/usr/share/gap/doc/ref/chap44.html is in gap-doc 4r8p8-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 | <?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (ref) - Chapter 44: Matrix Groups</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap44" onload="jscontent()">
<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chap26.html">26</a> <a href="chap27.html">27</a> <a href="chap28.html">28</a> <a href="chap29.html">29</a> <a href="chap30.html">30</a> <a href="chap31.html">31</a> <a href="chap32.html">32</a> <a href="chap33.html">33</a> <a href="chap34.html">34</a> <a href="chap35.html">35</a> <a href="chap36.html">36</a> <a href="chap37.html">37</a> <a href="chap38.html">38</a> <a href="chap39.html">39</a> <a href="chap40.html">40</a> <a href="chap41.html">41</a> <a href="chap42.html">42</a> <a href="chap43.html">43</a> <a href="chap44.html">44</a> <a href="chap45.html">45</a> <a href="chap46.html">46</a> <a href="chap47.html">47</a> <a href="chap48.html">48</a> <a href="chap49.html">49</a> <a href="chap50.html">50</a> <a href="chap51.html">51</a> <a href="chap52.html">52</a> <a href="chap53.html">53</a> <a href="chap54.html">54</a> <a href="chap55.html">55</a> <a href="chap56.html">56</a> <a href="chap57.html">57</a> <a href="chap58.html">58</a> <a href="chap59.html">59</a> <a href="chap60.html">60</a> <a href="chap61.html">61</a> <a href="chap62.html">62</a> <a href="chap63.html">63</a> <a href="chap64.html">64</a> <a href="chap65.html">65</a> <a href="chap66.html">66</a> <a href="chap67.html">67</a> <a href="chap68.html">68</a> <a href="chap69.html">69</a> <a href="chap70.html">70</a> <a href="chap71.html">71</a> <a href="chap72.html">72</a> <a href="chap73.html">73</a> <a href="chap74.html">74</a> <a href="chap75.html">75</a> <a href="chap76.html">76</a> <a href="chap77.html">77</a> <a href="chap78.html">78</a> <a href="chap79.html">79</a> <a href="chap80.html">80</a> <a href="chap81.html">81</a> <a href="chap82.html">82</a> <a href="chap83.html">83</a> <a href="chap84.html">84</a> <a href="chap85.html">85</a> <a href="chap86.html">86</a> <a href="chap87.html">87</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap43.html">[Previous Chapter]</a> <a href="chap45.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap44_mj.html">[MathJax on]</a></p>
<p><a id="X7CF51CB48610A07D" name="X7CF51CB48610A07D"></a></p>
<div class="ChapSects"><a href="chap44.html#X7CF51CB48610A07D">44 <span class="Heading">Matrix Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X86CEA60E7C04744C">44.1 <span class="Heading">IsMatrixGroup (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7E6093FF85F1C3A1">44.1-1 IsMatrixGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X7FD808E386FAF9B0">44.2 <span class="Heading">Attributes and Properties for Matrix Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7E55258C783C50CA">44.2-1 DimensionOfMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7D540083793CD496">44.2-2 DefaultFieldOfMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X78A9F0E580DA613A">44.2-3 FieldOfMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X832D18C77ED608DE">44.2-4 TransposedMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X84B36A827E5EFC35">44.2-5 IsFFEMatrixGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X7F4B0B397AAC7659">44.3 <span class="Heading">Actions of Matrix Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7BD4F38E8624735D">44.3-1 ProjectiveActionOnFullSpace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7F8EA8D583C1E9B2">44.3-2 ProjectiveActionHomomorphismMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X849C451A80B4A210">44.3-3 BlowUpIsomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X7934EED77891BE6B">44.4 <span class="Heading">GL and SL</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X781387AF7999EA99">44.4-1 IsGeneralLinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X86F9A27D7AFAEB5A">44.4-2 IsNaturalGL</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X816677CD821261FA">44.4-3 IsSpecialLinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X84134F08781EB943">44.4-4 IsNaturalSL</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7ED43D4F7E993A31">44.4-5 IsSubgroupSL</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X7CA4097C79F5BD90">44.5 <span class="Heading">Invariant Forms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7C08385A81AB05E1">44.5-1 InvariantBilinearForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X8652FBF781940AC3">44.5-2 IsFullSubgroupGLorSLRespectingBilinearForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X82F22079852130C9">44.5-3 InvariantSesquilinearForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7B35A8AF7D8F0313">44.5-4 IsFullSubgroupGLorSLRespectingSesquilinearForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7BCACC007EB9B613">44.5-5 InvariantQuadraticForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X84AB04A67DFC0274">44.5-6 IsFullSubgroupGLorSLRespectingQuadraticForm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X7FB0138F79E8C5E7">44.6 <span class="Heading">Matrix Groups in Characteristic 0</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X850821F78558C829">44.6-1 IsCyclotomicMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7FEDB2E17EE02674">44.6-2 IsRationalMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7F737FC4795F3E48">44.6-3 IsIntegerMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X86F9CC1E7DB97CB6">44.6-4 IsNaturalGLnZ</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7B0E70127F5D2EAF">44.6-5 IsNaturalSLnZ</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7DE412A37A6975B3">44.6-6 InvariantLattice</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7CC4D6DC81739698">44.6-7 NormalizerInGLnZ</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7DAFB71F86525DE7">44.6-8 CentralizerInGLnZ</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X8217762A863F1382">44.6-9 ZClassRepsQClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X84FD9FC97FB90795">44.6-10 IsBravaisGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7AAE301C83116451">44.6-11 BravaisGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X788C7D9C7C2301C5">44.6-12 BravaisSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7F5FF1A481E08AD5">44.6-13 BravaisSupergroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X79B7CD797A420720">44.6-14 NormalizerInGLnZBravaisGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X868288377CFA8D1B">44.7 <span class="Heading">Acting OnRight and OnLeft</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X7D1318A6780CD88B">44.7-1 CrystGroupDefaultAction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap44.html#X792D237385977BE6">44.7-2 SetCrystGroupDefaultAction</a></span>
</div></div>
</div>
<h3>44 <span class="Heading">Matrix Groups</span></h3>
<p>Matrix groups are groups generated by invertible square matrices.</p>
<p>In the following example we temporarily increase the line length limit from its default value 80 to 83 in order to get a nicer output format.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m1 := [ [ Z(3)^0, Z(3)^0, Z(3) ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ Z(3), 0*Z(3), Z(3) ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0*Z(3), Z(3), 0*Z(3) ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m2 := [ [ Z(3), Z(3), Z(3)^0 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ Z(3), 0*Z(3), Z(3) ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ Z(3)^0, 0*Z(3), Z(3) ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := Group( m1, m2 );</span>
Group(
[
[ [ Z(3)^0, Z(3)^0, Z(3) ], [ Z(3), 0*Z(3), Z(3) ],
[ 0*Z(3), Z(3), 0*Z(3) ] ],
[ [ Z(3), Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3) ],
[ Z(3)^0, 0*Z(3), Z(3) ] ] ])
</pre></div>
<p><a id="X86CEA60E7C04744C" name="X86CEA60E7C04744C"></a></p>
<h4>44.1 <span class="Heading">IsMatrixGroup (Filter)</span></h4>
<p>For most operations, <strong class="pkg">GAP</strong> only provides methods for finite matrix groups. Many calculations in finite matrix groups are done via so-called "nice monomorphisms" (see Section <a href="chap40.html#X7FFD731684606BC6"><span class="RefLink">40.5</span></a>) that represent a faithful action on vectors.</p>
<p><a id="X7E6093FF85F1C3A1" name="X7E6093FF85F1C3A1"></a></p>
<h5>44.1-1 IsMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMatrixGroup</code>( <var class="Arg">grp</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>The category of matrix groups.</p>
<p><a id="X7FD808E386FAF9B0" name="X7FD808E386FAF9B0"></a></p>
<h4>44.2 <span class="Heading">Attributes and Properties for Matrix Groups</span></h4>
<p><a id="X7E55258C783C50CA" name="X7E55258C783C50CA"></a></p>
<h5>44.2-1 DimensionOfMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DimensionOfMatrixGroup</code>( <var class="Arg">mat-grp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The dimension of the matrix group.</p>
<p><a id="X7D540083793CD496" name="X7D540083793CD496"></a></p>
<h5>44.2-2 DefaultFieldOfMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DefaultFieldOfMatrixGroup</code>( <var class="Arg">mat-grp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Is a field containing all the matrix entries. It is not guaranteed to be the smallest field with this property.</p>
<p><a id="X78A9F0E580DA613A" name="X78A9F0E580DA613A"></a></p>
