1 | //////////////////////////////////////////////////////////////// |
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2 | version="version lrcalc.lib 4.1.2.0 Feb_2019 "; //$Id$ |
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3 | category="Littlewood-Richardson coefficients"; |
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4 | info=" |
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5 | LIBRARY: lrcalc.lib An interface to the Littlewood-Richardson Calculator by Anders Buch |
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6 | |
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7 | AUTHOR: Oleksandr Iena, o.g.yena@gmail.com |
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8 | |
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9 | OVERVIEW: |
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10 | An interface to the documented functions of the Littlewood-Richardson Calculator |
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11 | by Anders Buch is implemented. |
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12 | The library requires the Littlewood-Richardson Calculator by Anders Buch, |
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13 | which is available at http://math.rutgers.edu/~asbuch/lrcalc/ |
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14 | |
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15 | REFERENCES: |
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16 | [1] http://math.rutgers.edu/~asbuch/lrcalc/ |
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17 | http://math.rutgers.edu/~asbuch/lrcalc/lrcalc-1.2/README |
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18 | |
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19 | PROCEDURES: |
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20 | LRinstall() installs the Littlewood-Richardson Calculator |
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21 | LRcoef(z, x, y); Littlewood-Richardson coefficient c^z_{x, y} |
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22 | LRskew(z, x) partitions y for which the Littlewood-Richardson coefficient |
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23 | c^z_{x,y} is non-zero together with that coefficient |
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24 | LRmult(x, y) partitions z for which the Littlewood-Richardson coefficient |
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25 | c^z_{x,y} is non-zero together with that coefficient |
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26 | LRcoprod(z) pairs of partitions x and y for which the Littlewood-Richardson |
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27 | coefficient c^z_{x,y} is non-zero together with that coefficient |
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28 | LRschubmult(x, y) expantion of a product of two Schubert polynomials |
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29 | in the basis of Schubert polynomials |
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30 | "; |
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31 | //---------------------------------------------------------- |
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32 | |
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33 | proc LRinstall() |
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34 | "USAGE: LRinstall(); |
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35 | RETURN: int (exit status of the shell) |
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36 | PURPOSE: installs the Littlewood-Richardson Calculator |
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37 | EXAMPLE: example LRinstall; shows an example |
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38 | NOTE: |
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39 | " |
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40 | { |
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41 | int i; |
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42 | string s; |
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43 | s = "wget math.rutgers.edu/~asbuch/lrcalc/lrcalc-1.2.tar.gz"; |
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44 | s = s + " && tar zxvf lrcalc-1.2.tar.gz && cd lrcalc-1.2"; |
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45 | s = s + " && ./configure && make && sudo make install"; |
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46 | i=system("sh", s); |
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47 | return(i); |
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48 | } |
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49 | example |
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50 | { |
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51 | "EXAMPLE:"; echo = 2; |
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52 | // In order to install the Littlewood-Richardson Calculator |
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53 | // type "LRinstall();" |
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54 | // This will execute the following commands: |
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55 | // wget math.rutgers.edu/~asbuch/lrcalc/lrcalc-1.2.tar.gz |
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56 | // tar zxvf lrcalc-1.2.tar.gz |
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57 | // cd lrcalc-1.2 |
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58 | // ./configure |
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59 | // make |
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60 | // sudo make install |
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61 | } |
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62 | //---------------------------------------------------------- |
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63 | |
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64 | proc LRcoef(list u, list l1, list l2) |
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65 | "USAGE: LRcoef(z, x, y); z, x, y lists of integers (partitions) |
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66 | RETURN: bigint |
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67 | PURPOSE: computes the Littlewood-Richardson coefficient c^z_{x, y} |
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68 | EXAMPLE: example LRcoef; shows an example |
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69 | NOTE: |
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70 | " |
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71 | { |
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72 | // construct a string with the required lrcalc command to be passed to shell |
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73 | string s="lrcalc coef"; |
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74 | int i; |
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75 | int sz; |
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76 | sz=size(u); |
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77 | for(i=1; i<=sz; i++) |
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78 | { |
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79 | s=s+" "+string(u[i]); |
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80 | } |
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81 | s=s+" -"; |
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82 | sz=size(l1); |
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83 | for(i=1; i<=sz; i++) |
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84 | { |
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85 | s=s+" "+string(l1[i]); |
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86 | } |
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87 | s=s+" -"; |
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88 | sz=size(l2); |
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89 | for(i=1; i<=sz; i++) |
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90 | { |
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91 | s=s+" "+string(l2[i]); |
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92 | } |
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93 | s=read("|: "+s); // execute the string in shell and return the output string back to Singular |
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94 | return( string2int(s) ); // return the integer represented by this string |
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95 | } |
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96 | example |
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97 | { |
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98 | "EXAMPLE:"; echo = 2; |
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99 | // Compute the Littlewood-Richardson coefficient c^z_{x, y} |
