A complete description of the conjugacy classes of GL(n, q) is given, where the theories of irreducible polynomials and partitions of i n} form the atoms from which each conjugacy class of GL(n, q) is constructed. The last section of this chapter is an application to the sign table for GL(3, q). The maximal parabolic subgroup M P(3, q) of GL(3, q) will produce a significant number of irreducible characters of GL(3) , q).
Preliminaries
The theories of representations and characters of finite groups were developed towards the end of the 19th century. Therefore, the set of all class functions of a group G forms an algebra, which we denote by C(G). The collection of all characters of Gforms asubalgebra of C(G). However, it may not be clear that the product is of two characters. is again a character.
Character Tables and Orthogonality Relations
We shall use the notation Irr(G) to denote the set of all regular irreducible characters in G. The number of irreducible characters in G is equal to the number of conjugacy classes of G.
Tensor Product of Characters
Now we show that by knowing the character tables of two groups K and H, then the tensor products can be used to obtain the character table of K×H. Note that ifχ, ψ∈Irr(G), then in generalχψ6∈Irr(G). In the special case when deg(ψ) = 1, we have the following proposition.
Lifting of Characters
One of the advantages of the character table of G is that it provides us with all normal subgroups of G. Therefore, the character table can be used to decide whether Gis is a simple group or not.
Restriction and Induction of Characters
Restriction of Characters
Induction of Characters
With the above, the representation φ∗ is called the representation of G induced from the representation φ of H and is denoted by φ∗ =φ↑GH. Let φ be a class function of H. Then φ↑GH, the induced class function on G, is defined by.
Permutation Character
Therefore, the permutation character of the action of G on Ω is the same as the permutation character of the action of G on the left cosets of Gω inG, namely 1↑GG. So χ is the permutation character of G on Ω. ii) Using Frobenius reciprocity we get wethχ, ψiG=. iii) Since χ is a transitive permutation character, this follows from Corollary.
Subgroups and Associated Groups
- The General and Special Linear Groups
- Upper Triangular, p−Sylow and Parabolic Subgroups
- Weyl Group of GL(n, F )
- Center and Derived Subgroups of GL(n, F) and SL(n, F)
- Groups Related To GL(n, F )
An important subgroup of the group U T(n,F) is U T(n,F)∩SL(n,F), denoted by SU T(n,F) and called the special upper triangular group. Any element of the group U T(n,F) can be transformed into an element of the group SU U T(n,F) by multiplying by some appropriate transvections.
The BN Pair Structure of The General Linear Group
Then the character table of the group SL(2, q), q even, is obtained directly from that of GL(2, q) as a result of Corollary 3.1.5. Also, in describing the conjugation classes and irreducible characters of the group GL(2, q), we mainly follow James [40] and Steinberg [72].
Conjugacy Classes of GL(2, q)
The centralizer of an element of the second type of classes of GL(2, q) therefore consists of the elements of the formt= a b. The centralizer of an element of the classes of third type forGL(2, q) therefore consists of the elements of the formt= a 0.
Irreducible Characters of GL(2, q)
The following table shows the values of the linear characters λk on the conjugation classes GL(2, q). However, we have a very natural correspondence between the conjugation classes and the irreducible characters of the group GL(2, q). Groups F∗q,F∗q2; and their character groups Ch(F∗q) and Ch(F∗q2); are used to parameterize the conjugacy classes and irreducible characters of GL(2, q) as follows.
Character Table of SL(2, q)
By counting the number of elements in the conjugation classes we have found so far, we get 1 + 1 + q2−1. This shows that we have found all conjugation classes of SL(2, q). The number of classes in this group is. Let (F∗q)2 denote the subset ofF∗q, consisting of the square elements ofF∗q. By Lemma 3.1.4 we have that.
It was shown in the unpublished paper by Berkovich and Kazarin (see Berckovich [6]) that if a set possesses only one irreducible nonlinear character of degree p0, then it is solvable.
This indicates that there can be no further classes of SU T(2, q). We list these classes in Table 4.18.
To see that any irreducible representation of a Frobenius group has one of the above forms, see Grove [29]. At this step, we can use elementary properties of the sign tables (like the orthogonality relations) to produce the values of χq on classes of SU T(2, q). We conclude this section of the sign table for the group SU T(2, q) by making a connection to the work done by Muktibodh [57].
He defined a new concept in group theory by saying that a groupG is called aCon-Cos group if it has a proper normal subgroupN and ∀x∈G\ {1G}, the cosetN x forms a conjugation class of G. Moreover, a groupG is called a 2-Con-Cos group if it is a Con-Cos group and N splits into exactly two conjugation classes in G. Moreover, he classified all 2-Con-Cos groups.
Examples
- GL(2, 3)
- GL(2, 4)
- SL(2, 3)
- SL(2, 4)
- SU T(2, 3)
- SU T(2, 4)
- U T(2, 3)
- U T(2, 4)
We explain how to calculate irreducible sign values on GL(2,3) conjugation classes. The order and size of conjugation classes and centralizers of the above representatives are given in Table 4.35. However, the idea here is to use the SU T(2, q) character table given in Table 4.19.
