Computation of Irreducible Characters of GL(5,q)

TL;DR

This paper computes the irreducible characters of the five-dimensional general linear group over a finite field, extending the understanding of its representation theory using Green's techniques.

Contribution

It provides the explicit determination of all irreducible characters of GL(5,q), a significant advancement in the representation theory of finite groups.

Findings

Complete set of irreducible characters for GL(5,q) determined

Methodology based on Green's techniques applied to higher dimensions

Enhances understanding of the structure of linear groups over finite fields

Abstract

In this paper we determine the ordinary irreducible characters of the five-dimensional full linear group over a Galois field of q elements. We use the techniques developed by J. A. Green.

Equations

$${\rho }_1=1^5,{\rho }_2=2.1^3,{\rho }_3=2^2.1,{\rho }_4=3.1^2,{\rho }_5=3.2,{\rho }_6=4.1,{\rho }_7=5.$$

$${\omega }_1={\omega }_2^{q+1}={\omega }_3^{q^2+q+1}={\omega }_4^{q^3+q^2+q+1}={\omega }_5^{q^4+q^3+q^2+q+1}.$$

$${\epsilon }_1={\epsilon }_2^{q+1}={\epsilon }_3^{q^2+q+1}={\epsilon }_4^{q^3+q^2+q+1}={\epsilon }_5^{q^4+q^3+q^2+q+1}$$

$$\sum _{g\in G}\mid \nu (g)\mid deg(g)=n$$

$$U(f)=\left(\begin{array}[]{cccccccc}&1&&&&&&\\ &&.&&&&&\\ &&&.&&&&\\ &&&&.&&&\\ &&&&&.&&\\ &&&&&&1&\\ a_{0}&a_{1}&a_{2}&.&.&.&.&a_{d-1}\end{array}\right)$$

$$U_{m}(f)=\left(\begin{array}[]{cccccccc}U(f)&I_{d}&&&&&\\ &U(f)&I_{d}&&&&\\ &&&.&&&&\\ &&&&&.&&\\ &&&&&&.&\\ &&&&&&&I_{d}\\ &&&&&&&U(f)\end{array}\right)$$

$$U_{\lambda }(f)=diag\{U_{{\ell }_1}(f),U_{{\ell }_2}(f),\dots ,U_{{\ell }_r}(f)\}.$$

$$(a_0+a_{1}X+a_2{X}^2+\cdots +a_r{X}^r)v=a_0+a_{1}Av+a_2{A}^{2}v+\cdots +a_r{A}^{r}v$$

$$k(\lambda,q)=\left\{\begin{array}[]{ll}(1-q)(1-q^{2})(1-q^{3})\cdots(1-q^{p-1})&\mbox{if $p>1$}\\ 0&\mbox{if $p=1$}\end{array}\right.$$

$$Q_{\rho }^{\lambda }(q)=\sum g_{{\lambda }_1,{\lambda }_2\cdots }^{\lambda }(q).k({\lambda }_1,q).k({\lambda }_2,q)\cdots$$

$$Q_{\rho +\sigma }^{\nu }(q)=\sum g_{\lambda ,\mu }^{\nu }(q)Q_{\rho }^{\lambda }(q)Q_{\sigma }^{\mu }(q)$$

$$\sum _{\mid \rho \mid =n}\frac{1}{z}{Q}_{\rho }^{\nu }(q)=1$$

$$\alpha:\left\{\begin{array}[]{lllll}x_{11},x_{12}\rightarrow f_{11}&&&&\hskip 36.135pt\rho(\alpha,f_{11})=1^{2}\\ x_{13},x_{21}\rightarrow f_{12}&&&&\hskip 36.135pt\rho(\alpha,f_{12})=1.2\end{array}\right.$$

$$\beta:\left\{\begin{array}[]{lllll}x_{21}\rightarrow f_{11}&&&&\hskip 14.45377pt\rho(\beta,f_{11})=2\\ x_{11},x_{12},x_{13}\rightarrow f_{12}&&&&\hskip 14.45377pt\rho(\beta,f_{12})=1^{3}\end{array}\right.$$

$$U_{\rho }({\xi }^{\rho })=U({\xi }_{11},{\xi }_{12},\cdots ,{\xi }_{1r_1},{\xi }_{21},\cdots ,{\xi }_{2r_2}.\cdots ).$$

$$U_{\rho }({\xi }^{\rho }m)=U_{\rho }({\xi }^{\rho }\alpha )U({\gamma }_{11},{\gamma }_{12},\cdots ,{\gamma }_{1r_1},{\gamma }_{21},\cdots ,{\gamma }_{2r_2}.\cdots ).$$

$$U(c)=\sum _{\rho }\sum _{m}Q(m,c)U_{\rho }(x^{\rho }m).$$

$$Q(m,c)=\prod _{f\in F}\frac{1}{z_{\rho (m,f)}}{Q}_{\rho (m,f)}^{{\nu }_c(f)}(q^{deg(f)})$$

