<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.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.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 [JM97] is included, as is Kleshchev's <a href="chapBib.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>
<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.html#biBDJ1">[DJ86]</a> <a href="chapBib.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 <spanclass="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">μ</span> of <span class="SimpleMath">n</span>, Dipper and James defined a <em>Specht module</em> <span class="SimpleMath">S(μ)</span>. Let <span class="SimpleMath">rad~S(μ)</span> be the radical of <span class="SimpleMath">S(μ)</span> and define <span class="SimpleMath">D(μ)=S(μ)/rad~S(μ)</span>. When <span class="SimpleMath">R</span> is a field, <span class="SimpleMath">D(μ)</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+...+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">μ=(μ_1,μ_2,...)</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">μ_i=μ_i+1=⋯= μ_i+e-1>0</span>; otherwise, <span class="SimpleMath">μ</span> is <em><span class="SimpleMath">e</span>-regular</em>. Dipper and James <a href="chapBib.html#biBDJ1">[DJ86]</a> showed that <span class="SimpleMath">D(ν)≠ 0</span> if and only if <span class="SimpleMath">ν</span> is <span class="SimpleMath">e</span>-regular and that the <span class="SimpleMath">D(ν)</span> give a complete set of non-isomorphic irreducible <span class="SimpleMath">H</span>-modules as <span class="SimpleMath">ν</span> runs over the <span class="SimpleMath">e</span>-regular partitions of <span class="SimpleMath">n</span>. Further, <span class="SimpleMath">S(μ)</span> and <span class="SimpleMath">S(ν)</span> belong to the same block if and only if <span class="SimpleMath">μ</span> and <span class="SimpleMath">ν</span> have the same <span class="SimpleMath">e</span>-core <a href="chapBib.html#biBDJ2">[DJ87]</a><a href="chapBib.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">μ</span> and <span class="SimpleMath">ν</span>, where <span class="SimpleMath">ν</span> is <span class="SimpleMath">e</span> -regular, let <span class="SimpleMath">d_μ,ν</span> be the composition multiplicity of <span class="SimpleMath">D(ν)</span> in <span class="SimpleMath">S(ν)</span>. The matrix <span class="SimpleMath">D=(d_μ,ν)</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(ν)</span> are indexed by <span class="SimpleMath">e</span> -regular partitions <span class="SimpleMath">ν</span>. By general arguments, <span class="SimpleMath">P(ν)</span> has the same composition factors as <span class="SimpleMath">∑_μ d_μ,ν S(μ)</span>; so these linear combinations of modules become identified in the Grothendieck ring of <span class="SimpleMath">H</span>. Similarly, <span class="SimpleMath">D(ν) = ∑_μ d_ν,μ^-1 S(μ)</span> in the Grothendieck ring. These observations are the basis for many of the computations in <strong class="pkg">Hecke</strong>.</p>
<h4>1.3 <span class="Heading">Two small examples</span></h4>
<p>Because of the algorithm of <a href="chapBib.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.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>
<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.html#biBLT">[LT96]</a> for computing the decomposition matrices of the <span class="SimpleMath">q</span>-Schur algebras over fields of characteristic zero.</p>
<h4>1.4 <span class="Heading">Overview over this manual</span></h4>
<p>Chapter <a href="chap2.html#X79845447824F3333"><span class="RefLink">2</span></a> describes the installation of this package. Chapter <a href="chap3.html#X7ED1AB5C7E41D277"><span class="RefLink">3</span></a> shows instructive examples for the usage of this package.</p>
<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>
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