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100 | // for z= (3, 2, 1), x=(2, 1), y=(2, 1) |
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101 | list z = 3, 2, 1; |
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102 | list x = 2, 1; |
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103 | list y = 2, 1; |
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104 | LRcoef(z, x, y); |
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105 | } |
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106 | //---------------------------------------------------------- |
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107 | |
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108 | proc LRskew(list I, list J, list #) |
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109 | "USAGE: LRskew(z, x [,s, r]); z, x lists of integers (partitions) |
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110 | s string equal to 'r', r non-negative integer |
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111 | RETURN: list of lists |
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112 | PURPOSE: computes the partitions y for which the Littlewood-Richardson |
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113 | coefficient c^z_{x,y} is non-zero together with that coefficient; |
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114 | only partitions up to length r are computed |
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115 | if the optional parameters age given |
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116 | EXAMPLE: example LRskew; shows an example |
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117 | NOTE: |
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118 | " |
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119 | { |
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120 | // construct a string with the required lrcalc command to be passed to shell |
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121 | string s="lrcalc skew"; |
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122 | int sz; |
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123 | // take care of the optional parameters |
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124 | sz = size(#); |
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125 | if(sz!=0) // if there are optional parameters |
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126 | { |
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127 | if(typeof(#[1])=="string") // if the first optional parameter is a string |
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128 | { |
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129 | if(#[1]=="r") // if it equals "r" |
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130 | { |
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131 | if(sz>1) // if there is a second optional parameter |
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132 | { |
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133 | if(typeof(#[2])=="int") // that is integer |
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134 | { |
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135 | if(#[2]>=0) // and non-negative |
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136 | { |
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137 | s=s+ " -r "+string(#[2]); // add the corresponding string to the lrcalc command |
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138 | } |
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139 | } |
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140 | } |
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141 | } |
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142 | } |
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143 | } |
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144 | int i; |
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145 | sz=size(I); |
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146 | for(i=1; i<=sz; i++) |
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147 | { |
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148 | s=s+" "+string(I[i]); |
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149 | } |
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150 | s=s+" /"; |
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151 | sz=size(J); |
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152 | for(i=1; i<=sz; i++) |
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153 | { |
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154 | s=s+" "+string(J[i]); |
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155 | } |
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156 | // execute the string in shell and return the output string back to Singular |
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157 | link L="|: "+s+" && echo end"; |
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158 | list rez; // the result will be computed here |
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159 | list T; |
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160 | int next; |
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161 | while(1) |
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162 | { |
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163 | s=read(L); |
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164 | if(s=="end") |
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165 | { |
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166 | break; |
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167 | } |
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168 | i=1; |
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169 | next=find(s," ",i); |
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170 | if(next ==0){break;} |
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171 | T=list(string2int(s[i,next-i])); |
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172 | i=next+1; |
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173 | next=find(s, "(", i); |
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174 | i=next+1; |
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175 | next=find(s, ")", i); |
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176 | T= T+list( string2list(s[i,next-i]) ); |
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177 | rez=rez+ list(T); |
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178 | } |
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179 | close(L); |
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180 | return( rez ); // return the result |
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181 | } |
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182 | example |
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183 | { |
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184 | "EXAMPLE:"; echo = 2; |
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185 | // Compute the partitions y for which the Littlewood-Richardson coefficient |
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186 | // c^z_{x,y} is non-zero together with that coefficient |
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187 | // for z= (3, 2, 1), x=(2, 1) |
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188 | list z = 3, 2, 1; |
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189 | list x = 2, 1; |
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190 | LRskew(z, x); |
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191 | // Now compute only the partitions with at most 2 entries |
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192 | LRskew(z, x, "r", 2); |
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193 | } |
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194 | //---------------------------------------------------------- |
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195 | |
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196 | proc LRmult(list I, list J, list #) |
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197 | "USAGE: LRmult(x, y); x, y lists of integers (partitions) |
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198 | LRmult(x, y [, s, r]); x, y lists of integers (partitions), |
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199 | s string equal to 'r', r integer |
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200 | LRmult(x, y [, s, m, k]); x, y lists of integers (partitions), |
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201 | s string equal to 'q' or 'f', m, k integers |
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202 | RETURN: list of lists |
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203 | PURPOSE: computes the partitions z for which the Littlewood-Richardson |