$$\left[\begin{array}[]{cc}A_{1}&0\\ \ast&A_{2}\end{array}\right]$$

$$Q(m,c)=\prod _{f\in F}\frac{1}{z_{\rho (m,f)}}{Q}_{\rho (m,f)}^{{\nu }_c(f)}(q^{deg(f)})$$

$$\chi (m,e)=\prod _{g\in G}\frac{1}{z_{\rho (m,e)}}{Q}_{\rho (m,e)}^{{\nu }_e(g)}(q^{deg(g)})$$

$$J_1(h_{11})\circ J_1(h_{12})\circ \cdots J_1(h_{1r_1})\circ J_2(h_{21})\circ \cdots J_2(h_{2r_2})\circ \cdots$$

$$\prod _d\{\sum S_d(h_{d1}:{\xi }_{d1})\cdots S_d(h_{d{r}_d}:{\xi }_{d{r}_d}\}$$

$$S_d(k:\xi )={\theta }^k(\xi )+{\theta }^{qk}(\xi )+\cdots +{\theta }^{q^{d-1}k}(\xi ).$$

$$B^{\rho }(h)(c)=\sum _{m}Q(m,c)B_{\rho }(h:{\xi }^{\rho }m)$$

$$B^{\rho }(h^{\rho }m:{\xi }^{\rho })=B^{\rho }(h_{11}m,h_{12}m,\cdots ,h_{1r_1}m,h_{21}m,\cdots ,h_{2r_2}m,\cdots :{\xi }^{\rho })$$

$$(-1{)}^{5-\sum \mid \nu (g)\mid }\sum _{\rho ,m}\chi (m,e)B^{\rho }(h^{\rho }m)$$

$${\Psi }_5(q)\prod _{g\in G}(-1{)}^{\mid \nu (g)\mid }\{\nu (g):q^{deg(g)}\}$$

$$\{\lambda :q\}=q^{{\ell }_2+2{\ell }_3+\cdots }\prod _{1\le r<s\le p}(1-q^{({\ell }_r-{\ell }_s-r+s)})/\prod _{r=1}^{r=p}{\Phi }_{{\ell }_r+p-r}(q)$$

$$\frac{1}{120}(q+1)(q^2+q+1)(q^3+q^2+q+1)(q^4+q^3+q^2+q+1){\epsilon }_1^{5ia}+\frac{1}{12}(q^2+q+1)(q^3+q^2+q+1)(q^5-1){\epsilon }_1^{5ia}+$$

Full text

Computation of Irreducible Characters of $GL(5,q)$

By Shiv K. Gupta

1 INTRODUCTION

111Mathematics Subject Classification. Primary 20C15; Secondary 20C30

In this paper we determine the ordinary irreducible characters of the 5-dimensional full linear group over a Galois field of $q$ elements. We use the techniques developed by Green [1].

The characters of $GL(2,q)$ have been known for long time. Those of $GL(3,q)$ and $GL(4,q)$ have been determined by Steinberg [11]. Steinberg has also developed some techniques to find some of the characters in the general case.

There are 42 “types” of characters of $GL(5,q)$. Each of these types contains a number of characters which is a function of $q$. The total number of characters of $GL(5,q)$ is $q^{5}-q^{2}-q+1$. There is a perfect symmetry in the description of conjugate classes and characters.

Thee 42 “types” of characters may in turn be grouped into 17 classes, such that the characters belonging to the same class involve the same basic constituents–the basic characters. For example, the characters of the first seven types $A^{\prime}_{1}$ to $A^{\prime}_{7}$ (each containing $q-1$ characters) form one class. Each of these types involve the same basic constituents–the basic characters of type $A^{\prime}$ , occurring with distinct coefficients. [These coefficients being the linear combinations which involve what are known as Green Polynomials, with (dual) functions which involve the characters of the symmetric groups.]

2 Definitions and Notation

We have used the standard notation throughout. $GF(q)$ will denote a finite field of $q$ elements; $GL(n,q)$ the group of $n\times n$ non-singular matrices over GF($q$), $S_{n}$ the symmetric group on $n$ letters. If $\lambda$ is a partition of $n$ then we shall write $\mid\lambda\mid=n$. Conjugate class “types” of $GL(5,q)$ have been denoted by the letters $A,B,C,\ldots,Q$ and the characters types by $A^{\prime},B^{\prime},C^{\prime},\ldots,Q^{\prime}$.

Let $\rho=\{\ell_{1},\ell_{2},\ldots\},\ell_{1}\geq\ell_{2}\cdots$ be a partition of n. Then the partition $\rho^{{}^{\prime}}=\{\ell_{1}^{{}^{\prime}},\ell_{2}^{{}^{\prime}},\ldots\}$ is said to be conjugate to the partition $\rho$ if $\ell_{r}^{{}^{\prime}}$ = number of parts of $\rho$ which are $\geq r$. A conjugate class $c$ of $GL(n,q)$ is said to be primary if its characteristic polynomial is a power of irreducible (over $GF(q)$) polynomial. We shall number the seven partitions for $n=7$ as follows:

[TABLE]

$\omega_{i},1\leq i\leq 5$ will denote the primitive elements of GF($q^{i}$) such that

