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<div class="ChapSects"><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>
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<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>
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<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>
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<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>
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<h3>1 <span class="Heading">Decomposition numbers of Hecke algebras of type A</span></h3>

<p><a id="X7BBCB13F82ACC213" name="X7BBCB13F82ACC213"></a></p>

<h4>1.1 <span class="Heading">Description</span></h4>

<p><strong class="pkg">Hecke</strong> is a port of the <strong class="pkg">GAP</strong> 3-package <strong class="pkg">Specht</strong> <span class="SimpleMath">\(2.4\)</span> to <strong class="pkg">GAP</strong> 4.</p>

<p>This package contains functions for computing the decomposition matrices for Iwahori-Hecke algebras of the symmetric groups. As the (modular) representation theory of these algebras closely resembles that of the (modular) representation theory of the symmetric groups (indeed, the latter is a special case of the former) many of the combinatorial tools from the representation theory of the symmetric group are included in the package.</p>

<p>These programs grew out of the attempts by Gordon James and Andrew Mathas <a href="chapBib_mj.html#biBJM1">[JM96]</a> to understand the decomposition matrices of Hecke algebras of type <em>A</em> when <span class="SimpleMath">\(q=-1\)</span>. The package is now much more general and its highlights include:</p>

<ol>
<li><p><strong class="pkg">Hecke</strong> provides a means of working in the Grothendieck ring of a Hecke algebra <span class="SimpleMath">\(H\)</span> using the three natural bases corresponding to the Specht modules, projective indecomposable modules, and simple modules.</p>

</li>
<li><p>For Hecke algebras defined over fields of characteristic zero, the algorithm of Lascoux, Leclerc, and Thibon <a href="chapBib_mj.html#biBLLT">[LLT96]</a> for computing decomposition numbers and <q>crystallized decomposition matrices</q> has been implemented. In principle, this gives all of the decomposition matrices of Hecke algebras defined over fields of characteristic zero.</p>

</li>
<li><p><strong class="pkg">Hecke</strong> provides a way of inducing and restricting modules. In addition, it is possible to <q>induce</q> decomposition matrices; this function is quite effective in calculating the decomposition matrices of Hecke algebras for small <span class="SimpleMath">\(n\)</span>.</p>

</li>
<li><p>The <span class="SimpleMath">\(q\)</span>-analogue of Schaper's theorem <a href="chapBib_mj.html#biBJM2">[JM97]</a> is included, as is Kleshchev's <a href="chapBib_mj.html#biBK">[Kle96]</a> algorithm of calculating the Mullineux map. Both are used extensively when inducing decomposition matrices.</p>

</li>
<li><p><strong class="pkg">Hecke</strong> can be used to compute the decomposition numbers of <span class="SimpleMath">\(q\)</span>-Schur algebras (and the general linear groups), although there is less direct support for these algebras. The decomposition matrices for the <span class="SimpleMath">\(q\)</span>-Schur algebras defined over fields of characteristic zero for <span class="SimpleMath">\(n<11\)</span> and all <span class="SimpleMath">\(e\)</span> are included in <strong class="pkg">Hecke</strong>.</p>

</li>
<li><p>The Littlewood-Richard rule, its inverse, and functions for many of the standard operations on partitions (such as calculating cores, quotients, and adding and removing hooks), are included.</p>

</li>
<li><p>The decomposition matrices for the symmetric groups <span class="SimpleMath">\(S_n\)</span> are included for <span class="SimpleMath">\(n<15\)</span> and for all primes.</p>

</li>
</ol>
<p><a id="X7A3BD2F77A4AC7EA" name="X7A3BD2F77A4AC7EA"></a></p>

<h4>1.2 <span class="Heading">The modular representation theory of Hecke algebras</span></h4>

<p>The <q>modular</q> representation theory of the Iwahori-Hecke algebras of type <em>A</em> was pioneered by Dipper and James <a href="chapBib_mj.html#biBDJ1">[DJ86]</a> <a href="chapBib_mj.html#biBDJ2">[DJ87]</a>; here the theory is briefly outlined, referring the reader to the references for details.</p>

<p>Given a commutative integral domain <span class="SimpleMath">\(R\)</span> and a non-zero unit <span class="SimpleMath">\(q\)</span> in <span class="SimpleMath">\(R\)</span>, let <span class="SimpleMath">\(H=H_{R, q}\)</span> be the Hecke algebra of the symmetric group <span class="SimpleMath">\(S_n\)</span> on <span class="SimpleMath">\(n\)</span> symbols defined over <span class="SimpleMath">\(R\)</span> and with parameter <span class="SimpleMath">\(q\)</span>. For each partition <span class="SimpleMath">\(\mu\)</span> of <span class="SimpleMath">\(n\)</span>, Dipper and James defined a <em>Specht module</em> <span class="SimpleMath">\(S(\mu)\)</span>. Let <span class="SimpleMath">\(rad~S(\mu)\)</span> be the radical of <span class="SimpleMath">\(S(\mu)\)</span> and define <span class="SimpleMath">\(D(\mu)=S(\mu)/rad~S(\mu)\)</span>. When <span class="SimpleMath">\(R\)</span> is a field, <span class="SimpleMath">\(D(\mu)\)</span> is either zero or absolutely irreducible. Henceforth, we will always assume that <span class="SimpleMath">\(R\)</span> is a field.</p>

