Fischer-Clifford matrices of the generalized symmetric group and some associated groups.

But we will use combinatorics to construct the Fischer-Clifford matrices for the groups discussed here. Examples include using the program 5.2.4 to construct the Fischer-Clifford matrices. Note, however, that these matrices are the equivalent form of the Fischer-Clifford matrices of B(2,12) and B(4,5).

In this thesis we constructed the character tables of groups B(2,6) and B(3,5) and some group extensions related to these two groups and B(3,3).

Notation

Contents

List of Tables

Preliminaries

Introduction

PRELIMINARIES 2 Fischer-Clifford matrices of G together with the fusion maps and the character tables

PRELIMINARIES 3 ces of group extensions of a faithful irreducible constituent of the elementary abelian

PRELIMINARIES

Group Extensions

Conjugation classes of a finite group give a lot of important information about the structure of the group. The use of computational methods to determine group conjugation classes is another aspect that should be mentioned. A coset analysis technique used to determine the conjugacy classes of the elements of split and non-split extensions of G = N.G.

For any conjugation class [g] of G, the conjugation classes of G in which N is abelian will be determined by the action by conjugation of Cy on the elements of Ng.

PRELIMINARIES 9

Representations and Characters of Finite Groups

In this section, we discuss some group representation and sign results that are applicable to the Fischer-Clifford matrix technique, which will be described in Section 1.5. We will not deal much with modular signs, so we will mainly devote our discussion to ordinary representations and signs, that is, representations and signs of a finite group. Consider a complex field C. Proofs are omitted in most of the following material, but reference is made to [28] for an extensive treatment of sign theory.

PRELIMINARIES 12 It is clear that equivalent representations afford the same character, since similar

PRELIMINARIES 13 Lemma 1.3.6 Let G be a group, p be a representation of G which affords the char-

We now discuss the relationships between the characters of a group G and those of its subgroups. If P is a representation of G, then the restriction from P to H is a representation of H, denoted by PH or P-l. If X is a character of G given by P, then the restriction of X to H is a character of H given by the representation PH, and is denoted by XH or X-l.H.

But under certain conditions the irreducibility of X implies that XH is irreducible, as in the following result.

PRELIMINARIES 15

PRELIMINARIES 16

PRELIMINARIES 17 Definition 1.3.18 Let G be a group, N a normal subgroup of G and X be a character

PRELIMINARIES 18

Clifford Theory

Then the automorphism a of G causes a permutation on the conjugation classes of G and therefore also on the columns of X. Consequently, a permutation on the irreducible characters Xi of G and therefore also on the rows of X causes a permutation permutation on the conjugacy classes of G and thus causes a permutation on the columns of X.

Let d1 , d2 be the number of orbits of K in the irreducible characters and in the conjugation classes of G respectively.

PRELIMINARIES 22

PRELIMINARIES 23

Since by assumption p is a common representation in Hand tt 1E H we have p(ttI) = p(t)p(tI). Since Gis is a semidirect product of N by G, then each x E G can be uniquely expressed as x = ng, where n E N, 9 E G. We now claim that eH = Lpf3(1c)f3'ljJ, where f3 ranges over all irreducible characters of H containing N in their nuclei.

PRELIMINARIES 27

Fischer-Clifford Matrices

PRELIMINARIES 28

PRELIMINARIES 29 conjugacy classes of G which correspond to the coset Ng. The matrix M(g) is called

PRELIMINARIES 30

Then, according to Theorem 1.4.4, group A acts on the conjugation classes of elements of K and on the irreducible characters of K, resulting in the same number of orbits. The Fischer-Clifford matrix M(g) is divided row-wise into blocks, each block corresponding to an inertia group. The rows of M(g) are indexed by R(g) and on the left side of each row we write jCHi(Yk)l, where Yk merges into [g] in G.

Combinatorics

Compositions
COMBINATORICS
COMBINATORICS 38

Partitions

COMBINATORICS 39
COMBINATORICS 40

Magic Matrices

COMBINATORICS 42
COMBINATORICS 43

We obtain sets with the following respective numbers of 4-compositions afn grouped into sets for each 3-composition afn: {n+I, n, n-I,. We have developed the following programs m GAP [22] for calculating m-compositions of n- and multinomial coefficients, respectively. Examples 2.1.5 Below we give examples of sets of m-compositions of n and corresponding multinomial coefficients.

However, for most of the results here, we only require the partial magic matrices M/3,d,>"s'. Using Definition 2.3.1, we have constructed the following program which is used to compute the partial magic matrices M/3 , 5,>"s for each As = 1, ..,n.

