1025 | for (i = n; i > 0; i--) |
---|
1026 | { |
---|
1027 | p = 1; |
---|
1028 | for (j = 1; j <= P[i]; j++){p = lpMult(p,var(i));} |
---|
1029 | I = p - 1,I; |
---|
1030 | } |
---|
1031 | |
---|
1032 | I = simplify(I,2); |
---|
1033 | export(I); |
---|
1034 | if (baseringdef == 1) {setring save;} |
---|
1035 | return(R); |
---|
1036 | } |
---|
1037 | example { |
---|
1038 | "EXAMPLE:"; echo = 2; |
---|
1039 | intvec P = 1,2,3; |
---|
1040 | def R = dyckGroup3(3,5,P); setring R; |
---|
1041 | I; |
---|
1042 | } |
---|
1043 | |
---|
1044 | //////////////////////////////////////////////////////////////////// |
---|
1045 | // Fibonacci Group ///////////////////////////////////////////////// |
---|
1046 | // from Grischa Studzinski ///////////////////////////////////////// |
---|
1047 | //////////////////////////////////////////////////////////////////// |
---|
1048 | |
---|
1049 | proc fibonacciGroup(int m, int d) |
---|
1050 | "USAGE: fibonacciGroup(m,d); m an integer, d an integer |
---|
1051 | RETURN: ring |
---|
1052 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
1053 | @* - The Fibonacci group F(2, m) with the following presentation |
---|
1054 | @* < x_1, x_2, ... , x_m | x_i * x_(i + 1) = x_(i + 2) > |
---|
1055 | @* - d gives the degreebound for the Letterplace ring |
---|
1056 | @* |
---|
1057 | @* This is a family |
---|
1058 | @* |
---|
1059 | " |
---|
1060 | // TODO: basefield Q oder F2? |
---|
1061 | // TODO: inverse Elemente! |
---|
1062 | { |
---|
1063 | if (m < 3) {ERROR("At least three generators are required!");} |
---|
1064 | if (d < 2) {ERROR("Degree bound must be at least 2!");} |
---|
1065 | int baseringdef,i; |
---|
1066 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
1067 | { |
---|
1068 | def save = basering; |
---|
1069 | baseringdef = 1; |
---|
1070 | } |
---|
1071 | ring r = 2,(x(1..m),Y(1..m)),dp; |
---|
1072 | def R = makeLetterplaceRing(d); |
---|
1073 | setring R; |
---|
1074 | ideal I; poly p; |
---|
1075 | for (i = 1; i < m-1; i++) |
---|
1076 | { |
---|
1077 | p = lpMult(var(i),var(i+1))-var(i+2); |
---|
1078 | I = I,p; |
---|
1079 | } |
---|
1080 | for (i = 1; i <= m; i++) |
---|
1081 | { |
---|
1082 | p = lpMult(var(i),var(i+m))-1; |
---|
1083 | I = I,p; |
---|
1084 | p = lpMult(var(i+m),var(i))-1; |
---|
1085 | I = I,p; |
---|
1086 | } |
---|
1087 | I = simplify(I,2); |
---|
1088 | export(I); |
---|
1089 | if (baseringdef == 1) {setring save;} |
---|
1090 | return(R); |
---|
1091 | } |
---|
1092 | example { |
---|
1093 | "EXAMPLE:"; echo = 2; |
---|
1094 | def R = fibonacciGroup(3,5); setring R; |
---|
1095 | I; |
---|
1096 | } |
---|
1097 | |
---|
1098 | |
---|
1099 | //////////////////////////////////////////////////////////////////// |
---|
1100 | // Tetrahedron Groups /////////////////////////////////////////////// |
---|
