<!--  chap1.xml    SpinSym GAP package    Lukas Maas  -->

<Chapter Label="chap1">
<Heading>Introduction</Heading>

The purpose of this  &GAP; package is to make a collection of 
<M>p</M>-modular character tables (Brauer tables) of 
spin-symmetric groups (and some related groups) available in 
&GAP;, thereby extending  Thomas Breuer's  &GAP; Character Table Library 
<Cite Key="ctbllib"/>.
The &SpinSym; package is based on <Cite Key="Maas2011"/> which serves as the 
general reference here. 
If you are interested in computing with &SpinSym; I would like to refer you to 
<Cite Key="Maas2011"/> 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/>

The term `spin-symmetric' refers to the groups 
<Alt Only="LaTeX">
	<Display Mode="M">	\tilde{S}_n=\langle\ z,t_1,\ldots,t_{n-1}\ :\ 
	z^2=1,\ t_i^2=(t_it_{i+1})^3=z,\ (t_jt_k)^2=z\ \rangle
	</Display>
</Alt> 
<Alt Not="LaTeX">
	<Display Mode="M">	2.Sym(n)= &lt; z,t_1,...,t_{n-1} : 
	z^2=1, t_i^2=(t_it_{i+1})^3=z, (t_jt_k)^2=z &gt; 
	</Display>
</Alt> 
and 
<Alt Only="LaTeX">
	<Display Mode="M">\hat{S}_n=\langle\ z,t_1,\ldots,t_{n-1}\ :\ 
	z^2=1,\ t_i^2=(t_it_{i+1})^3=1,\ (t_jt_k)^2=z,\  zt_i=t_iz\ \rangle
	</Display>
</Alt>
<Alt Not="LaTeX">
	<Display Mode="M">(2^+).Sym(n)= &lt; z,t_1,...,t_{n-1} : 
	z^2=1, t_i^2=(t_it_{i+1})^3=1, (t_jt_k)^2=z,  zt_i=t_iz &gt;
	</Display>
</Alt>

where the relations are imposed for all admissable <M>i,j,k</M> 
with <M>|j-k|>1</M>. Provided <M>n\geq 4</M>, these groups are 
double covers of the symmetric group <Alt Only="LaTeX"><M>S_n</M></Alt>
<Alt Not="LaTeX"><M>Sym(n)</M></Alt> 
on <M>n</M> letters. Although 
<Alt Not="LaTeX"><M>2.Sym(n)</M></Alt>
<Alt Only="LaTeX"><M>\tilde{S}_n</M></Alt> 
and 
<Alt Not="LaTeX"><M>(2^+).Sym(n)</M></Alt>
<Alt Only="LaTeX"><M>\hat{S}_n</M></Alt> 
are non-isomorphic groups for <M>n\neq 6</M>, they are isoclinic 
and their representation theory is very similar. 
By <E>choice</E>, we restrict the attention to  
<Alt Not="LaTeX"><M>2.Sym(n)</M></Alt>
<Alt Only="LaTeX"><M>\tilde{S}_n</M></Alt>. 
(However, if you are interested in character tables of 
<Alt Not="LaTeX"><M>(2^+).Sym(n)</M></Alt>
<Alt Only="LaTeX"><M>\hat{S}_n</M></Alt> then have a look at 
<C>CharacterTableIsoclinic()</C> in the &GAP; Reference Manual.) <P/>
The natural epimorphism
<Alt Not="LaTeX"><M>\pi: 2.Sym(n) \to Sym(n), 
t_i\mapsto (i,i+1)</M></Alt>
<Alt Only="LaTeX"><M>\pi: \tilde{S}_n\to S_n,\ 
t_i\mapsto (i,i+1)</M></Alt>, 
whose kernel is generated by the central involution <M>z</M>, 
gives rise to the double cover 
<Alt Not="LaTeX"><M>2.Alt(n)=Alt(n)^{{\pi^{-1}}}</M></Alt>
<Alt Only="LaTeX"><M>\tilde{A}_n= A_n^{\pi^{-1}}</M></Alt>
of the alternating group 
<Alt Only="LaTeX"><M>A_n</M></Alt>
<Alt Not="LaTeX"><M>Alt(n)</M></Alt>
as the preimage of 
<Alt Only="LaTeX"><M>A_n</M></Alt>
<Alt Not="LaTeX"><M>Alt(n)</M></Alt> 
under <M>\pi</M>. Irreducible faithful representations of 
<Alt Not="LaTeX"><M>2.Sym(n)</M></Alt>
<Alt Only="LaTeX"><M>\tilde{S}_n</M></Alt> 
or 
<Alt Not="LaTeX"><M>2.Alt(n)</M></Alt>
<Alt Only="LaTeX"><M>\tilde{A}_n</M></Alt>
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 
<Alt Only="LaTeX"><M>S_n</M></Alt>
<Alt Not="LaTeX"><M>Sym(n)</M></Alt>
or 
<Alt Only="LaTeX"><M>A_n</M></Alt>
<Alt Not="LaTeX"><M>Alt(n)</M></Alt>, 
respectively. 