Partitions
Although we do not attempt to fully describe the set Irr(GL(n, q)), we can find all irreducible characters of GL(3, q). We did this in Section 5.7. Green [27] showed that there is a complete duality between conjugation classes and irreducible characters GL(n, q). This means that each conjugation class can be associated with an irreducible character, a property that not many groups have. We attach to each partition λ an integer (λ) that basically depends on λ0. This integer is given by .
Conjugacy Classes of GL(n, q)
Representatives of Conjugacy Classes of GL(n, q)
Therefore, each conjugation class of GL(n, q) is defined by the data ({fi},{di},{zi},{νi}). The integer is called the length of the data. The integer t(n) is given by the coefficient of xn in the expansion of the series. Note that in definition 3.1.5 we defined the unipotency of an elementA∈GL(n, q) and of a subgroupH≤GL(n, q).
Sizes of Conjugacy Classes of GL(n, q)
If :V −→V is the linear transformation corresponding to P, then we claim that f is an Fq[t]− isomorphism between VT and VS. From Lemma 5.2.3, it suffices to show sef(T v) =S(f(v )), ∀v∈V, i.e., f T =Sf. In terms of matrices, this represents P A=BP, which we have. From Lemma 5.2.3, we have Sf =f T. Since f is an isomorphism, it follows that S =f T f−1. If P is the matrix corresponding to the linear transformation f, then B =P AP−1; that is A and B are similar ma-. Note that it is the length of the data parameter c. The term length of the conjugation class has a different meaning.
Regular Semisimple Elements and Primary Classes of GL(n, q)
The number of regular semisimple classes of type λ, which we denote by F(λ), is given by. Note that by Theorems 5.2.8 and 5.2.10 the number of regular semisimple classes of GL(n, q) is given by X. Now we give the main theorem of this subsection which counts the number of regular semisimple elements of GL(n, q).
Thus, each ordered pair (α, β), α, β ∈F∗q, α6=β represents a new conjugation class of this type and therefore there are (q−1)(q−2) distinct conjugation classes of this type. The class type of this format will be marked with T(5). As in the previous subcase, the number of distinct conjugation classes of this type is (q−1)(q−2). Since α∈F∗q (there areq−1 possibilities for α) andr∈Fq2\Fq (there are q2−q choices for r), it follows that there are q(q−1)2 classes of this type.
Cuspidal Characters
Parametrization of the Cuspidal Characters
Let N D(F∗qn) denote the set of all nondecomposable signs of F∗qn. Since Γ acts on the sign group of F∗qn, it follows that Γ acts on N D(F∗qn), and if θ0, θ ∈N D(F∗qn), then we have. Grøn [27] established a canonical bijection between the set S of all distinct simplexes S of degree ≤n and the set F of all distinct irreducible polynomials over Fq of degree ≤n. The number of cuspid characters in GL(n, q) is the same as the number of regular semisimple classes of GL(n, q) of the type λ = (n) ` n, which is equal to the number of irreducible polynomials of degree n over Fq . This number is given by n1 X.
Values of the Cuspidal Characters on Classes of GL(n, q)
The following examples illustrate the indexing and values of the string characters GL(2, q), GL(3, q), and GL(4, q). We calculate the cuspid characters of GL(4, q). First, we determine the indecomposable characters of F∗q4. To evaluate the base characters on the classes GL(4, q), we use Theorem 5.4.4 and similar steps used in the calculation of the values of the base characters GL(2, q).
Steinberg Characters
A network of subspaces V(n+ 1, q) with dimension at least 1 is called a projective geometry and is denoted by P G(n, q); i.e. a structure containing points, lines, planes, hyperplanes, etc. Note that the ST character values we recently obtained are the same as the Steinberg S character values found on page 50. To obtain the five Steinberg characters GL(4, q) we can use a method similar to that used in calc. Steinberg symbols GL(3, q) via geometric entities or to determine values directly.
Construction of the Characters
With the exception of casen=q = 2, the following statement shows that these q−1 linear characters are all linear characters of GL(n, q). According to Theorem 2.3.4, the number of linear characters of a group G is [G : G0], where G0 is the derived subgroup. It follows that the number of linear characters of GL(n, q) is q−1. Since equation (5.26) gives us q−1 distinct linear characters, we therefore know that there cannot be any further linear characters.
Application: Character Table of GL(3, q)
Principal Series Characters of GL(3, q)
Here we use the parabolic subgroup P(2,1) defined in Example 3.1.2 to construct some characters of GL(3, q). Through similar steps we can define those separating and non-separating classes of GL(3, q) in M P(3, q) and calculate the size of each class of M P(3, q). Note that the operation of obtaining the characters of GL(3, q) in this way is the parabolic induction described earlier.
Discrete Series Characters of GL(3, q)
Eachχ(7)k,l is therefore a J−product of the pointed characters πl of GL(2, q) and the pointed characters χk of GL(1, q). Here we give the character table of GL(3,3) and the conjugation classes of GL(3,4). We list these tables in the appendix. We conclude this chapter by mentioning that there is a complete duality between the irreducible characters and the conjugation classes of GL(n, q).