1101 | // from Grischa Studzinski ///////////////////////////////////////// |
---|
1102 | //////////////////////////////////////////////////////////////////// |
---|
1103 | |
---|
1104 | proc tetrahedronGroup(int g, int d) |
---|
1105 | "USAGE: tetrahedronGroup(g,d); g an integer, d an integer |
---|
1106 | RETURN: ring |
---|
1107 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
1108 | @* - g gives the number of the example (1 - 5) |
---|
1109 | @* - d gives the degreebound for the Letterplace ring |
---|
1110 | @* |
---|
1111 | @* This is a family |
---|
1112 | @* |
---|
1113 | The examples are found in |
---|
1114 | Classification of the finite generalized tetrahedron groups |
---|
1115 | by Gerhard Rosenberger and Martin Scheer. |
---|
1116 | The 5 examples are denoted in Proposition 1.9 and concern |
---|
1117 | finite generalized tetrahedron group in the Tsarnarov-case, which are |
---|
1118 | not equivalent to a presentation for an ordinary tetrahedron group. |
---|
1119 | @* |
---|
1120 | " |
---|
1121 | { |
---|
1122 | if (g < 1 || g > 5) {ERROR("There are only 5 examples!");} |
---|
1123 | if ((g == 1 && d < 6)||(g == 2 && d < 6)||(g == 3 && d < 5)||(g == 4 && d < 4)||(g == 5 && d < 5)) |
---|
1124 | {ERROR("Degreebound is to small for choosen example!");} |
---|
1125 | |
---|
1126 | int baseringdef,i,j; |
---|
1127 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
1128 | { |
---|
1129 | def save = basering; |
---|
1130 | baseringdef = 1; |
---|
1131 | } |
---|
1132 | ring r = 2,(x,y,z),dp; |
---|
1133 | def R = makeLetterplaceRing(d); |
---|
1134 | setring R; |
---|
1135 | ideal I; |
---|
1136 | if (g == 1) |
---|
1137 | {I = x(1)*x(2)*x(3)*x(4)*x(5)-1, y(1)*y(2)-1, z(1)*z(2)*z(3)-1, x(1)*y(2)*x(3)*y(4)*x(5)*y(6)-1, x(1)*x(2)*z(3)*x(4)*x(5)*z(6)-1, |
---|
1138 | y(1)*z(2)*y(3)*z(4)-1; |
---|
1139 | } |
---|
1140 | if (g == 2) |
---|
1141 | {I = x(1)*x(2)*x(3)-1, y(1)*y(2)*y(3)-1, z(1)*z(2)*z(3)*z(4)*z(5)-1,x(1)*y(2)*x(3)*y(4)-1,x(1)*z(2)*x(3)*z(4)-1, |
---|
1142 | y(1)*z(2)*z(3)*y(4)*z(5)*z(6)-1; |
---|
1143 | } |
---|
1144 | if (g == 3) |
---|
1145 | {I = x(1)*x(2)*x(3)-1, y(1)*y(2)*y(3)-1, z(1)*z(2)*z(3)-1, x(1)*y(2)*x(3)*y(4)-1, x(1)*z(2)*x(3)*z(4)-1, y(1)*z(2)*y(3)*z(4)-1; |
---|
1146 | } |
---|
1147 | if (g == 4) |
---|
1148 | {I = x(1)*x(2)*x(3)-1, y(1)*y(2)*y(3)-1, z(1)*z(2)*z(3)*z(4)-1,x(1)*y(2)*x(3)*y(4)-1, x(1)*z(2)*x(3)*z(4)-1, y(1)*z(2)*y(3)*z(4)-1; |
---|
1149 | } |
---|
1150 | if (g ==5) |
---|
1151 | {I = x(1)*x(2)*x(3)-1, y(1)*y(2)*y(3)-1, z(1)*z(2)*z(3)*z(4)*z(5)-1,x(1)*y(2)*x(3)*y(4)-1, x(1)*z(2)*x(3)*z(4)-1, y(1)*z(2)*y(3)*z(4)-1; |
---|
1152 | } |
---|
1153 | |
---|
1154 | I = simplify(I,2); |
---|
1155 | export(I); |
---|
1156 | if (baseringdef == 1) {setring save;} |
---|
1157 | return(R); |
---|
1158 | } |
---|
1159 | example { |
---|