<h5>44.2-3 FieldOfMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FieldOfMatrixGroup</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The smallest field containing all the matrix entries of all elements of the matrix group <var class="Arg">matgrp</var>. As the calculation of this can be hard, this should only be used if one <em>really</em> needs the smallest field, use <code class="func">DefaultFieldOfMatrixGroup</code> (<a href="chap44.html#X7D540083793CD496"><span class="RefLink">44.2-2</span></a>) to get (for example) the characteristic.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DimensionOfMatrixGroup(m);</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">DefaultFieldOfMatrixGroup(m);</span>
GF(3)
</pre></div>
<p><a id="X832D18C77ED608DE" name="X832D18C77ED608DE"></a></p>
<h5>44.2-4 TransposedMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransposedMatrixGroup</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the transpose of the matrix group <var class="Arg">matgrp</var>. The transpose of the transpose of <var class="Arg">matgrp</var> is identical to <var class="Arg">matgrp</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := Group( [[0,-1],[1,0]] );</span>
Group([ [ [ 0, -1 ], [ 1, 0 ] ] ])
<span class="GAPprompt">gap></span> <span class="GAPinput">T := TransposedMatrixGroup( G );</span>
Group([ [ [ 0, 1 ], [ -1, 0 ] ] ])
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIdenticalObj( G, TransposedMatrixGroup( T ) );</span>
true
</pre></div>
<p><a id="X84B36A827E5EFC35" name="X84B36A827E5EFC35"></a></p>
<h5>44.2-5 IsFFEMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFFEMatrixGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>tests whether all matrices in <var class="Arg">G</var> have finite field element entries.</p>
<p><a id="X7F4B0B397AAC7659" name="X7F4B0B397AAC7659"></a></p>
<h4>44.3 <span class="Heading">Actions of Matrix Groups</span></h4>
<p>The basic operations for groups are described in Chapter <a href="chap41.html#X87115591851FB7F4"><span class="RefLink">41</span></a>, special actions for <em>matrix</em> groups mentioned there are <code class="func">OnLines</code> (<a href="chap41.html#X86DC2DD5829CAD9A"><span class="RefLink">41.2-12</span></a>), <code class="func">OnRight</code> (<a href="chap41.html#X7960924D84B5B18F"><span class="RefLink">41.2-2</span></a>), and <code class="func">OnSubspacesByCanonicalBasis</code> (<a href="chap41.html#X85124D197F0F9C4D"><span class="RefLink">41.2-15</span></a>).</p>
<p>For subtleties concerning multiplication from the left or from the right, see <a href="chap44.html#X868288377CFA8D1B"><span class="RefLink">44.7</span></a>.</p>
<p><a id="X7BD4F38E8624735D" name="X7BD4F38E8624735D"></a></p>
<h5>44.3-1 ProjectiveActionOnFullSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveActionOnFullSpace</code>( <var class="Arg">G</var>, <var class="Arg">F</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be a group of <var class="Arg">n</var> by <var class="Arg">n</var> matrices over a field contained in the finite field <var class="Arg">F</var>. <code class="func">ProjectiveActionOnFullSpace</code> returns the image of the projective action of <var class="Arg">G</var> on the full row space <span class="SimpleMath"><var class="Arg">F</var>^<var class="Arg">n</var></span>.</p>
<p><a id="X7F8EA8D583C1E9B2" name="X7F8EA8D583C1E9B2"></a></p>
<h5>44.3-2 ProjectiveActionHomomorphismMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveActionHomomorphismMatrixGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an action homomorphism for a faithful projective action of <var class="Arg">G</var> on the underlying vector space. (Note: The action is not necessarily on the full space, if a smaller subset can be found on which the action is faithful.)</p>
<p><a id="X849C451A80B4A210" name="X849C451A80B4A210"></a></p>
<h5>44.3-3 BlowUpIsomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlowUpIsomorphism</code>( <var class="Arg">matgrp</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a matrix group <var class="Arg">matgrp</var> and a basis <var class="Arg">B</var> of a field extension <span class="SimpleMath">L / K</span>, say, such that the entries of all matrices in <var class="Arg">matgrp</var> lie in <span class="SimpleMath">L</span>, <code class="func">BlowUpIsomorphism</code> returns the isomorphism with source <var class="Arg">matgrp</var> that is defined by mapping the matrix <span class="SimpleMath">A</span> to <code class="code">BlownUpMat</code><span class="SimpleMath">( A, <var class="Arg">B</var> )</span>, see <code class="func">BlownUpMat</code> (<a href="chap24.html#X85923C107A4569D0"><span class="RefLink">24.13-3</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= GL(2,4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= CanonicalBasis( GF(4) );; BasisVectors( B );</span>
[ Z(2)^0, Z(2^2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">iso:= BlowUpIsomorphism( g, B );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Image( iso, [ [ Z(4), Z(2) ], [ 0*Z(2), Z(4)^2 ] ] ) );</span>
. 1 1 .
1 1 . 1
. . 1 1
. . 1 .