[TABLE]

$\varepsilon_{i},1\leq i\leq 5$ are the primitive ($q^{i}$-1)-$th$ roots of unity such that

[TABLE]

Let $f$ be an irreducible polynomial of degree $s$ over GF($q$). Let $\omega$ be a generating element of GF($q^{s}$). Then a root of $f$ has the form $\omega^{k}$ where $k$ is uniquely determined modulo $(q^{s}-1)$, and the set $\{\omega^{k},\omega^{kq},\omega^{kq^{2}},\cdots,\omega^{kq^{s-1}}\}$ consists of all the roots of $f$ since $k,kq,kq^{2},\ldots,kq^{s-1}$ are distinct residue mod $(q^{s}-1)$. Following Green [1] we shall call the set $\{k,kq,kq^{2},\ldots,kq^{s-1}\}$ an $s$-$simplex$ with $k,kq,kq^{2},\ldots,kq^{s-1}$ as its roots and $s$ its degree. $\chi_{\rho}^{\lambda}$ will denote the value of the irreducible character of $S_{\mid\lambda\mid}$ corresponding to the partition $\lambda$ of $\mid\lambda\mid$ , on the class corresponding to the partition $\rho$. $\sum_{i,j,k\ldots}$ will mean that the summation is taken over all the permutations of $i,j,k\ldots$.

Let $G$ denote the set of all simplexes of degrees $\leq n$, and $P$ be the set of all partitions of numbers $\leq n$ . Then a function $\nu:G\rightarrow P$ such that

[TABLE]

specifies what we shall call a dual class and will designate it by the symbol $e=(\ldots g^{\nu(g)}\ldots)$. To each symbol of a dual class there will correspond a character which we shall denote by the same symbol.

3 Description of Conjugate Classes

Let $f(t)=t^{d}-a_{d-1}-\cdots-a_{0}$ be a polynomial over $GF(q)$. Following Green [1] we define

[TABLE]

and

[TABLE]

where $I_{d}$ is an $d\times d$ identity matrix and there are $m$ blocks of $U(f)$.

Let $\lambda=\{\ell_{1},\ell_{2},\ldots,\ell_{p}\}\ \ell_{1}\geq\ell_{2}\ldots\geq\ell_{p}$ , be a partition of $n$. We define

[TABLE]

Then $U_{\lambda}(f)$ is a matrix with characteristic polynomial $f(t)^{\mid\lambda\mid}$. Let $A\in$ GL($n,q$) have characteristic polynomial $f_{1}^{n_{1}}.f_{2}^{n_{2}}\cdots f_{k}^{n_{k}}$ where $f_{1},f_{2},\cdots,f_{k}$ are distinct irreducibles polynomials over GF($q$). Then $\sum_{i=1}^{k}n_{i}deg(f_{i})=n$, where deg($f_{i}$) is the degree of the polynomial $f_{i}$. The matrix $A\in GL(n,q)$ is conjugate to a matrix diag{$U_{\nu_{1}}(f_{1}),U_{\nu_{2}}(f_{2}),\cdots,U_{\nu_{k}}(f_{k})$}, where $\nu_{1},\nu_{2},\cdots,\nu_{k}$ are certain partitions of $n_{1},n_{2},\cdots,n_{k}$ respectively.

Conversely, let $F$ denote the set of all irreducible polynomials over GF($q$) of degree $\leq n$. A conjugate class of GL($n,q$) can be specified by a partition valued function $\nu:F\rightarrow P$ such that $\sum_{f\in F}\mid\nu(f)\mid deg(f)=n$. We designate this class by the symbol $(\cdots f^{\nu(f)}\cdots)$.

In the case of $GL(5,q)$ we are thus able to describe the conjugate classes systematically. There are $q^{5}-q^{2}-q+1$ conjugate classes in $GL(5,q)$. These are divided into 42 “types” and all these types do appear if $q\geq 7$. Each type consists of a number of conjugate classes and this number is a function of $q$. All the classes of a particular type have similar minimal polynomial. In $GL(5,2),GL(5,3),GL(5,4)$ and $GL(5,5)$ there are respectively 17, 32, 39 and 41 types of classes. We give a list of the canonical forms of the various types of conjugate classes in table 1. The table 2 gives the number of classes in each type and also the number of elements in each class.

4 Uniform Functions and Basic Uniform Functions

Let $A$ denote a square matrix of order $n$ with coefficients in GF($q$). Let $V=V_{n}$ denote an n-dimensional vector space over GF($q$). $V$ can be considered as a GF($q$)[X]-module denoted by $V_{A}$ by defining:

[TABLE]

where $a_{i}\in$ GF$(q),v\in V$. Let $\lambda$ be a fixed partition of $n,f(t)=t$ and $A=U_{\lambda}(f)$. Set $V_{\lambda}=V_{U_{\lambda}}(f)$. If the partition $\lambda$ has $p$ parts then we define

[TABLE]