<p>Given a non-negative integer <span class="SimpleMath">\(i\)</span>, let <span class="SimpleMath">\([i]_q=1+q+\ldots+q^{i-1}\)</span>. Define <span class="SimpleMath">\(e\)</span> to be the smallest non-negative integer such that <span class="SimpleMath">\([e]_q=0\)</span>; if no such integer exists, we set <span class="SimpleMath">\(e\)</span> equal to <span class="SimpleMath">\(0\)</span>. Many of the functions in this package depend upon e; the integer <span class="SimpleMath">\(e\)</span> is the Hecke algebras analogue of the characteristic of the field in the modular representation theory of finite groups.</p>

<p>A partition <span class="SimpleMath">\(\mu=(\mu_1,\mu_2,\ldots)\)</span> is <em><span class="SimpleMath">\(e\)</span>-singular</em> if there exists an integer <span class="SimpleMath">\(i\)</span> such that <span class="SimpleMath">\(\mu_i=\mu_{i+1}=\cdots= \mu_{i+e-1}>0\)</span>; otherwise, <span class="SimpleMath">\(\mu\)</span> is <em><span class="SimpleMath">\(e\)</span>-regular</em>. Dipper and James <a href="chapBib_mj.html#biBDJ1">[DJ86]</a> showed that <span class="SimpleMath">\(D(\nu)\neq 0\)</span> if and only if <span class="SimpleMath">\(\nu\)</span> is <span class="SimpleMath">\(e\)</span>-regular and that the <span class="SimpleMath">\(D(\nu)\)</span> give a complete set of non-isomorphic irreducible <span class="SimpleMath">\(H\)</span>-modules as <span class="SimpleMath">\(\nu\)</span> runs over the <span class="SimpleMath">\(e\)</span>-regular partitions of <span class="SimpleMath">\(n\)</span>. Further, <span class="SimpleMath">\(S(\mu)\)</span> and <span class="SimpleMath">\(S(\nu)\)</span> belong to the same block if and only if <span class="SimpleMath">\(\mu\)</span> and <span class="SimpleMath">\(\nu\)</span> have the same <span class="SimpleMath">\(e\)</span>-core <a href="chapBib_mj.html#biBDJ2">[DJ87]</a><a href="chapBib_mj.html#biBJM2">[JM97]</a>. Note that these results depend only on <span class="SimpleMath">\(e\)</span> and not directly on <span class="SimpleMath">\(R\)</span> or <span class="SimpleMath">\(q\)</span>.</p>

<p>Given two partitions <span class="SimpleMath">\(\mu\)</span> and <span class="SimpleMath">\(\nu\)</span>, where <span class="SimpleMath">\(\nu\)</span> is <span class="SimpleMath">\(e\)</span> -regular, let <span class="SimpleMath">\(d_{\mu,\nu}\)</span> be the composition multiplicity of <span class="SimpleMath">\(D(\nu)\)</span> in <span class="SimpleMath">\(S(\nu)\)</span>. The matrix <span class="SimpleMath">\(D=(d_{\mu,\nu})\)</span> is the <em> decomposition matrix</em> of <span class="SimpleMath">\(H\)</span>. When the rows and columns are ordered in a way compatible with dominance, <span class="SimpleMath">\(D\)</span> is lower unitriangular.</p>

<p>The indecomposable <span class="SimpleMath">\(H\)</span>-modules <span class="SimpleMath">\(P(\nu)\)</span> are indexed by <span class="SimpleMath">\(e\)</span> -regular partitions <span class="SimpleMath">\(\nu\)</span>. By general arguments, <span class="SimpleMath">\(P(\nu)\)</span> has the same composition factors as <span class="SimpleMath">\(\sum_{\mu} d_{\mu,\nu} S(\mu)\)</span>; so these linear combinations of modules become identified in the Grothendieck ring of <span class="SimpleMath">\(H\)</span>. Similarly, <span class="SimpleMath">\(D(\nu) = \sum_{\mu} d_{\nu,\mu}^{-1} S(\mu)\)</span> in the Grothendieck ring. These observations are the basis for many of the computations in <strong class="pkg">Hecke</strong>.</p>