<span class="GAPprompt">gap></span> <span class="GAPinput">img:= Image( iso, g );</span>
<matrix group with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Index( GL(4,2), img );</span>
112
</pre></div>
<p><a id="X7934EED77891BE6B" name="X7934EED77891BE6B"></a></p>
<h4>44.4 <span class="Heading">GL and SL</span></h4>
<p>(See also section <a href="chap50.html#X8674AAA578FE4AEE"><span class="RefLink">50.2</span></a>.)</p>
<p><a id="X781387AF7999EA99" name="X781387AF7999EA99"></a></p>
<h5>44.4-1 IsGeneralLinearGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralLinearGroup</code>( <var class="Arg">grp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGL</code>( <var class="Arg">grp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>The General Linear group is the group of all invertible matrices over a ring. This property tests, whether a group is isomorphic to a General Linear group. (Note that currently only a few trivial methods are available for this operation. We hope to improve this in the future.)</p>
<p><a id="X86F9A27D7AFAEB5A" name="X86F9A27D7AFAEB5A"></a></p>
<h5>44.4-2 IsNaturalGL</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNaturalGL</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property tests, whether a matrix group is the General Linear group in the right dimension over the (smallest) ring which contains all entries of its elements. (Currently, only a trivial test that computes the order of the group is available.)</p>
<p><a id="X816677CD821261FA" name="X816677CD821261FA"></a></p>
<h5>44.4-3 IsSpecialLinearGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSpecialLinearGroup</code>( <var class="Arg">grp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSL</code>( <var class="Arg">grp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>The Special Linear group is the group of all invertible matrices over a ring, whose determinant is equal to 1. This property tests, whether a group is isomorphic to a Special Linear group. (Note that currently only a few trivial methods are available for this operation. We hope to improve this in the future.)</p>
<p><a id="X84134F08781EB943" name="X84134F08781EB943"></a></p>
<h5>44.4-4 IsNaturalSL</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNaturalSL</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property tests, whether a matrix group is the Special Linear group in the right dimension over the (smallest) ring which contains all entries of its elements. (Currently, only a trivial test that computes the order of the group is available.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNaturalGL(m);</span>
false
</pre></div>
<p><a id="X7ED43D4F7E993A31" name="X7ED43D4F7E993A31"></a></p>
<h5>44.4-5 IsSubgroupSL</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSubgroupSL</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property tests, whether a matrix group is a subgroup of the Special Linear group in the right dimension over the (smallest) ring which contains all entries of its elements.</p>
<p><a id="X7CA4097C79F5BD90" name="X7CA4097C79F5BD90"></a></p>
<h4>44.5 <span class="Heading">Invariant Forms</span></h4>
<p><a id="X7C08385A81AB05E1" name="X7C08385A81AB05E1"></a></p>
<h5>44.5-1 InvariantBilinearForm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InvariantBilinearForm</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This attribute describes a bilinear form that is invariant under the matrix group <var class="Arg">matgrp</var>. The form is given by a record with the component <code class="code">matrix</code> which is a matrix <span class="SimpleMath">F</span> such that for every generator <span class="SimpleMath">g</span> of <var class="Arg">matgrp</var> the equation <span class="SimpleMath">g ⋅ F ⋅ g^tr = F</span> holds.</p>
<p><a id="X8652FBF781940AC3" name="X8652FBF781940AC3"></a></p>
<h5>44.5-2 IsFullSubgroupGLorSLRespectingBilinearForm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFullSubgroupGLorSLRespectingBilinearForm</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property tests, whether a matrix group <var class="Arg">matgrp</var> is the full subgroup of GL or SL (the property <code class="func">IsSubgroupSL</code> (<a href="chap44.html#X7ED43D4F7E993A31"><span class="RefLink">44.4-5</span></a>) determines which it is) respecting the form stored as the value of <code class="func">InvariantBilinearForm</code> (<a href="chap44.html#X7C08385A81AB05E1"><span class="RefLink">44.5-1</span></a>) for <var class="Arg">matgrp</var>.</p>
<p><a id="X82F22079852130C9" name="X82F22079852130C9"></a></p>