Let $\lambda_{1},\lambda_{2},\cdots,\lambda_{k}$ be any partitions such that $\sum_{i=1}^{i=k}\mid\lambda_{i}\mid=\mid\lambda\mid$ Then $g^{\lambda}_{\lambda_{1},\lambda_{2},\ldots,\lambda_{k}}$ is the number of chains $V_{\lambda}=V^{0}\supset V^{1}\supset V^{2}\supset\cdots\supset V^{k-1}\supset V^{k}=0$ of sub-modules of $V_{\lambda}$ in which the factor $V^{(i-1)}/V^{i}$ is isomorphic to $V_{\lambda_{i}},i=1,2,\cdots,k.$

4.1 Green Polynomial

Let $\lambda,\rho$ be partitions of $n$. Assume that $\rho=1^{r_{1}}.2^{r_{2}}.3^{r_{3}}\cdots$  the polynomials

[TABLE]

the sum being taken over all sequences ($\lambda_{1},\lambda_{2},\cdots$) of partitions of $\rho$ such that $\lambda_{1},\lambda_{2},\cdots,\lambda_{r_{1}}$ are partitions of 1 and $\lambda_{r_{1}+1},\lambda_{r_{1}+2},\cdots,\lambda_{r_{2}}$ are partitions of $2$ etc. are known as Green Polynomials. The Green Polynomials are closely related with the characters of the symmmetric groups. In fact it is known [3] that if $\overline{\lambda}=(\overline{\lambda_{1}},\overline{\lambda_{2}},\cdots)$ is a partition conjugate to $\lambda$ and $n_{\lambda}=\frac{1}{2}\sum\overline{\lambda_{i}}(\overline{\lambda_{i}}$-1) then $\chi^{\lambda}_{\rho}$ is equal to the coefficient of $q^{n_{\lambda}}$ in $Q^{\lambda}_{\rho}$. Green Polynomials satisfy the following orthogonality relations:

If $\nu$ be any partition of $n$ then

[TABLE]

where the summation is over all pairs $\lambda,\mu$ of partitions $\ell$ and $m$, where $\ell+m=n$. Here if $\rho=1^{r_{1}}.2^{r_{2}}.3^{r_{3}}\cdots$ and $\sigma=1^{s_{1}}.2^{s_{2}}.3^{s_{3}}\cdots$ then the partition $\rho+\sigma$ is defined to be the partition $1^{r_{1}+s_{1}}.2^{r_{2}+s_{2}}.3^{r_{3}+s_{3}}\cdots$ 2. 2.

Let $\nu,\rho$ be partitions of $n$ then

[TABLE]

where $z_{\rho}=1^{r_{1}}(r_{1}!.2^{r_{2}}(r_{2}!)\cdots$ is the order of the centralizer (in $S_{n}$) of an element of an element in $S_{n}$ corresponding to the partition $\rho$.

Green [1] gives a table of the polynomials $Q^{\lambda}_{\rho}$ for $1\leq n\leq 5$. Morris’s paper [7] deals mainly with these polynomials and a method of their determination. He also gives a table of these polynomials for $n=6,7$.

4.2 $\rho$-Variables and $\rho$-Functions

Let $\rho=1^{r_{1}}.2^{r_{2}}\cdots$ be a partition of $\rho$. Then a formal set $X^{\rho}=\{x_{11},\cdots,x_{1r_{1}},x_{21},\cdots,x_{2r_{2}}\cdots\}$ of variables is called a set of $\rho$-variables, and each member of the set is called a $\rho$-variable. For each positive integer $d$ there are $r_{d}\ \rho$-variables namely $x_{d1},x_{d2},\cdots,x_{dr_{d}}$. Each of these $r_{d}\ \rho$-variables is said to have degree deg($x_{di})=d$, $i=1,2,\cdots,r_{d}$. To each of these $\rho$-variables of degree $d$ there correspond $d$ variables called the roots of $x_{di}$ or simply $\rho$-roots and are written as: $\xi_{di},\xi^{q}_{di},\cdots,\xi^{d-1}_{di},i=1,2,\cdots,r_{d};d=1,2\cdots$. These $\rho$-roots can be thought of as the characteristic roots of a typical class $c=(f_{11},f_{12},\cdots,f_{1r_{1}},f_{21},\cdots,f_{2r_{2}}\cdots,f_{dr_{d}},\cdots)$ of $GL(n,q)$. Let $F$ denote the set of all irreducible monic polynomials over GF($q$). A mapping $\alpha:X^{\rho}\rightarrow F$ such that $deg(x\alpha)$ divides $deg(x)$ for each $x\in X^{\rho}$ is called a substitution of $X^{\rho}$ into $F$. We can apply a substitution $\alpha$ to $\rho$-roots as well. For each $\xi_{di}\in X^{\rho}$ we choose any root $\gamma_{di}$ of $x_{di}\alpha$ and define $xi^{q^{u}}_{di}\alpha=\gamma^{q^{u}}_{di}$. Two substitutions $\alpha$ and $\alpha^{{}^{\prime}}$ of $X^{\rho}$ into $F$ are said to be equivalent if there is degree preserving permutation $\phi$ of $X^{\rho}$ such that $\alpha\circ\phi=\phi^{{}^{\prime}}$. A class of equivalent substitutions is called mode of substitution. Let $\alpha$ be a substitution of $X^{\rho}$ into $F$ and let $f\in F$ be of degree $d$. For each positive integer $i$ let $r_{i}(\alpha,f)=\#\{x\in X^{\rho}:deg(x)=id\ and\ x\alpha=f\}.$. Then we define $\rho(\alpha,f)=1^{r_{1}(\alpha,f)}.2^{r_{2}(\alpha,f)}\cdots$ It is easily seen that two substitutions $\alpha$ and $\alpha^{{}^{\prime}}$ are equivalent if and only if $\rho(\alpha,f)=\rho(\alpha^{{}^{\prime}},f)$ for all $f\in F$. So if $m$ denotes the equivalence class of substitutions to which $\alpha$ belongs, we can write without ambiguity $\rho(\alpha,f)=\rho(m,f)$. We shall also talk of substitutions of $X^{\rho}$ into a class $c$ of $GL(5,q)$. If $c=(\cdots f^{\nu_{c}(f)}\cdots)$ is a conjugate class of $GL(5,q)$ then a substitution $\alpha$ of $X^{\rho}$ into $F$ satisfying $\mid\rho(\alpha,f)\mid=\mid\nu_{c}(f)\mid$ for all $f\in F$ will be called a substitution of $X^{\rho}$ into the class $c$.