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204 | coefficient c^z_{x,y} is non-zero together with that coefficient; |
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205 | partitions up to length r |
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206 | EXAMPLE: example LRmult; shows an example |
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207 | NOTE: |
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208 | " |
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209 | { |
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210 | // construct a string with the required lrcalc command to be passed to shell |
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211 | string s="lrcalc mult"; |
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212 | int i; |
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213 | int sz; |
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214 | // take care of the optional parameters |
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215 | sz = size(#); |
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216 | if(sz!=0) // if there are optional parameters |
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217 | { |
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218 | if(typeof(#[1])=="string") // if the first optional parameter is a string |
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219 | { |
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220 | if(#[1]=="r") // if the first optional parameter is "r" |
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221 | { |
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222 | if(sz>1) // if there is a second optional parameter |
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223 | { |
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224 | if(typeof(#[2])=="int") // which is an integer |
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225 | { |
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226 | if(#[2]>=0) // and non-negative |
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227 | { |
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228 | s=s+ " -r "+string(#[2]); // add the corresponding string to the lrcalc command |
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229 | } |
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230 | } |
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231 | } |
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232 | } |
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233 | if( (#[1]=="q") || (#[1]=="f") ) // if the first optional parameter is "q" or "f" |
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234 | { |
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235 | if(sz>2) // if there are a second and a third parameters |
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236 | { |
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237 | if( ( typeof(#[2])=="int" ) && ( typeof(#[3])=="int" ) ) // that are integers |
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238 | { |
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239 | if( (#[2]>0)&&(#[3]>0) ) // and positive |
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240 | { |
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241 | // add the corresponding string to the lrcalc command |
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242 | s=s+ " -"+#[1]+" "+string(#[2])+","+string(#[3]); |
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243 | } |
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244 | } |
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245 | } |
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246 | } |
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247 | } |
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248 | } |
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249 | sz=size(I); |
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250 | for(i=1; i<=sz; i++) |
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251 | { |
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252 | s=s+" "+string(I[i]); |
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253 | } |
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254 | s=s+" -"; |
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255 | sz=size(J); |
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256 | for(i=1; i<=sz; i++) |
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257 | { |
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258 | s=s+" "+string(J[i]); |
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259 | } |
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260 | // execute the string in shell and return the output string back to Singular |
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261 | link L="|: "+s+" && echo end"; |
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262 | list rez; // the result will be computed here |
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263 | list T; |
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264 | int next; |
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265 | while(1) |
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266 | { |
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267 | s=read(L); |
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268 | if(s=="end") |
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269 | { |
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270 | break; |
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271 | } |
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272 | i=1; |
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273 | next=find(s," ",i); |
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274 | if(next ==0){break;} |
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275 | T=list(string2int(s[i,next-i])); |
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276 | i=next+1; |
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277 | next=find(s, "(", i); |
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278 | i=next+1; |
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279 | next=find(s, ")", i); |
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280 | T= T+list( string2list(s[i,next-i]) ); |
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281 | rez=rez+ list(T); |
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282 | } |
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283 | close(L); |
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284 | return( rez ); // return the result |
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285 | } |
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286 | example |
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287 | { |
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288 | "EXAMPLE:"; echo = 2; |
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289 | // Compute the partitions z for which the Littlewood-Richardson coefficient |
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290 | // c^z_{x,y} is non-zero together with that coefficient |
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291 | // for x= (2, 1), y=(2, 1) |
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292 | list x = 2, 1; |
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293 | list y = 2, 1; |
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294 | LRmult(x, y); |
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295 | // Now compute only the partitions with at most 2 entries |
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296 | LRmult(x, y, "r", 2); |
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297 | // Now compute the product in the quantum cohomology ring of the Grassmannian Gr(3,3+2). |
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298 | LRmult(x, y, "q", 3, 2); |
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299 | // Compute the same product with the output given in fusion ring notation |
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300 | LRmult(x, y, "f", 3, 2); |
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301 | } |
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302 | //---------------------------------------------------------- |
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303 | |
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304 | proc LRcoprod(list I) |
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305 | "USAGE: LRcoprod(z); z list of integers (partition) |
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306 | RETURN: list of lists |
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307 | PURPOSE: computes the pairs of partitions x and y for which |
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308 | the Littlewood-Richardson coefficient c^z_{x,y} is non-zero |