<h5>44.5-3 InvariantSesquilinearForm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InvariantSesquilinearForm</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This attribute describes a sesquilinear form that is invariant under the matrix group <var class="Arg">matgrp</var> over the field <span class="SimpleMath">F</span> with <span class="SimpleMath">q^2</span> elements, say. The form is given by a record with the component <code class="code">matrix</code> which is is a matrix <span class="SimpleMath">M</span> such that for every generator <span class="SimpleMath">g</span> of <var class="Arg">matgrp</var> the equation <span class="SimpleMath">g ⋅ M ⋅ (g^tr)^f = M</span> holds, where <span class="SimpleMath">f</span> is the automorphism of <span class="SimpleMath">F</span> that raises each element to its <span class="SimpleMath">q</span>-th power. (<span class="SimpleMath">f</span> can be obtained as a power of the <code class="func">FrobeniusAutomorphism</code> (<a href="chap59.html#X8758E4AB7D0A1955"><span class="RefLink">59.4-1</span></a>) value of <span class="SimpleMath">F</span>.)</p>
<p><a id="X7B35A8AF7D8F0313" name="X7B35A8AF7D8F0313"></a></p>
<h5>44.5-4 IsFullSubgroupGLorSLRespectingSesquilinearForm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFullSubgroupGLorSLRespectingSesquilinearForm</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property tests, whether a matrix group <var class="Arg">matgrp</var> is the full subgroup of GL or SL (the property <code class="func">IsSubgroupSL</code> (<a href="chap44.html#X7ED43D4F7E993A31"><span class="RefLink">44.4-5</span></a>) determines which it is) respecting the form stored as the value of <code class="func">InvariantSesquilinearForm</code> (<a href="chap44.html#X82F22079852130C9"><span class="RefLink">44.5-3</span></a>) for <var class="Arg">matgrp</var>.</p>
<p><a id="X7BCACC007EB9B613" name="X7BCACC007EB9B613"></a></p>
<h5>44.5-5 InvariantQuadraticForm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InvariantQuadraticForm</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a matrix group <var class="Arg">matgrp</var>, <code class="func">InvariantQuadraticForm</code> returns a record containing at least the component <code class="code">matrix</code> whose value is a matrix <span class="SimpleMath">Q</span>. The quadratic form <span class="SimpleMath">q</span> on the natural vector space <span class="SimpleMath">V</span> on which <var class="Arg">matgrp</var> acts is given by <span class="SimpleMath">q(v) = v Q v^tr</span>, and the invariance under <var class="Arg">matgrp</var> is given by the equation <span class="SimpleMath">q(v) = q(v M)</span> for all <span class="SimpleMath">v ∈ V</span> and <span class="SimpleMath">M</span> in <var class="Arg">matgrp</var>. (Note that the invariance of <span class="SimpleMath">q</span> does <em>not</em> imply that the matrix <span class="SimpleMath">Q</span> is invariant under <var class="Arg">matgrp</var>.)</p>
<p><span class="SimpleMath">q</span> is defined relative to an invariant symmetric bilinear form <span class="SimpleMath">f</span> (see <code class="func">InvariantBilinearForm</code> (<a href="chap44.html#X7C08385A81AB05E1"><span class="RefLink">44.5-1</span></a>)), via the equation <span class="SimpleMath">q(λ x + μ y) = λ^2 q(x) + λ μ f(x,y) + μ^2 q(y)</span>, see <a href="chapBib.html#biBCCN85">[CCN+85, Chapter 3.4]</a>. If <span class="SimpleMath">f</span> is represented by the matrix <span class="SimpleMath">F</span> then this implies <span class="SimpleMath">F = Q + Q^tr</span>. In characteristic different from <span class="SimpleMath">2</span>, we have <span class="SimpleMath">q(x) = f(x,x)/2</span>, so <span class="SimpleMath">Q</span> can be chosen as the strictly upper triangular part of <span class="SimpleMath">F</span> plus half of the diagonal part of <span class="SimpleMath">F</span>. In characteristic <span class="SimpleMath">2</span>, <span class="SimpleMath">F</span> does not determine <span class="SimpleMath">Q</span> but still <span class="SimpleMath">Q</span> can be chosen as an upper (or lower) triangular matrix.</p>
<p>Whenever the <code class="func">InvariantQuadraticForm</code> value is set in a matrix group then also the <code class="func">InvariantBilinearForm</code> (<a href="chap44.html#X7C08385A81AB05E1"><span class="RefLink">44.5-1</span></a>) value can be accessed, and the two values are compatible in the above sense.</p>
<p><a id="X84AB04A67DFC0274" name="X84AB04A67DFC0274"></a></p>
<h5>44.5-6 IsFullSubgroupGLorSLRespectingQuadraticForm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFullSubgroupGLorSLRespectingQuadraticForm</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property tests, whether the matrix group <var class="Arg">matgrp</var> is the full subgroup of GL or SL (the property <code class="func">IsSubgroupSL</code> (<a href="chap44.html#X7ED43D4F7E993A31"><span class="RefLink">44.4-5</span></a>) determines which it is) respecting the <code class="func">InvariantQuadraticForm</code> (<a href="chap44.html#X7BCACC007EB9B613"><span class="RefLink">44.5-5</span></a>) value of <var class="Arg">matgrp</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Sp( 2, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= InvariantBilinearForm( g ).matrix;</span>