Example: Let $\rho=1^{3}.2$ and $c=f^{1^{2}}_{11}f^{1^{3}}_{12}$ where both $f_{11}$ and $f_{12}$ are of degree one. So in this case we have $X^{\rho}=\{x_{11},x_{12},x_{13},x_{21}\}$ and there are two modes of substitutions of $X^{\rho}$ into $c$, the representative substitutions being:

[TABLE]

[TABLE]

Substitutions can also be applied to $\rho$-roots. For each $x_{di}\in X^{\rho}$ choose any root $\gamma_{di}$ of $x_{di}\alpha$ and define $\xi^{q^{u}}_{di}\alpha=\gamma^{q^{u}}_{di}(u=0,1,2\cdots,d-1)$. For our purposes it would be irrelevant what root $\gamma_{di}$ is chosen.

4.3 $\rho$-Functions

A $\rho$-function $U_{\rho}(x^{\rho})=U_{\rho}(x_{11},x_{12},\cdots,x_{1r_{1}},x_{21},\cdots,x_{2r_{2}},\cdots)$ is a function of $\rho$-variables. It is a complex-valued function on the set of modes of substitutions of $X^{\rho}$ into $F$. It can also be considered as a function of $\rho$-roots if we write

[TABLE]

Let $\gamma\in GF(q^{d})^{*}$ be a root of $x_{di}\alpha$. Then

[TABLE]

where $m$ is the mode of the substitution $\alpha$.

4.4 Uniform Functions

For each partition $\rho$ of $n$ let there be a given a $\rho$-function $U_{\rho}(x^{\rho})$. Then a uniform function U on $GL(n,q)$ is the class function whose value on the class $c$ is given by:

[TABLE]

Here summation is taken over all partitions $\rho$ of $n$ and all modes $m$ of substitutions of $X^{\rho}$ into the class $c$ , and

[TABLE]

The functions $U_{\rho}(x^{\rho})$ are called the principal parts of $U$ and $U_{\rho}=U_{\rho}(x^{\rho})$ is its $\rho$-part. A uniform function $U$ whose principal parts are all zero except for $U_{\rho}$ is called Basic Uniform Function of type $\rho$. Consider the subgroup of $GL(m+n,q)$ consisting of the matrices of the form:

[TABLE]

where $A_{1}\in GL(n,q)$ and $A_{2}\in GL(m,q)$. Green [1] has shown that if $U$ and $V$ are uniform functions on $GL(m,q)$ and $GL(n,q)$ respectively then the function induced on $GL(m+n,q)$ by the function $\Psi(A)=U(A_{1})V(A_{2})$ is a uniform function on $GL(m+n,q)$. This induced uniform function on $GL(m+n,q)$ will be denoted by $U\circ V$.

5 Calculations of $Q(m,c)$ And $\chi(m,e)$

In this section we shall describe the computations of the values of the functions $Q(m,c)$ and $\chi(m,e)$. These functions are in a way dual to each other with respect to characters of $GL(n,q)$. Their definition is similar in as much as the Green Polynomials occurring in the definitions $Q(m,c)$ are replaced by the characters of the symmetric groups in the definition of $\chi(m,e)$, whereas, of course, the class $c$ is replaced by the dual class $e$. For a given partition $\rho$ and for a given class $c$ and dual class $e$ we have:

[TABLE]

and

[TABLE]

As has been pointed out earlier here $m$ is the mode of substitution of $X^{\rho}$ into the class $c$ and $z_{\rho(m,f)}$ is the order of centralizer (in the symmetric group) of an element represented by the partition $\rho(m,f)$. If there are more than one modes of substitutions we get more than one such functions (as happens in the case of $\rho=1^{3}.2$ and the classes $C,E$).

