<p>The purpose of this <strong class="pkg">GAP</strong> package is to make a collection of <span class="SimpleMath">p</span>-modular character tables (Brauer tables) of spin-symmetric groups (and some related groups) available in <strong class="pkg">GAP</strong>, thereby extending Thomas Breuer's GAP Character Table Library [1]. The SpinSym package is based on [2] which serves as the general reference here. If you are interested in computing with SpinSym I would like to refer you to [2] for further references and a more thorough description of some of the topics below. And, of course, I would like to hear from you about more or less successful attempts in using the present functionalities.
<p>where the relations are imposed for all admissable <span class="SimpleMath">i,j,k</span> with <span class="SimpleMath">|j-k|>1</span>. Provided <span class="SimpleMath">n≥ 4</span>, these groups are double covers of the symmetric group <span class="SimpleMath">Sym(n)</span> on <span class="SimpleMath">n</span> letters. Although <span class="SimpleMath">2.Sym(n)</span> and <span class="SimpleMath">(2^+).Sym(n)</span> are non-isomorphic groups for <span class="SimpleMath">n≠ 6</span>, they are isoclinic and their representation theory is very similar. By <em>choice</em>, we restrict the attention to <span class="SimpleMath">2.Sym(n)</span> . (However, if you are interested in character tables of <span class="SimpleMath">(2^+).Sym(n)</span> then have a look at <code class="code">CharacterTableIsoclinic()</code> in the <strong class="pkg">GAP</strong> Reference Manual.)</p>
<p>The natural epimorphism <span class="SimpleMath">π: 2.Sym(n) -> Sym(n), t_i↦ (i,i+1)</span> , whose kernel is generated by the central involution <span class="SimpleMath">z</span>, gives rise to the double cover <span class="SimpleMath">2.Alt(n)=Alt(n)^{π^-1}</span> of the alternating group <span class="SimpleMath">Alt(n)</span> as the preimage of <span class="SimpleMath">Alt(n)</span> under <span class="SimpleMath">π</span>. Irreducible faithful representations of <span class="SimpleMath">2.Sym(n)</span> or <span class="SimpleMath">2.Alt(n)</span> are called spin representations and a similar `spin' terminology is used for all related faithful objects, to set them apart from the non-faithful objects that belong esssentially to Sym(n) or Alt(n), respectively.
<h4>1.1 <span class="Heading">The data part</span></h4>
<p>The package contains complete Brauer tables of <span class="SimpleMath">2.Sym(n)</span> and <span class="SimpleMath">2.Alt(n)</span> up to degree <span class="SimpleMath">n=18</span> in characteristic <span class="SimpleMath">p=3,5,7</span>. Thus it includes the corresponding Brauer tables of <span class="SimpleMath">Sym(n)</span> and <span class="SimpleMath">Alt(n)</span>. Moreover, Brauer tables of <span class="SimpleMath">Sym(n)</span> and <span class="SimpleMath">Alt(n)</span> up to degree <span class="SimpleMath">n=19</span> in characteristic <span class="SimpleMath">p=2</span> are part of the package too.</p>
<p>Every Brauer table comes with lists of character parameters (row labels) and class parameters (column labels), see <a href="chap2.html#X7FFBFA7E7E31EA14"><span class="RefLink">2.2</span></a> and <a href="chap2.html#X857C89397E32A4E1"><span class="RefLink">2.3</span></a>. I would like to mention that only some of the data is `new', large portions date back to the work of James, Morris, Yaseen, and the Modular Atlas Project. Detailed references are to be found in [2]. The 2-modular tables of Sym(n) and Alt(n) for n=18,19 were computed jointly by Jürgen Müller and the author.
<p>Please note that some of our Brauer tables differ to some extent from those contained in the <strong class="pkg">GAP</strong> Character Table Library <a href="chapBib.html#biBctbllib">[1]</a> (for example, in terms of the ordering of conjugacy classes and characters or in terms of their parameters). Therefore it seemed appropriate to collect these tables in their own package - so here we are.</p>
<p>I'm grateful to Thomas Breuer for supporting the idea of writing this package and for converting my tables into the right GAP Character Table Library format.
<p>Besides Brauer tables, the package provides some related functionalities such as functions that determine class fusions of subgroup character tables and functions that compute character tables of some Young subgroups of <span class="SimpleMath">2.Sym(n)</span> .</p>
<h4>1.3 <span class="Heading">Installation and loading</span></h4>
<p>To install this package, download the archive file <code class="keyw">spinsym-1.5.2.tar.gz</code> and unpack it inside the <code class="keyw">pkg</code> subdirectory of your <strong class="pkg">GAP</strong> installation. It creates a subdirectory called <code class="keyw">spinsym</code>. Then load the package using the <code class="keyw">LoadPackage</code> command.</p>
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