GAP (SpinSym) - Chapter 1: Introduction

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1 Introduction
1.1 The data part
1.2 The functions part
1.3 Installation and loading

### 1 Introduction

The purpose of this GAP package is to make a collection of \(p\)-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 [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.

The term `spin-symmetric' refers to the groups

\[ 2.Sym(n)= < z,t_1,...,t_{n-1} : z^2=1, t_i^2=(t_it_{i+1})^3=z, (t_jt_k)^2=z > \]

and

\[(2^+).Sym(n)= < z,t_1,...,t_{n-1} : z^2=1, t_i^2=(t_it_{i+1})^3=1, (t_jt_k)^2=z, zt_i=t_iz > \]

where the relations are imposed for all admissable \(i,j,k\) with \(|j-k|>1\). Provided \(n\geq 4\), these groups are double covers of the symmetric group \(Sym(n)\) on \(n\) letters. Although \(2.Sym(n)\) and \((2^+).Sym(n)\) are non-isomorphic groups for \(n\neq 6\), they are isoclinic and their representation theory is very similar. By choice, we restrict the attention to \(2.Sym(n)\) . (However, if you are interested in character tables of \((2^+).Sym(n)\) then have a look at `CharacterTableIsoclinic()` in the GAP Reference Manual.)

The natural epimorphism \(\pi: 2.Sym(n) \to Sym(n), t_i\mapsto (i,i+1)\) , whose kernel is generated by the central involution \(z\), gives rise to the double cover \(2.Alt(n)=Alt(n)^{{\pi^{-1}}}\) of the alternating group \(Alt(n)\) as the preimage of \(Alt(n)\) under \(\pi\). Irreducible faithful representations of \(2.Sym(n)\) or \(2.Alt(n)\) 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.

#### 1.1 The data part

The package contains complete Brauer tables of \(2.Sym(n)\) and \(2.Alt(n)\) up to degree \(n=18\) in characteristic \(p=3,5,7\). Thus it includes the corresponding Brauer tables of \(Sym(n)\) and \(Alt(n)\). Moreover, Brauer tables of \(Sym(n)\) and \(Alt(n)\) up to degree \(n=19\) in characteristic \(p=2\) are part of the package too.

Every Brauer table comes with lists of character parameters (row labels) and class parameters (column labels), see 2.2 and 2.3. 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.

Please note that some of our Brauer tables differ to some extent from those contained in the GAP Character Table Library [1] (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.

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.

#### 1.2 The functions part

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 \(2.Sym(n)\) .

#### 1.3 Installation and loading

To install this package, download the archive file `spinsym-1.5.2.tar.gz` and unpack it inside the `pkg` subdirectory of your GAP installation. It creates a subdirectory called `spinsym`. Then load the package using the `LoadPackage` command.

    gap> LoadPackage("spinsym");

The SpinSym package banner should appear on the screen. You may want to run a quick test of the installation:

    gap> dir:= DirectoriesPackageLibrary( "spinsym", "tst" )[1];;
    gap> tst:= Filename( dir, "testall.tst" );;
    gap> Test( tst );
    true

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