1160 | "EXAMPLE:"; echo = 2; |
---|
1161 | def R = tetrahedronGroup(3,5); setring R; |
---|
1162 | I; |
---|
1163 | } |
---|
1164 | |
---|
1165 | |
---|
1166 | //////////////////////////////////////////////////////////////////// |
---|
1167 | // Triangular Groups /////////////////////////////////////////////// |
---|
1168 | // from Grischa Studzinski ///////////////////////////////////////// |
---|
1169 | //////////////////////////////////////////////////////////////////// |
---|
1170 | |
---|
1171 | proc triangularGroup(int g, int d) |
---|
1172 | "USAGE: triangularGroup(g,d); g an integer, d an integer |
---|
1173 | RETURN: ring |
---|
1174 | NOTE: - the ring contains the ideal I, which contains the required relations |
---|
1175 | @* - g gives the number of the example (1 - 14) |
---|
1176 | @* - d gives the degreebound for the Letterplace ring |
---|
1177 | @* |
---|
1178 | @* This is a family |
---|
1179 | @* |
---|
1180 | The examples are found in |
---|
1181 | Classification of the finite generalized tetrahedron groups |
---|
1182 | by Gerhard Rosenberger and Martin Scheer. |
---|
1183 | The 14 examples are denoted in theorem 2.12 |
---|
1184 | @* |
---|
1185 | " |
---|
1186 | { |
---|
1187 | if (g < 1 || g > 14) {ERROR("There are only 14 examples!");} |
---|
1188 | if ((g == 1 && d < 20)||(g == 2 && d < 21)||(g == 3 && d < 10)||(g == 4 && d < 12)||(g == 5 && d < 10)||(g == 6 && d < 18)||(g == 7 && d < 20)||(g == 8 && d < 16)||(g == 9 && d < 10)||(g == 10 && d < 14)||(g == 11 && d < 16)||(g == 12 && d < 24)||(g == 13 && d < 28)||(g == 14 && d < 37)) |
---|
1189 | {ERROR("Degreebound is to small for choosen example!");} |
---|
1190 | |
---|
1191 | int baseringdef; |
---|
1192 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
1193 | { |
---|
1194 | def save = basering; |
---|
1195 | baseringdef = 1; |
---|
1196 | } |
---|
1197 | ring r = 2,(a,b),dp; |
---|
1198 | def R = makeLetterplaceRing(d); |
---|
1199 | setring R; |
---|
1200 | ideal I; |
---|
1201 | |
---|
1202 | if (g == 1) |
---|
1203 | {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, |
---|
1204 | a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*a(8)*b(9)*b(10)*a(11)*b(12)*a(13)*b(14)*a(15)*b(16)*b(17)*a(18)*b(19)*b(20)-1; |
---|
1205 | } |
---|
1206 | if (g == 2) |
---|
1207 | {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, |
---|
1208 | a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*a(8)*b(9)*a(10)*b(11)*a(12)*b(13)*b(14)*a(15)*b(16)*a(17)*b(18)*a(19)*b(20)*b(21)-1; |
---|
1209 | } |
---|
1210 | if (g == 3) |
---|
1211 | {I = a(1)*a(2)*a(3)-1, b(1)*b(2)*b(3)-1, |
---|
1212 | a(1)*b(2)*a(3)*b(4)*b(5)*a(6)*b(7)*a(8)*b(9)*b(10)-1; |
---|
1213 | } |
---|
1214 | if (g == 4) |
---|
1215 | {I = a(1)*a(2)*a(3)-1, b(1)*b(2)*b(3)-1, |
---|
1216 | a(1)*b(2)*a(3)*a(4)*b(5)*b(6)*a(7)*b(8)*a(9)*a(10)*b(11)*b(12)-1; |
