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<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="chapInd.html">Ind</a>  </div>

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<h1>RepnDecomp</h1>


<h2>Decompose representations of finite groups into irreducibles</h2>

<p>
    1.3.1</p>

<p>
    10 September 2025
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<p><b>
    Kaashif Hymabaccus



  </b>
<br />Email: <span class="URL"><a href="mailto:kaashif@kaashif.co.uk">kaashif@kaashif.co.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://kaashif.co.uk">https://kaashif.co.uk</a></span>
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<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X789233A47A277072">1.1 <span class="Heading">Getting started with RepnDecomp</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8360C04082558A12">1.1-1 <span class="Heading">Installation</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X792C0F507B4A3B89">1.1-2 <span class="Heading">Note on what is meant by a representation</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8315479878D25E37">1.1-3 <span class="Heading">API Overview</span></a>
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<div class="ContChap"><a href="chap2.html#X7D9B253E794EF912">2 <span class="Heading">Isomorphisms between representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7AEE81C2809E0B98">2.1 <span class="Heading">Finding explicit isomorphisms</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7F0D3CFB7800149A">2.1-1 LinearRepresentationIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X841DE7D08491325F">2.1-2 LinearRepresentationIsomorphismSlow</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X85DAC9E583D8EFB9">2.2 <span class="Heading">Testing isomorphisms</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X86EB9DD586958473">2.2-1 AreRepsIsomorphic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X81080E1B7917B361">2.2-2 IsLinearRepresentationIsomorphism</a></span>
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<div class="ContChap"><a href="chap3.html#X83B4D1DB7F92BD3A">3 <span class="Heading">Algorithms for unitary representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X870B3D0D80CFADB1">3.1 <span class="Heading">Unitarising representations</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X86B2367A79BE5B9F">3.1-1 UnitaryRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87D121227C027253">3.1-2 IsUnitaryRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X78F7DFD186A4E7CA">3.1-3 LDLDecomposition</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7974D0C580C833D1">3.2 <span class="Heading">Decomposing unitary representations</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8175C1167A31C3D6">3.2-1 IrreducibleDecompositionDixon</a></span>
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<div class="ContChap"><a href="chap4.html#X8346542B8387968B">4 <span class="Heading">Miscellaneous useful functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X87196FDB78749ECA">4.1 <span class="Heading">Predicates for representations</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8631A1417C3C1D88">4.1-1 IsFiniteGroupLinearRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X826D5ADF7FA87782">4.1-2 IsFiniteGroupPermutationRepresentation</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8271F7A386CFEA63">4.2 <span class="Heading">Efficient summing over groups</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85E8A5FC844DC09A">4.2-1 GroupSumBSGS</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X86F42D257CFB192D">4.3 <span class="Heading">Space-efficient representation of tensors of matrices</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84335C447DE377B0">4.3-1 IsTensorProductOfMatricesObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X868B9CB1873C93AC">4.3-2 IsTensorProductPairRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85AD76DB7D5F8B12">4.3-3 IsTensorProductKroneckerRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X86671AD582FF77E2">4.3-4 TensorProductOfMatrices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79D0379D79F5DF9B">4.3-5 CharacterOfTensorProductOfRepresentations</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X83F160967ED7EE14">4.4 <span class="Heading">Matrices and homomorphisms</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D17785482F143B0">4.4-1 ComposeHomFunction</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7C3EDA5E7A24196C">4.5 <span class="Heading">Representation theoretic functions</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X841424DF824E258B">4.5-1 TensorProductRepLists</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84EAE3DB7FA8102C">4.5-2 DirectSumOfRepresentations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85147CF97B912CC3">4.5-3 DegreeOfRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B14287E7BFC548D">4.5-4 PermToLinearRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E67A4817A5E4879">4.5-5 IsOrthonormalSet</a></span>
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<div class="ContChap"><a href="chap5.html#X7F968DF987DE4A6E">5 <span class="Heading">Computing decompositions of representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7E29883984400D2C">5.1 <span class="Heading">Block diagonalizing</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8361AD057AD282AC">5.1-1 BlockDiagonalBasisOfRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X86EB837579C1416D">5.1-2 BlockDiagonalRepresentation</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X863A16A179A7486B">5.2 <span class="Heading">Algorithms due to the authors</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X831574AD864C94A8">5.2-1 REPN_ComputeUsingMyMethod</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7DB659DB7E48D502">5.2-2 REPN_ComputeUsingMyMethodCanonical</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7C22F13E80A74438">5.3 <span class="Heading">Algorithms due to Serre</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E95B0367992BEC4">5.3-1 CanonicalDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X795C63F386C45308">5.3-2 IrreducibleDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X87E91CBE7992D126">5.3-3 IrreducibleDecompositionCollected</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7C1CF0547D72D354">5.3-4 REPN_ComputeUsingSerre</a></span>
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<div class="ContChap"><a href="chap6.html#X7A0EF2C67E2DB726">6 <span class="Heading">Centralizer (commutant) rings</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7E70A3D881CD5FFA">6.1 <span class="Heading">Finding a basis for the centralizer</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7901B6A7860D35C3">6.1-1 CentralizerBlocksOfRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X86B19E2B877121E9">6.1-2 CentralizerOfRepresentation</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X83C4F8C17DA976EE">6.2 <span class="Heading">Using the centralizer for computations</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X87E5BAEB82DC00C3">6.2-1 ClassSumCentralizer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X78719DC8868B0744">6.2-2 ClassSumCentralizerNC</a></span>
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