[ [ 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">[ 0, 1 ] * m * [ 1, -1 ]; # evaluate the bilinear form</span>
Z(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFullSubgroupGLorSLRespectingBilinearForm( g );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= SU( 2, 4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= InvariantSesquilinearForm( g ).matrix;</span>
[ [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, 0*Z(2) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">[ 0, 1 ] * m * [ 1, 1 ]; # evaluate the bilinear form</span>
Z(2)^0
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFullSubgroupGLorSLRespectingSesquilinearForm( g );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= GO( 1, 2, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= InvariantBilinearForm( g ).matrix;</span>
[ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, 0*Z(3) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">[ 0, 1 ] * m * [ 1, 1 ]; # evaluate the bilinear form</span>
Z(3)^0
<span class="GAPprompt">gap></span> <span class="GAPinput">q:= InvariantQuadraticForm( g ).matrix;</span>
[ [ 0*Z(3), Z(3)^0 ], [ 0*Z(3), 0*Z(3) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">[ 0, 1 ] * q * [ 0, 1 ]; # evaluate the quadratic form</span>
0*Z(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFullSubgroupGLorSLRespectingQuadraticForm( g );</span>
true
</pre></div>
<p><a id="X7FB0138F79E8C5E7" name="X7FB0138F79E8C5E7"></a></p>
<h4>44.6 <span class="Heading">Matrix Groups in Characteristic 0</span></h4>
<p>Most of the functions described in this and the following section have implementations which use functions from the <strong class="pkg">GAP</strong> package <strong class="pkg">Carat</strong>. If <strong class="pkg">Carat</strong> is not installed or not compiled, no suitable methods are available.</p>
<p><a id="X850821F78558C829" name="X850821F78558C829"></a></p>
<h5>44.6-1 IsCyclotomicMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCyclotomicMatrixGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>tests whether all matrices in <var class="Arg">G</var> have cyclotomic entries.</p>
<p><a id="X7FEDB2E17EE02674" name="X7FEDB2E17EE02674"></a></p>
<h5>44.6-2 IsRationalMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRationalMatrixGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>tests whether all matrices in <var class="Arg">G</var> have rational entries.</p>
<p><a id="X7F737FC4795F3E48" name="X7F737FC4795F3E48"></a></p>
<h5>44.6-3 IsIntegerMatrixGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIntegerMatrixGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>tests whether all matrices in <var class="Arg">G</var> have integer entries.</p>
<p><a id="X86F9CC1E7DB97CB6" name="X86F9CC1E7DB97CB6"></a></p>
<h5>44.6-4 IsNaturalGLnZ</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNaturalGLnZ</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>tests whether <var class="Arg">G</var> is <span class="SimpleMath">GL_n(ℤ)</span> in its natural representation by <span class="SimpleMath">n × n</span> integer matrices. (The dimension <span class="SimpleMath">n</span> will be read off the generating matrices.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNaturalGLnZ( GL( 2, Integers ) );</span>
true
</pre></div>
<p><a id="X7B0E70127F5D2EAF" name="X7B0E70127F5D2EAF"></a></p>
<h5>44.6-5 IsNaturalSLnZ</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNaturalSLnZ</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>tests whether <var class="Arg">G</var> is <span class="SimpleMath">SL_n(ℤ)</span> in its natural representation by <span class="SimpleMath">n × n</span> integer matrices. (The dimension <span class="SimpleMath">n</span> will be read off the generating matrices.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNaturalSLnZ( SL( 2, Integers ) );</span>
true
</pre></div>
<p><a id="X7DE412A37A6975B3" name="X7DE412A37A6975B3"></a></p>
<h5>44.6-6 InvariantLattice</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InvariantLattice</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a matrix <span class="SimpleMath">B</span>, whose rows form a basis of a <span class="SimpleMath">ℤ</span>-lattice that is invariant under the rational matrix group <var class="Arg">G</var> acting from the right. It returns <code class="keyw">fail</code> if the group is not unimodular. The columns of the inverse of <span class="SimpleMath">B</span> span a <span class="SimpleMath">ℤ</span>-lattice invariant under <var class="Arg">G</var> acting from the left.</p>