Given the partition $\rho$ and the class $c$ we first find the mode(s) of substitutions of $X^{\rho}$ into $c$. For each mode we evaluate the coorresponding $\rho(m,f)$ and $z_{\rho(m,f)}$. We read the value of $Q_{\rho(m,f)}^{\nu_{c}(f)}(q)$ from the table in Green [1] for each irreducible polynomial appearing in class $c$, and can thus compute the value of $Q(m,c)$.

Examples

Let $\rho=1^{5}$ and $c=B_{1}=f_{1a}^{1^{4}}.f_{1b}^{1}$ , so $X^{\rho}=\{x_{11},x_{12},x_{13},x_{14},x_{15}\}$. There is only one mode of substitution of $X^{\rho}$ into $c$ whose representative substitution being $x_{11},x_{12},x_{13},x_{14}\rightarrow f_{1a}$ and $x_{15}\rightarrow f_{1b}$. So $\rho(m,f_{1a})=1,\rho(m,f_{1a})=1,\rho(m,f_{1b})=1,z_{\rho(m,f_{1a})}=4.1,z_{\rho(m,f_{1b})}=1$. Now

$Q(m,c)=\frac{1}{z_{\rho(m,f_{1a})}}Q^{\nu_{c}(f_{1a})}_{\rho(m,f_{1a})}(q)\frac{1}{z_{\rho(m,f_{1b})}}Q^{\nu_{c}(f_{1b})}_{\rho(m,f_{1b})}(q)=\frac{1}{4!}Q^{1^{4}}_{1^{4}}(q)Q^{1}_{1}(q)$

$=\frac{1}{24}(q+1)(q^{2}+q+1)(q^{3}+q^{2}+q+1)$ 2. 2.

Let $\rho=2.1^{3},c=f^{2.1}_{1a}f^{1^{2}}_{1b}$. So $X^{\rho}=\{x_{21},x_{11},x_{12},x_{13}\}$. There are two modes of substitutions of $X^{\rho}$ into $c$ namely: The mode $m_{1}:x_{11},x_{12},x_{13}\rightarrow f_{1a},\ x_{21}\rightarrow f_{1b}$ and the mode $m_{2}:x_{11},x_{21}\rightarrow f_{1a},\ x_{12},x_{13}\rightarrow f_{1b}$. We have $\rho(m_{1},f_{1a})=1^{3},\rho(m_{1},f_{1b})=2,\rho(m_{2},f_{1a})=1.2,\rho(m_{2},f_{1b})=1^{2}$. So $Q(m_{1},c)=\frac{1}{3!}\frac{1}{2}Q^{1^{3}}_{1^{3}}(q)Q^{1^{2}}_{2}(q)=-\frac{1}{12}(q+1)(q^{3}+1)$ and $Q(m_{2},c)=\frac{1}{2}\frac{1}{2}Q^{1^{3}}_{1.2}(q)Q^{1^{2}}_{1^{2}}(q)=-\frac{1}{4}(q+1)(q^{3}-1)$.

The determination of the functions $\chi(m,e)$ is similar excepting for the fact that the class $c$ is replaced by the dual class $e$ and the irreducible polynomials are replaced by the simplexes $g$ and the Green Polynomials $Q^{\nu}_{\rho}(q)$ are replaced by the characters $\chi^{\nu}_{\rho}$ of the symmetric groups. As a matter of fact in most cases the value of $\chi(m,e)$ is equal to the coefficient of the highest power of $q$ occurring in the corresponding $Q(m,c)$. The table 3 and table 4 (resp.) give the values of $Q(m,c)$ and $\chi(m,e)$ (resp.) for all classes $c$ and dual classes $e$, and for all partitions $\rho_{i},1\leq i\leq 7$ of 5.

6 Basic characters

The Basic Characters are the basic constituents of the irreducible characters of $GL(5,q)$. The 42 types of irreducible characters of $GL(5,q)$ can be grouped into 17 categories such that the characters in the same category involve the same basic constituents namely the basic characters. Roughly speaking these basic characters express the irrationalities involved in the irreducible characters.

Let $\rho=1^{r_{1}}.2^{r_{2}}\cdots$ be a partition of $n$ and let $h$ denote the vector $(h_{11},h_{12},\cdots,h_{1r_{1}},h_{21},\cdots,h_{2r_{2}},\cdots)$ whose components (which are integers) correspond to parts of $\rho$. Then $B^{\rho}(h)$ the basic character of type $\rho$ is defined to be the character

[TABLE]

of $GL(n,q)$ where $J_{d}(k)$ is a character of $GL(d,q)$ defined as: $J_{d}(k)(c)=0$ if the class $c$ is not primary and $J_{d}(k)(f^{\lambda})=k(\lambda,q^{d(f)}\{\theta^{k}(\gamma)+\theta^{kq}(\gamma)+\theta^{kq^{2}}(\gamma)+\cdots+\theta^{q^{k(d(f)-1)}}(\gamma)\}$ where $\gamma$ is a root of the polynomial $f$ and $\theta$ is a generating character of the the group $GF(q^{n!})^{*}$ and we have abbreviated $deg(f)$ as $d(f)$. $B^{\rho}(h)$ is a basic uniform function whose $\rho$-part $B_{\rho}(h:\xi^{\rho})$ is given by the expression

[TABLE]

where the summation is over all permutations of $1,2,\cdots,r_{d}$ and

[TABLE]

Actually $\theta^{k}(\xi)$ makes sense only if $\xi$ is a root of an irreducible polynomial over $GF(q)$. However when a substitution is applied to $\xi$, a $\rho$-root is transformed into a root of an irreducible polynomial over $GF(q)$. The value of $B^{\rho}$ at a class $c$ is given by

[TABLE]

where the summation is taken over all modes of substiitutions of $X^{\rho}$ into the class $c$.