<p><a id="X8226F6F77ACF26D8" name="X8226F6F77ACF26D8"></a></p>

<h4>1.3 <span class="Heading">Two small examples</span></h4>

<p>Because of the algorithm of <a href="chapBib_mj.html#biBLLT">[LLT96]</a>, in principle, all of decomposition matrices for all Hecke algebras defined over fields of characteristic zero are known and available using <strong class="pkg">Hecke</strong>. The algorithm is recursive; however, it is quite quick and, as with a car, you need never look at the engine:</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">H:=Specht(4);   # e=4, 'R' a field of characteristic 0</span>
<Hecke algebra with e = 4>
<span class="GAPprompt">gap></span> <span class="GAPinput">RInducedModule(MakePIM(H,12,2));</span>
<direct sum of 5 P-modules>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(last);</span>
P(13,2) + P(12,3) + P(12,2,1) + P(10,3,2) + P(9,6)
</pre></div>

<p>The <a href="chapBib_mj.html#biBLLT">[LLT96]</a> algorithm was applied 24 times during this calculation.</p>

<p>For Hecke algebras defined over fields of positive characteristic the major tool provided by <strong class="pkg">Hecke</strong>, apart from the decomposition matrices contained in the libraries, is a way of <q>inducing</q> decomposition matrices. This makes it fairly easy to calculate the associated decomposition matrices for <q>small</q> <span class="SimpleMath">\(n\)</span>. For example, the <strong class="pkg">Hecke</strong> libraries contain the decomposition matrices for the symmetric groups <span class="SimpleMath">\(S_n\)</span> over fields of characteristic <span class="SimpleMath">\(3\)</span> for <span class="SimpleMath">\(n<15\)</span>. These matrices were calculated by <strong class="pkg">Hecke</strong> using the following commands:</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">H:=Specht(3,3);   # e=3, 'R' field of characteristic 3</span>
<Hecke algebra with e = 3>
<span class="GAPprompt">gap></span> <span class="GAPinput">d:=DecompositionMatrix(H,5);  # known for n<2e</span>
<7x5 decomposition matrix>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(last);</span>
5    | 1         
4,1  | . 1       
3,2  | . 1 1     
3,1^2| . . . 1   
2^2,1| 1 . . . 1
2,1^3| . . . . 1
1^5  | . . 1 . . 
<span class="GAPprompt">gap></span> <span class="GAPinput">for n in [6..14] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">      d:=InducedDecompositionMatrix(d); SaveDecompositionMatrix(d);</span>
<span class="GAPprompt">></span> <span class="GAPinput">   od;</span>
# Inducing..
# Inducing..
# Inducing...
# Inducing...
# Inducing...
# Inducing....
</pre></div>

<p>The function <code class="code">InducedDecompositionMatrix</code> contains almost every trick for computing decomposition matrices (except using the spin groups).</p>

<p><strong class="pkg">Hecke</strong> can also be used to calculate the decomposition numbers of the <span class="SimpleMath">\(q\)</span>-Schur algebras; although, as yet, here no additional routines for calculating the projective indecomposables indexed by <span class="SimpleMath">\(e\)</span>-singular partitions. Such routines may be included in a future release, together with the (conjectural) algorithm <a href="chapBib_mj.html#biBLT">[LT96]</a> for computing the decomposition matrices of the <span class="SimpleMath">\(q\)</span>-Schur algebras over fields of characteristic zero.</p>

<p><a id="X786BACDB82918A65" name="X786BACDB82918A65"></a></p>

<h4>1.4 <span class="Heading">Overview over this manual</span></h4>

<p>Chapter <a href="chap2_mj.html#X79845447824F3333"><span class="RefLink">2</span></a> describes the installation of this package. Chapter <a href="chap3_mj.html#X7ED1AB5C7E41D277"><span class="RefLink">3</span></a> shows instructive examples for the usage of this package.</p>

<p><a id="X8779AFAF8411A26A" name="X8779AFAF8411A26A"></a></p>

<h4>1.5 <span class="Heading">Credits</span></h4>

<p>I would like to thank Anne Henke for offering me the interesting project of porting <strong class="pkg">Specht</strong> <span class="SimpleMath">\(2.4\)</span> to the current <strong class="pkg">GAP</strong> version, Max Neunhöffer for giving me an excellent introduction to the <strong class="pkg">GAP</strong> 4-style of programming and Benjamin Wilson for supporting the project and helping me to understand the mathematics behind <strong class="pkg">Hecke</strong>.</p>

<p>Also I thank Andrew Mathas for allowing me to use his <strong class="pkg">GAP</strong> 3-code of the <strong class="pkg">Specht</strong> <span class="SimpleMath">\(2.4\)</span> package.</p>

<p>The lastest version of <strong class="pkg">Hecke</strong> can be obtained from</p>

<p><span class="URL"><a href="https://home.in.tum.de/~traytel/hecke/">https://home.in.tum.de/~traytel/hecke/</a></span>.</p>

<p>Dmitriy Traytel</p>

<p>traytel@in.tum.de</p>

<p>Technische Universität München, 2010.</p>


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