<p><a id="X7CC4D6DC81739698" name="X7CC4D6DC81739698"></a></p>
<h5>44.6-7 NormalizerInGLnZ</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalizerInGLnZ</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is an attribute used to store the normalizer of <var class="Arg">G</var> in <span class="SimpleMath">GL_n(ℤ)</span>, where <var class="Arg">G</var> is an integer matrix group of dimension <var class="Arg">n</var>. This attribute is used by <code class="code">Normalizer( GL( n, Integers ), G )</code>.</p>
<p><a id="X7DAFB71F86525DE7" name="X7DAFB71F86525DE7"></a></p>
<h5>44.6-8 CentralizerInGLnZ</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CentralizerInGLnZ</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is an attribute used to store the centralizer of <var class="Arg">G</var> in <span class="SimpleMath">GL_n(ℤ)</span>, where <var class="Arg">G</var> is an integer matrix group of dimension <var class="Arg">n</var>. This attribute is used by <code class="code">Centralizer( GL( n, Integers ), G )</code>.</p>
<p><a id="X8217762A863F1382" name="X8217762A863F1382"></a></p>
<h5>44.6-9 ZClassRepsQClass</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ZClassRepsQClass</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The conjugacy class in <span class="SimpleMath">GL_n(ℚ)</span> of the finite integer matrix group <var class="Arg">G</var> splits into finitely many conjugacy classes in <span class="SimpleMath">GL_n(ℤ)</span>. <code class="code">ZClassRepsQClass( <var class="Arg">G</var> )</code> returns representative groups for these.</p>
<p><a id="X84FD9FC97FB90795" name="X84FD9FC97FB90795"></a></p>
<h5>44.6-10 IsBravaisGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBravaisGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>test whether <var class="Arg">G</var> coincides with its Bravais group (see <code class="func">BravaisGroup</code> (<a href="chap44.html#X7AAE301C83116451"><span class="RefLink">44.6-11</span></a>)).</p>
<p><a id="X7AAE301C83116451" name="X7AAE301C83116451"></a></p>
<h5>44.6-11 BravaisGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BravaisGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the Bravais group of a finite integer matrix group <var class="Arg">G</var>. If <span class="SimpleMath">C</span> is the cone of positive definite quadratic forms <span class="SimpleMath">Q</span> invariant under <span class="SimpleMath">g ↦ g Q g^tr</span> for all <span class="SimpleMath">g ∈ <var class="Arg">G</var></span>, then the Bravais group of <var class="Arg">G</var> is the maximal subgroup of <span class="SimpleMath">GL_n(ℤ)</span> leaving the forms in that same cone invariant. Alternatively, the Bravais group of <var class="Arg">G</var> can also be defined with respect to the action <span class="SimpleMath">g ↦ g^tr Q g</span> on positive definite quadratic forms <span class="SimpleMath">Q</span>. This latter definition is appropriate for groups <var class="Arg">G</var> acting from the right on row vectors, whereas the former definition is appropriate for groups acting from the left on column vectors. Both definitions yield the same Bravais group.</p>
<p><a id="X788C7D9C7C2301C5" name="X788C7D9C7C2301C5"></a></p>
<h5>44.6-12 BravaisSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BravaisSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the subgroups of the Bravais group of <var class="Arg">G</var>, which are themselves Bravais groups.</p>
<p><a id="X7F5FF1A481E08AD5" name="X7F5FF1A481E08AD5"></a></p>
<h5>44.6-13 BravaisSupergroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BravaisSupergroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the subgroups of <span class="SimpleMath">GL_n(ℤ)</span> that contain the Bravais group of <var class="Arg">G</var> and are Bravais groups themselves.</p>
<p><a id="X79B7CD797A420720" name="X79B7CD797A420720"></a></p>
<h5>44.6-14 NormalizerInGLnZBravaisGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalizerInGLnZBravaisGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the normalizer of the Bravais group of <var class="Arg">G</var> in the appropriate <span class="SimpleMath">GL_n(ℤ)</span>.</p>
<p><a id="X868288377CFA8D1B" name="X868288377CFA8D1B"></a></p>
<h4>44.7 <span class="Heading">Acting OnRight and OnLeft</span></h4>