6.1 Simplex and Dual Classes

Let $f\in F$ be an irreducible polynomial of degree $s$ over $GF(q)$. Let $\omega$ be a generator of $GF(q^{s})^{*}$. Then a root of $f$ has the form $\omega^{k}$ where $k$ is uniquely determined modulo($q^{s}-1$), and the set {$\omega^{k},\omega^{kq},\cdots,\omega^{kq^{s-1}}$} consists all the roots of $f$. Recall that the set {$k,kq,kq^{2},\cdots,kq^{s-1}$ } is an $s$-simplex with $k,kq,kq^{2},\cdots,kq^{s-1}$ as its roots and $s$ its degree. It is clear that there are as many simplexes of degree $s$ as there are irreducible polynomials $f\in F$ of degree $s$ over $GF(q)$. As before the set of all s-simplexes for $1\leq s\leq n$ will be denoted by $G$ and a particular simplex by the letter $g$. For each partition $\rho=1^{r_{1}}.2^{r_{2}}\cdots$, let $Y^{\rho}$ denote the set of $\rho$-variables {$y_{11},y_{12},\cdots,y_{1r_{1}},y_{21},\cdots,y_{2r_{2}},\cdots$}. The $\rho$-variable $y_{ij}$ is considered to be of degree $i$. A mapping $\alpha:Y^{\rho}\rightarrow G$ is called a substitution of $Y^{\rho}$ into $G$ if $deg(y\alpha)$ divides $deg(y)$. The notions of equivalenc of two substitutions, mode of a substitution and the partitions $\rho(\alpha,g)$ are defined exactly in the same manner as for the substitutions of $X^{\rho}$ into $F$. A function $\nu\rightarrow P$ such that $\sum_{g\in G}\mid\nu(g)\mid deg(g)=n$ specifies what is called a dual class and will be denoted by the symbol $e=(\cdots,g^{\nu(g)},\cdots)$. To each symbol of dual class there will correspond a character which will be denoted by the same symbol. For a fixed partition $\rho$ of $n$ let $m$ be the mode of certain substitution $\alpha$ of $Y^{\rho}$ into the set $G$ of all simplexes. Set $h_{di}m=c_{di}(q^{d}-1)/(q^{s_{di}}-1)$, where $c_{di}$ is a root of the simplex $y_{di}\alpha$ and $s_{di}$ is the degree of $y_{di}\alpha$

We shall now describe the computation of $\rho$-parts of the basic characters of $GL(5,q)$ for a partition $\rho$ of $5$ corresponding to a given symbol $e$ of a dual class of $GL(5,q)$. First we find the modes of substitution of $Y^{\rho}$ into $e$, then for each mode $m$ we find the integers $h_{di}m$ defined above. Then the $\rho$-parts of the basic character for the mode $m$ corresponding to the irreducible character given by the symbol $e$ of the dual class is given by

[TABLE]

where now $h_{11}m,h_{12}m,\cdots$ are all integers and so the value of the above expression can be calculated.

For example let $\rho=1.2^{2},e=g^{4}_{1i}g_{1j}$. Then $Y^{\rho}=\{y_{11},y_{21},y_{22}\}$. It is easily seen that there is only one mode of substitution of $Y^{\rho}$ into $e$ and a substitution $\alpha:Y^{\rho}\rightarrow e$ belonging to this mode is given by $y_{21},y_{22}\rightarrow g_{1i},y_{11}\rightarrow g_{1j}$. Also $h^{\rho}=\{h_{11},h_{21},h_{22}\},h_{11}m=j$, where $j$ is a root of $g_{1i}$, and $h_{21}m=h_{22}m=i(q+1)$ where $i$ is root of $g_{1i}$. So $B^{\rho}(h^{\rho}m)=B^{\rho}(j,i(q+1),i(q+1))$ and the $\rho$-part of the basic character corresponding to the symbol $e=g^{4}_{1i}g_{1j}$ of the dual class is:

$B^{\rho}(h^{\rho}m:\xi^{\rho})=S_{1}(j,\xi_{11})[\sum S_{2}(i(q+1),\xi_{21})S_{2}(i(q+1),\xi_{22})]$

