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<h1>hecke</h1>


<h2>Calculating decomposition matrices of Hecke algebras</h2>

<p>
    1.5.4</p>

<p>
    27 August 2024
  </p>

</div>
<p><b>
    Dmitriy Traytel



  </b>
<br />Email: <span class="URL"><a href="mailto:traytel@in.tum.de">traytel@in.tum.de</a></span>
<br />Homepage: <span class="URL"><a href="https://home.in.tum.de/~traytel/hecke/">https://home.in.tum.de/~traytel/hecke/</a></span>
</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2010–2013 by Dmitriy Traytel</p>

<p>This package may be distributed under the terms and conditions of the GNU Public License Version 2 or higher.</p>

<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p><strong class="pkg">Hecke</strong> is a port of the <strong class="pkg">GAP</strong> 3 package <strong class="pkg">Specht</strong> 2.4 to <strong class="pkg">GAP</strong> 4. <strong class="pkg">Specht</strong> 2.4 was written by Andrew Mathas, who allowed Dmitriy Traytel to use his source code as the basis for <strong class="pkg">hecke</strong>.</p>

<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>

<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1_mj.html#X7CD78FC183A57690">1 <span class="Heading">Decomposition numbers of Hecke algebras of type A</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7BBCB13F82ACC213">1.1 <span class="Heading">Description</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7A3BD2F77A4AC7EA">1.2 <span class="Heading">The modular representation theory of Hecke algebras</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X8226F6F77ACF26D8">1.3 <span class="Heading">Two small examples</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X786BACDB82918A65">1.4 <span class="Heading">Overview over this manual</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X8779AFAF8411A26A">1.5 <span class="Heading">Credits</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2_mj.html#X79845447824F3333">2 <span class="Heading">Installation of the <strong class="pkg">hecke</strong>-Package</span></a>
</div>
<div class="ContChap"><a href="chap3_mj.html#X7ED1AB5C7E41D277">3 <span class="Heading">Specht functionality</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X78AA2DBD7D5D3F02">3.1 <span class="Heading">Porting notes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8340A4F97986693C">3.1-1 <span class="Heading">Structure of <strong class="pkg">Hecke</strong></span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8120A27282B82CC8">3.1-2 <span class="Heading">Renamings</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7A7DF4FC796EF66F">3.2 <span class="Heading">Specht functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7FE26921867C440A">3.2-1 Specht</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8037763587274161">3.2-2 <span class="Heading">Simple information access</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C92700882971537">3.2-3 <span class="Heading">The functions MakeSpecht, MakePIM and MakeSimple</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X86F599A07A7C1C33">3.2-4 <span class="Heading">Decomposition numbers of the symmetric groups</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X838BEC0382BF87EA">3.2-5 <span class="Heading">Hecke algebras over fields of positive characteristic</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X83009CE685621BD4">3.2-6 <span class="Heading">The Fock space and Hecke algebras over fields of characteristic zero
</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B57DF517F73F00D">3.2-7 Schur</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84F0F9E47D5EEBCF">3.2-8 DecompositionMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F616CCE808FA11E">3.2-9 CrystalDecompositionMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X829A23A97EE4C20E">3.2-10 DecompositionNumber</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7C5B169286EFC900">3.3 <span class="Heading">Partitions in <strong class="pkg">Hecke</strong></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X87A6E8DD85F3F020">3.4 <span class="Heading">Inducing and restricting modules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X81D7F7A4812BB04D">3.4-1 RInducedModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X780709B3865BC344">3.4-2 SInducedModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X783BC74E81A7D0E6">3.4-3 RRestrictedModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8041ABFA86D7A3EF">3.4-4 SRestrictedModule</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X79F430837BA7BAD2">3.5 <span class="Heading">Operations on decomposition matrices</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7D257389845738DB">3.5-1 InducedDecompositionMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X803A99987E501AC9">3.5-2 IsNewIndecomposable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X802E811683E611EE">3.5-3 InvertDecompositionMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87B18FD97B2D8E80">3.5-4 AdjustmentMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X78B0FF2079269138">3.5-5 SaveDecompositionMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84DD2D517FC1F905">3.5-6 CalculateDecompositionMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7FDC65328102C1B9">3.5-7 MatrixDecompositionMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X86EBEBF680EBC98E">3.5-8 DecompositionMatrixMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B05627D83E6977E">3.5-9 AddIndecomposable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79FA055E8250E6A2">3.5-10 RemoveIndecomposable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8434DC7C8364CB54">3.5-11 MissingIndecomposables</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7A697AAA799BA7D4">3.6 <span class="Heading">Calculating dimensions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X828528747E4AC4C9">3.6-1 SimpleDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B98631580E193BB">3.6-2 SpechtDimension</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X78F1DC277875BAFD">3.7 <span class="Heading">Combinatorics on Young diagrams</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X820A908F8337F59C">3.7-1 Schaper</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7FB82B3184287362">3.7-2 IsSimpleModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A6262B684185E3D">3.7-3 MullineuxMap</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7CC6C04482DD1E9D">3.7-4 MullineuxSymbol</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7826922879DD8D8A">3.7-5 PartitionMullineuxSymbol</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7CE4D6487FD009B1">3.7-6 GoodNodes</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X783B52458335975F">3.7-7 NormalNodes</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X85B290977A17D9EE">3.7-8 GoodNodeSequence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B704FE781A311E5">3.7-9 PartitionGoodNodeSequence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A9DC101850008A2">3.7-10 GoodNodeLatticePath</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7918D9DE7ACE2294">3.7-11 LittlewoodRichardsonRule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B9901427D1CF6F4">3.7-12 InverseLittlewoodRichardsonRule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X790D4ACF7930340F">3.7-13 EResidueDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7DE3773C78BC324C">3.7-14 HookLengthDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F2ACCBF788A62E8">3.7-15 RemoveRimHook</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7CEA98C779BDBD1A">3.7-16 AddRimHook</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X8350934A7F9AB5BE">3.8 <span class="Heading">Operations on partitions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X867496487DC35776">3.8-1 ECore</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8236220C87814790">3.8-2 IsECore</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8538AAAF8628A725">3.8-3 EQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F357B417D495B6F">3.8-4 CombineEQuotientECore</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C460635829E7ED0">3.8-5 EWeight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X86308F6C818B220C">3.8-6 ERegularPartitions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7BEDA8F286ED5F20">3.8-7 IsERegular</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7D131AF0839089BD">3.8-8 ConjugatePartition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8711CC56792711A7">3.8-9 PartitionBetaSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7EC4D0FA81B55391">3.8-10 ETopLadder</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X820388EF7C8333BA">3.8-11 Dominates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84DB1DD37AF227CF">3.8-12 LengthLexicographic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8480188D81ECBD92">3.8-13 Lexicographic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X78F41DF77D6F8292">3.8-14 ReverseDominance</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X83890936806E3A34">3.9 <span class="Heading">Miscellaneous functions on modules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A8E810C85A62DD6">3.9-1 Specialized</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8232C0A1846A27FB">3.9-2 ERegulars</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X822E8193835DD1D9">3.9-3 SplitECores</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E92948B80075E46">3.9-4 Coefficient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79FB3FE67D55BCFA">3.9-5 InnerProduct</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7D473E167C866CEC">3.10 <span class="Heading">Semi-standard and standard tableaux</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F0F9663796E6978">3.10-1 Tableau</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79ADB1B980D12A14">3.10-2 SemiStandardTableaux</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E51D6107DBE2A74">3.10-3 StandardTableaux</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7869DA9A8198BD28">3.10-4 ConjugateTableau</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E5351C27C9253D9">3.10-5 ShapeTableau</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7CABF92D7BF07DD1">3.10-6 TypeTableau</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapBib_mj.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd_mj.html"><span class="Heading">Index</span></a></div>
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Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.