<Section Label="chap1:The data part"> 
<Heading>The data part</Heading>
The package contains complete Brauer tables of 
<Alt Not="LaTeX"><M>2.Sym(n)</M></Alt>
<Alt Only="LaTeX"><M>\tilde{S}_n</M></Alt> 
and 
<Alt Not="LaTeX"><M>2.Alt(n)</M></Alt>
<Alt Only="LaTeX"><M>\tilde{A}_n</M></Alt> up to degree 
<M>n=18</M> in characteristic <M>p=3,5,7</M>. Thus it includes 
the corresponding Brauer tables of 
<Alt Only="LaTeX"><M>S_n</M></Alt>
<Alt Not="LaTeX"><M>Sym(n)</M></Alt>
and 
<Alt Only="LaTeX"><M>A_n</M></Alt>
<Alt Not="LaTeX"><M>Alt(n)</M></Alt>. Moreover, Brauer tables of 
<Alt Only="LaTeX"><M>S_n</M></Alt>
<Alt Not="LaTeX"><M>Sym(n)</M></Alt>
and 
<Alt Only="LaTeX"><M>A_n</M></Alt>
<Alt Not="LaTeX"><M>Alt(n)</M></Alt> up to degree <M>n=19</M> 
in characteristic <M>p=2</M> are part of the package too. <P/>
Every Brauer table comes with lists of character parameters 
(row labels) and class parameters (column labels), see 
<Ref Sect="chap2:Character parameters"/> and 
<Ref Sect="chap2:Class parameters"/>.
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 <Cite Key="Maas2011"/>. 
The <M>2</M>-modular tables of 
<Alt Only="LaTeX"><M>S_n</M></Alt>
<Alt Not="LaTeX"><M>Sym(n)</M></Alt>
and <Alt Only="LaTeX"><M>A_n</M></Alt>
<Alt Not="LaTeX"><M>Alt(n)</M></Alt> for <M>n=18,19</M> 
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 &GAP; Character Table Library 
<Cite Key="ctbllib"/> (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/>
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.
</Section>

<Section Label="chap1:The functions part"> 
<Heading>The functions part</Heading>
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 
<Alt Not="LaTeX"><M>2.Sym(n)</M></Alt>
<Alt Only="LaTeX"><M>\tilde{S}_n</M></Alt>. 
</Section>

<Section Label="chap1:Installation and loading">
<Heading>Installation and loading</Heading>
To install this package, download the archive file
<K>spinsym-1.5.2.tar.gz</K> and unpack it inside the <K>pkg</K> 
subdirectory of your &GAP; installation. It creates a 
subdirectory called <K>spinsym</K>. Then load the package
using the <K>LoadPackage</K> command.
<Log>
<![CDATA[
gap> LoadPackage("spinsym");
]]>
</Log>
The <Package>SpinSym</Package> package banner should appear 
on the screen. You may want to run a quick test of the 
installation: 
<Log>
<![CDATA[
gap> dir:= DirectoriesPackageLibrary( "spinsym", "tst" )[1];;
gap> tst:= Filename( dir, "testall.tst" );;
gap> Test( tst );
true
]]>
</Log>
</Section>
</Chapter>