The Group B(m, n) and Some Associated Groups

THE GROUP B(M, N) AND SOME ASSOCIATED GROUPS 45

The Generalized Symmetric Group B(m, n)

The crown product of Zm with Sn is a split extension of N through Sn called the generalized symmetric group, B(m,n).

THE GROUP B(M, N) AND SOME ASSOCIATED GROUPS 46

THE GROUP B(M,N) AND SOME ASSOCIATED GROUPS 48

THE GROUP B(M, N) AND SOME ASSOCIATED GROUPS 50 will also determine the number of arrangements of symbols of n together with

THE GROUP B(M,N) AND SOME ASSOCIATED GROUPS 50 will also determine the number of arrangements of symbols of together with. THE GROUP B(M, N) AND SOME ASSOCIATED GROUPS 51Remark 3.1.8 The formula for orders of centralizers in B(m,n) can be ex-.

THE GROUP B(M, N) AND SOME ASSOCIATED GROUPS 51 Remark 3.1.8 The formula for the orders of centralizers in B(m, n) may be ex-

The Actions of Sn on Nand Irr(N)

THE GROUP B(M, N) AND SOME ASSOCIATED GROUPS 55

Since each d E dSn defines the same m-composition of n, we will equivalently say that the orbit dSn defines (k1 ,k2, .. ,km)' Let A(n, m)dSn denote the m-composition of n defined is by dSn. In [47], Mpono developed a computer program in CAYLEY [10] that calculates the elements of the orbits of the action of Sn on N, but this program does not explicitly give the lengths of the orbits or any information about the inertia factor. groups. We choose not to distinguish between the ei taking the same value (l(i-l) on elements of different copies of Zm.

So an irreducible character of N can be written as e = TIief9Ui, where some ei's can be the same for different i's.

The Groups Bs(m, n) and BQ(m, n)

The Actions of Sn on Sand Irr(S)

The Actions of Sn on Q and Irr(Q)

Fischer-Clifford Matrices of B(m, n)

Fischer-Clifford Theory Applied to B(m, n)

It is also clear that the inertia factor group Xl is a young subgroup, IXI = Skl X. Now by Theorem 1.4.11 (Mackey's Theorem (1958)) and since B(m, n) is a split extension of the abelian group N with Sn, each irreducible character N = Z:; extends to the irreducible character of its inertial group in B(m,n). And by Theorem 1.4.6 (Gallagher's Theorem), all irreducible signs B(m, n) are of the form CXI'l/J) t B(m, n), where Xl is the extension of the irreducible sign Xl from N to its inertia group Ixl, 'l/J is a sign of IX1, so N ~ Ker('l/J) and Xl'l/J is the tensor product of the signs.

Defining Fischer-Clifford Matrices of B(m, n)

The conjugation classes of the inertia factor groups, which merge with a conjugation class of Sn of type (1A12A2 ..nAn ), are of type. The conjugation classes of the elements of IxI' which are mapped to a conjugation class [YkJ of the inertia factor group IXl under the natural homomorphism ( : B(m, n) -t Sn) and merge with a conjugation class [bkJ ofB(m, n) ) , of type is. The character table of B(m, n) is constructed by multiplying the partial character tables of the inertia factor groups IXl of B(m,n) with the corresponding rows of the Fischer-Clifford matrices of B(m, n), according to the mergers of the conjugation classes of IXI in [gJ as indicated above.

Fischer-Clifford Matrices of B (rn, n) - a Combinato- rial Approach

FISCHER-CLIFFORD MATRICES OF B(M, N)

With the necessary changes, the proof of the following theorem is similar to the proof of Theorem 4.3.1.

Examples

The number seven of the 2 splitting groups [16J found above shows that the Fischer-Clifford matrix F(2,16) has order 7. However since using Theorem 4.3.1 involves constructing a number of magic matrices for each entry of a Fischer-Clifford matrix we show how to calculate only the entry of F(2, 16).

FISCHER-CLIFFORD MATRICES OF B(M, N) Intersecting the sets M({3, 6) and M(b,6) above, we obtain that

However, we can efficiently compute the entries of the Fischer-Clifford matrices B(2, 6) using Theorems 4.3.2 and 4.3.5.