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309 | together with that coefficient |
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310 | EXAMPLE: example LRcoprod; shows an example |
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311 | NOTE: |
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312 | " |
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313 | { |
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314 | // construct a string with the required lrcalc command to be passed to shell |
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315 | string s="lrcalc coprod"; |
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316 | int i; |
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317 | int sz; |
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318 | sz=size(I); |
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319 | for(i=1; i<=sz; i++) |
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320 | { |
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321 | s=s+" "+string(I[i]); |
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322 | } |
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323 | // execute the string in shell and return the output string back to Singular |
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324 | link L="|: "+s+" && echo end"; |
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325 | list rez; // the result will be computed here |
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326 | list T; |
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327 | int next; |
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328 | while(1) |
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329 | { |
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330 | s=read(L); |
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331 | if(s=="end") |
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332 | { |
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333 | break; |
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334 | } |
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335 | i=1; |
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336 | next=find(s," ",i); |
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337 | if(next ==0){break;} |
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338 | T=list(string2int(s[i,next-i])); |
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339 | i=next+1; |
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340 | next=find(s, "(", i); |
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341 | i=next+1; |
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342 | next=find(s, ")", i); |
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343 | T= T+list( string2list(s[i,next-i]) ); |
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344 | i=next+1; |
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345 | next=find(s, "(", i); |
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346 | i=next+1; |
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347 | next=find(s, ")", i); |
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348 | T= T+list( string2list(s[i,next-i]) ); |
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349 | rez=rez+ list(T); |
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350 | } |
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351 | close(L); |
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352 | return( rez ); // return the result |
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353 | } |
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354 | example |
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355 | { |
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356 | "EXAMPLE:"; echo = 2; |
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357 | // Compute the pairs of partitions x and y for which the Littlewood-Richardson |
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358 | // coefficient c^z_{x,y} is non-zero together with that coefficient |
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359 | // for z= (3, 2, 1) |
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360 | list z = 3, 2, 1; |
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361 | LRcoprod(z); |
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362 | } |
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363 | //---------------------------------------------------------- |
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364 | |
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365 | proc LRschubmult(list I, list J) |
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366 | "USAGE: LRschubmult(x, y); x, y lists of integers |
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367 | RETURN: list of lists |
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368 | PURPOSE: computes the expantion of a product |
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369 | of two Schubert polynomials in the basis of Schubert polynomials |
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370 | EXAMPLE: example LRschubmult; shows an example |
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371 | NOTE: |
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372 | " |
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373 | { |
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374 | // construct a string with the required lrcalc command to be passed to shell |
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375 | string s="schubmult"; |
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376 | int i; |
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377 | int sz; |
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378 | sz=size(I); |
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379 | for(i=1; i<=sz; i++) |
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380 | { |
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381 | s=s+" "+string(I[i]); |
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382 | } |
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383 | s=s+" -"; |
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384 | sz=size(J); |
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385 | for(i=1; i<=sz; i++) |
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386 | { |
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387 | s=s+" "+string(J[i]); |
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388 | } |
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389 | // execute the string in shell and return the output string back to Singular |
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390 | link L="|: "+s+" && echo end"; |
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391 | list rez; // the result will be computed here |
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392 | list T; |
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393 | int next; |
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394 | while(1) |
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395 | { |
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396 | s=read(L); |
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397 | if(s=="end") |
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398 | { |
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399 | break; |
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400 | } |
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401 | i=1; |
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402 | next=find(s," ",i); |
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403 | if(next ==0){break;} |
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404 | T=list(string2int(s[i,next-i])); |
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405 | i=next+1; |
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406 | next=find(s, "(", i); |
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407 | i=next+1; |
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408 | next=find(s, ")", i); |
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409 | T= T+list( string2list(s[i,next-i]) ); |
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410 | rez=rez+ list(T); |
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411 | } |
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412 | close(L); |
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413 | return( rez ); // return the result |
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414 | } |
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415 | example |
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416 | { |
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417 | "EXAMPLE:"; echo = 2; |