<p>In <strong class="pkg">GAP</strong>, matrices by convention act on row vectors from the right, whereas in crystallography the convention is to act on column vectors from the left. The definition of certain algebraic objects important in crystallography implicitly depends on which action is assumed. This holds true in particular for quadratic forms invariant under a matrix group. In a similar way, the representation of affine crystallographic groups, as they are provided by the <strong class="pkg">GAP</strong> package <strong class="pkg">CrystGap</strong>, depends on which action is assumed. Crystallographers are used to the action from the left, whereas the action from the right is the natural one for <strong class="pkg">GAP</strong>. For this reason, a number of functions which are important in crystallography, and whose result depends on which action is assumed, are provided in two versions, one for the usual action from the right, and one for the crystallographic action from the left.</p>
<p>For every such function, this fact is explicitly mentioned. The naming scheme is as follows: If <code class="code">SomeThing</code> is such a function, there will be functions <code class="code">SomeThingOnRight</code> and <code class="code">SomeThingOnLeft</code>, assuming action from the right and from the left, respectively. In addition, there is a generic function <code class="code">SomeThing</code>, which returns either the result of <code class="code">SomeThingOnRight</code> or <code class="code">SomeThingOnLeft</code>, depending on the global variable <code class="func">CrystGroupDefaultAction</code> (<a href="chap44.html#X7D1318A6780CD88B"><span class="RefLink">44.7-1</span></a>).</p>
<p><a id="X7D1318A6780CD88B" name="X7D1318A6780CD88B"></a></p>
<h5>44.7-1 CrystGroupDefaultAction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CrystGroupDefaultAction</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>can have either of the two values <code class="code">RightAction</code> and <code class="code">LeftAction</code>. The initial value is <code class="code">RightAction</code>. For functions which have variants OnRight and OnLeft, this variable determines which variant is returned by the generic form. The value of <code class="func">CrystGroupDefaultAction</code> can be changed with with the function <code class="func">SetCrystGroupDefaultAction</code> (<a href="chap44.html#X792D237385977BE6"><span class="RefLink">44.7-2</span></a>).</p>
<p><a id="X792D237385977BE6" name="X792D237385977BE6"></a></p>
<h5>44.7-2 SetCrystGroupDefaultAction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SetCrystGroupDefaultAction</code>( <var class="Arg">action</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>allows one to set the value of the global variable <code class="func">CrystGroupDefaultAction</code> (<a href="chap44.html#X7D1318A6780CD88B"><span class="RefLink">44.7-1</span></a>). Only the arguments <code class="code">RightAction</code> and <code class="code">LeftAction</code> are allowed. Initially, the value of <code class="func">CrystGroupDefaultAction</code> (<a href="chap44.html#X7D1318A6780CD88B"><span class="RefLink">44.7-1</span></a>) is <code class="code">RightAction</code>.</p>
<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap43.html">[Previous Chapter]</a> <a href="chap45.html">[Next Chapter]</a> </div>
<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chap26.html">26</a> <a href="chap27.html">27</a> <a href="chap28.html">28</a> <a href="chap29.html">29</a> <a href="chap30.html">30</a> <a href="chap31.html">31</a> <a href="chap32.html">32</a> <a href="chap33.html">33</a> <a href="chap34.html">34</a> <a href="chap35.html">35</a> <a href="chap36.html">36</a> <a href="chap37.html">37</a> <a href="chap38.html">38</a> <a href="chap39.html">39</a> <a href="chap40.html">40</a> <a href="chap41.html">41</a> <a href="chap42.html">42</a> <a href="chap43.html">43</a> <a href="chap44.html">44</a> <a href="chap45.html">45</a> <a href="chap46.html">46</a> <a href="chap47.html">47</a> <a href="chap48.html">48</a> <a href="chap49.html">49</a> <a href="chap50.html">50</a> <a href="chap51.html">51</a> <a href="chap52.html">52</a> <a href="chap53.html">53</a> <a href="chap54.html">54</a> <a href="chap55.html">55</a> <a href="chap56.html">56</a> <a href="chap57.html">57</a> <a href="chap58.html">58</a> <a href="chap59.html">59</a> <a href="chap60.html">60</a> <a href="chap61.html">61</a> <a href="chap62.html">62</a> <a href="chap63.html">63</a> <a href="chap64.html">64</a> <a href="chap65.html">65</a> <a href="chap66.html">66</a> <a href="chap67.html">67</a> <a href="chap68.html">68</a> <a href="chap69.html">69</a> <a href="chap70.html">70</a> <a href="chap71.html">71</a> <a href="chap72.html">72</a> <a href="chap73.html">73</a> <a href="chap74.html">74</a> <a href="chap75.html">75</a> <a href="chap76.html">76</a> <a href="chap77.html">77</a> <a href="chap78.html">78</a> <a href="chap79.html">79</a> <a href="chap80.html">80</a> <a href="chap81.html">81</a> <a href="chap82.html">82</a> <a href="chap83.html">83</a> <a href="chap84.html">84</a> <a href="chap85.html">85</a> <a href="chap86.html">86</a> <a href="chap87.html">87</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>
|