Constructing the Fischer-Clifford Matrices of B(3, 5)

Computing Fischer-Clifford Matrices of B(m, n)

Introduction

Programmes Written for GAP

COMPUTING FISCHER-CLIFFORD MATRICES OF B(M, N)100

COMPUTING FISCHER-CLIFFORD MATRICES OF B(M, N)102 All of the above problems except (iv) have been completely solved (see Programme

Examples

COMPUTING FISCHER-CLIFFORD MATRICES OF B(M,N)111 5.3.2 Computing the Fischer-Clifford Matrices of B(3, 5)

COMPUTING FISCHER-CLIFFORD MATRICES OF B(M, N)116

It is well known that the character table of the generalized symmetric group B(m, n) can be computed in GAP, where B(m, n) is considered the crown product of the cyclic group Zm of order m with the symmetric group Sn. For example, we can use the following code in GAP to calculate the sign table of B(m, n). In fact, in [50] Pfeiffer has given programs in GAP which compute the grade tables for wreath products with symmetric groups.

However, it may not be necessary to get hold of the complete character table of a group, we may instead want to get a partial character table of the group. Furthermore, due to limited workspace in GAP, we can only compute character tables of B(m,n) for small values of m and n. It is for these reasons, among others, that the technique of Fischer-Clifford matrices is sometimes used to construct the character tables of generalized symmetric groups.

In this chapter we use the Fischer-Clifford matrices constructed in Chapters 4 and 5 to construct the sign tables for the groups B(2,6) and B(3,5). As indicated in section 4.2, the character table of B(m, n) is constructed by multiplying the partial character tables of the inertia factor groups SkI xSk2x· .XSkm with the corresponding rows of the Fischer-Clifford matrices in accordance with the fusions of the conjugation classes ofSkI x Sk2 X. THE CHARACTER TABLE OFB( M, N) 118 the multiplication using GAP is performed, the rows within the blocks of Fischer-.

The Group B(2, 6)

THE CHARACTER TABLE OF B(M, N)

THE CHARACTER TABLE OF B(M,N)

The Group B(3,5)

In this section we construct the sign table for another example of the generalized symmetric group B(m, n) but for prime numbers m f: 2 and in particular we consider the group B(3,5) which has a higher order than B (3, 4) performed in [5]. We construct the sign table for B(3,5) using the Fischer-Clifford matrices given in Tables 5.2 through 5.5. The conjugation classes for B(3,5) including the respective orders of centralizers are given in Tables 6.5, 6.6 and 6.7.

THE CHARACTER TABLE OF B(M,N) 128

A Note on the Character tables of B(2, 12) and B( 4,5)
Fischer-Clifford Matrices of Bs(p, n)
The Group Bs(2, 6)
The Group Bs(3,3)
The Group Bs(3, 5)

The Fischer-Clifford Matrices of Bs (3, 5)

Some Preliminary Results
Fischer-Clifford Matrices of EQ (rn, n)

To construct the character table of the group 26:SP(6,2), which is a maximal subgroup of the Fischer group Fin', using the method of Fischer-Clifford matrices, we need the character table of Bs( 2,6) (see We also use the Fischer-Clifford matrices of the groups B5(2,6) to construct its character table This was done by the Fischer-Clifford matrices of the associated group from the Fischer-Clifford matrices of B(2, n) to construct.

We now consider the rows of the Fischer-Clifford matrices Bs(p, n), whose indices, as we know, must correspond to groups of inertia factors Bs(p, n). Now, in the following sections, we will construct the Fischer-Clifford matrices and sign tables of the groups Bs(2,6), Bs(3,3) and Bs(3,5). In [47], Mpono used the properties of Fischer-Clifford matrices given in Section 1.5.2 to construct the Fischer-Clifford matrices of the Bs(2,6) group.

We note that most of the details of the Fischer-Clifford matrices of Bs(2,6) were calculated while constructing the Fischer-Clifford matrices of B(2,6) in Chapters 4 and 5. In this subsection we use Proposition 7.1. 16 and the Fischer-Clifford matrices of B(3,3) given in Table 7.6 to construct the Fischer-Clifford matrices of the group Bs(3,3). The character table of Bs(3,5) is constructed by multiplying the partial character tables of the inertia factor groups of Bs(3,5) by the corresponding rows of the Fischer-Clifford matrices in Table 7.13 according to the mergers given in Table 6.9 ).

In this chapter we develop a method for constructing the Fischer-Clifford matrices of the group BQ(m,n). This is done by choosing data from the appropriate rows and columns of the Fischer-Clifford matrices of B(2,n). In this section we discuss some results that we use in section 8.2 for the construction of the Fischer-Clifford matrices of BQ(m, n).

Then /3 (or equivalently the set of g-equivalent m-sets of partition [A]) indexes a column of the Fischer.

Table 7.1: Conjugacy classes of Bs(2, 6)

THE GROUP BQ(M,N) 175

THE GROUP BQ(M,N) 176 of length 2 each with no negative sign, a21 = 4 cycles of length 1 each with

THE GROUP BQ(M,N) Example 8.2.7

THE GROUP BQ(M,N) 178