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418 | // Compute the expantion of a square of the Schubert polynomial |
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419 | // corresponding to (1 3 2) in the basis of Schubert polynomials |
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420 | list x = 1, 3, 2; |
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421 | LRschubmult(x, x); |
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422 | } |
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423 | //---------------------------------------------------------------------------------------- |
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424 | // The procedures below are for the internal usage only |
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425 | //---------------------------------------------------------------------------------------- |
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426 | |
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427 | static proc string2list(string s) |
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428 | "USAGE: string2list(s); s string |
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429 | RETURN: list of integers |
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430 | PURPOSE: converts a string representing integers separated by commas |
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431 | into a list of integers |
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432 | EXAMPLE: example string2list; shows an example |
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433 | NOTE: |
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434 | " |
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435 | { |
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436 | list l; // the result will be computed here |
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437 | if(size(s)==0) // if the string is empty |
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438 | { |
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439 | return(list(0)); // return zero |
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440 | } |
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441 | // otherwise form the corresponding list |
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442 | execute("l="+s+";") |
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443 | return(l); // return the result |
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444 | } |
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445 | example |
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446 | { |
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447 | "EXAMPLE:"; echo = 2; |
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448 | // Convert the string "3, 2, 1" into the corresponding list of integers |
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449 | string s= "3, 2, 1"; |
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450 | string2list(s); |
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451 | } |
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452 | //---------------------------------------------------------- |
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453 | |
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454 | static proc string2int(string s) |
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455 | "USAGE: string2int(s); s string |
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456 | RETURN: biging |
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457 | PURPOSE: converts a string representing a non-negative integer into integer |
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458 | EXAMPLE: example string2int; shows an example |
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459 | NOTE: |
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460 | " |
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461 | { |
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462 | bigint rez; |
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463 | int sz=size(s); |
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464 | if(sz==0) // if the string s is empty, return zero |
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465 | { |
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466 | return(bigint(0)); |
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467 | } |
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468 | // read the first character of the string and transform it to the corresponding digit |
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469 | while(1) |
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470 | { |
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471 | if(s[1]=="0") |
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472 | { |
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473 | rez=0; break; |
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474 | } |
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475 | if(s[1]=="1") |
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476 | { |
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477 | rez=1; break; |
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478 | } |
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479 | if(s[1]=="2") |
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480 | { |
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481 | rez=2; break; |
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482 | } |
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483 | if(s[1]=="3") |
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484 | { |
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485 | rez=3; break; |
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486 | } |
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487 | if(s[1]=="4") |
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488 | { |
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489 | rez=4; break; |
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490 | } |
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491 | if(s[1]=="5") |
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492 | { |
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493 | rez=5; break; |
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494 | } |
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495 | if(s[1]=="6") |
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496 | { |
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497 | rez=6; break; |
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498 | } |
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499 | if(s[1]=="7") |
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500 | { |
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501 | rez=7; break; |
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502 | } |
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503 | if(s[1]=="8") |
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504 | { |
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505 | rez=8; break; |
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506 | } |
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507 | if(s[1]=="9") |
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508 | { |
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509 | rez=9; break; |
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510 | } |
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511 | } |
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512 | if(sz==1) // if the string is of length 1 |
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513 | { |
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514 | return(bigint(rez)); // return the result |
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515 | } |
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516 | // otherwise compute the result recursively |
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517 | return( rez*bigint(10)^(sz-1) + string2int(s[2,sz-1]) ); |
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518 | } |
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519 | example |
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520 | { |
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521 | "EXAMPLE:"; echo = 2; |
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522 | // Convert the string "728" into the corresponding integer |
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523 | string s= "728"; |
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524 | string2int(s); |
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525 | } |
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526 | //---------------------------------------------------------- |
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