---|
1217 | } |
---|
1218 | if (g == 5) |
---|
1219 | {I = a(1)*a(2)-1, b(1)*b(2)*b(3)*b(4)*b(5)-1, |
---|
1220 | a(1)*b(2)*a(3)*b(4)*b(5)*a(6)*b(7)*a(8)*b(9)*b(10)-1; |
---|
1221 | } |
---|
1222 | if (g == 6) |
---|
1223 | {I = a(1)*a(2)-1, b(1)*b(2)*b(3)*b(4)*b(5)-1, |
---|
1224 | a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*b(8)*b(9)*a(10)*b(11)*a(12)*b(13)*a(14)*b(15)*b(16)*b(17)*b(18)-1; |
---|
1225 | } |
---|
1226 | if (g == 7) |
---|
1227 | {I = a(1)*a(2)-1, b(1)*b(2)*b(3)*b(4)*b(5)-1, |
---|
1228 | a(1)*b(2)*a(3)*b(4)*b(5)*a(6)*b(7)*b(8)*b(9)*b(10)*a(11)*b(12)*a(13)*b(14)*b(15)*a(16)*b(17)*b(18)*b(19)*b(20)-1; |
---|
1229 | } |
---|
1230 | if (g == 8) |
---|
1231 | {I = a(1)*a(2)-1, b(1)*b(2)*b(3)*b(4)-1, |
---|
1232 | a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*b(8)*a(9)*b(10)*a(11)*b(12)*a(13)*b(14)*b(15)*b(16)-1; |
---|
1233 | } |
---|
1234 | if (g == 9) |
---|
1235 | {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, |
---|
1236 | a(1)*b(2)*a(3)*b(4)*b(5)*a(6)*b(7)*a(8)*b(9)*b(10)-1; |
---|
1237 | } |
---|
1238 | if (g == 10) |
---|
1239 | {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, |
---|
1240 | a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*a(8)*b(9)*a(10)*b(11)*a(12)*b(13)*b(14)-1; |
---|
1241 | } |
---|
1242 | if (g == 11) |
---|
1243 | {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, |
---|
1244 | a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*a(7)*b(8)*b(9)*a(10)*b(11)*a(12)*b(13)*a(14)*b(15)*b(16)-1; |
---|
1245 | } |
---|
1246 | if (g == 12) |
---|
1247 | {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, |
---|
1248 | a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*a(8)*b(9)*a(10)*b(11)*b(12)*a(13)*b(14)*a(15)*b(16)*a(17)*b(18)*b(19)*a(20)*b(21)*a(22)*b(23)*b(24)-1; |
---|
1249 | } |
---|
1250 | if (g == 13) |
---|
1251 | {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, |
---|
1252 | a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*a(7)*b(8)*a(9)*b(10)*b(11)*a(12)*b(13)*b(14)*a(15)*b(16)*a(17)*b(18)*a(19)*b(20)*a(21)*b(22)*a(23)*b(24)*b(25)*a(26)*b(27)*b(28)-1; |
---|
1253 | } |
---|
1254 | if (g == 14) |
---|
1255 | {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1, |
---|
1256 | a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*a(7)*b(8)*b(9)*a(10)*b(11)*b(12)*a(13)*b(14)*a(15)*b(16)*b(17)*a(18)*b(19)*b(20)*a(21)*b(22)*a(23)*b(24)*a(25)*b(26)*a(27)*b(28)*b(29)*a(30)*b(31)*a(32)*b(33)*b(34)*a(35)*b(36)*b(37)-1; |
---|
1257 | } |
---|
1258 | |
---|
1259 | I = simplify(I,2); |
---|
1260 | export(I); |
---|
1261 | if (baseringdef == 1) {setring save;} |
---|
1262 | return(R); |
---|
1263 | } |
---|
1264 | example { |
---|
1265 | "EXAMPLE:"; echo = 2; |
---|
1266 | def R = triangularGroup(3,10); setring R; |
---|
1267 | I; |
---|
1268 | } |
---|