$=2S_{1}(j,\xi_{11})S_{2}(i(q+1),\xi_{21})S_{2}(i(q+1),\xi_{22})$

Let us determine the value of this basic character on class $c$ of type $A$ say $f_{1a}^{5}$. We first find the modes of substitutions of $X^{\rho}$ into $f_{1a}^{5}$. As $X^{\rho}=\{x_{11},x_{21},x_{22}\}$ there is only one mode of substitution whose representative substitution maps each of the $\rho$-variable into $f_{1a}$. According to our notation the root of the polynomial can be expressed as $\omega_{1}^{a}$. So we have $\xi_{11}m=\xi_{21}m=\xi_{22}m=\omega_{1}^{a}$ and $B^{\rho}(h^{\rho}m:\xi^{\rho})(c)=2S_{1}(j,\omega_{1}^{a})(S_{2}(i(q+1),\omega_{1}^{a})^{2}=2\theta^{j}\omega_{1}^{a}(\theta^{i(q+1)}\omega_{1}^{a}+\theta^{iq(q+1)}\omega_{1}^{a})^{2}=2\varepsilon_{1}^{ja}(\varepsilon_{1}^{i(q+1)a}+\varepsilon_{1}^{iq(q+1)a})^{2}=8\varepsilon_{1}^{(4i+j)a}$ ,where we recall $\varepsilon_{1}$ is a primitive $(q-1)$-th root of unity. The value of this basic charcater is same on every class of type $A$ namely on each of the classes $A_{i},1\leq i\leq 7$.

Table 6 gives the value of the $\rho$-part of each of the $17$ types of basic characters on each of the $17$ types of classes corresponding to each partition $\rho$ of $5$ for the $GL(5,q)$. Whenever there is more than one mode of substitution we have mentioned it explicitly. If the value of a basic character corresponding to a partition $\rho$ is missing then it should be understood that there is no mode of substitution - either into the symbol $e$ of a dual class or into the class $c$.

7 Irreducible Characters of $GL(5,q)$

The character of $GL(5,q)$ corresponding to the dual class $e=(\ldots,g^{\nu(g)},\ldots)$ of $GL(5,q)$ is given Green [1] by the expression

[TABLE]

where the summation is over all the $7$ partitions $\rho_{i}$ , $1\leq i\leq 7$ of $5$ and over all modes of substitutuions of $Y^{\rho{i}}$ into the dual class $e$. The degree of this character is given by the expression

[TABLE]

where $\Psi_{n}(q)=\prod_{i=1}^{i=n}(q^{i}-1)$ and if $\lambda$ is a partition of $n$ with parts $\ell_{1}\geq\ell_{2}\geq\cdots\ell_{p}\geq$ then

[TABLE]

where $\Phi_{n}(q)=(-1)^{n}\Psi_{n}(q)$. Now to calculate the value of $\sum_{m}\chi(m,e)B^{\rho}(h^{\rho}m)$, for a specific partition $\rho$, at a class $c$ of $GL(5,q)$ we write $B^{\rho}(h^{\rho}m)$ as simply $B^{\rho}(h)$ and keep in mind that now $h=h^{\rho}m$ is a specific sequence of integers whose value is uniquely determined by the partition $\rho$ of and the mode $m$ of substitution of $Y^{\rho}$ into $e$. Recall that the value of $B^{\rho}$ at a class $c$ of $GL(5,q)$ is given by the expression $\sum_{m}Q(m,c)B_{\rho}(h:\xi^{\rho}m)$, where the summation over all modes of substitutions of $X^{\rho}$ into the class $c$ and $B_{\rho}(h:\xi^{\rho}m)$ is the $\rho$-part of the basic character corresponding to the dual class $e$ whose value has already been determined (table [6] ). The values of the functions $Q(m,c)$ have been determined earlier (table [3] ). We do this for each partition $\rho_{i},1\leq i\leq 7$ of $5$ and for all modes of substitutions of $Y^{\rho_{i}}$ into the dual class $e$ and further for all modes of substitutions of $X^{\rho}$ into the class $c$. It may appear that it is an involved process but in the case of $GL(5,q)$ the computations are managebale. We shall give two example of explicit computation of values of characters of $GL(5,q)$.

Examples

Let us determine the value of the character of type $A_{1}^{{}^{\prime}}$ which corresponds to the dual class $e=g^{1^{5}}_{1i},1\leq i\leq q-1$ on the class $c=A_{1}$ (class symbol $f^{1^{5}}_{1a},1\leq a\leq q-1)$. For each $\rho_{i},1\leq i\leq 7$, there is only one mode of substitution of $Y^{\rho_{i}}$ into the dual class $e$ namely each variable mapping into $g_{1i}$. Similarly for each $\rho_{i},1\leq i\leq 7$, there is only one mode of substitution of $X^{\rho_{i}}$ into the class $c$. So in this case for each $\rho_{i},1\leq i\leq 7$, we find the value of $\chi(m,e)$ corresponding to the dual class $e=A_{1}^{{}^{\prime}}$, multiply it with the value of $Q(m,c)$ for corresponding $\rho_{i}$ and $c=A_{1}$, and with the $\rho_{i}$-part of the basic character of type $A^{{}^{\prime}}$ and add these up. Thus we get the following value for this character on given class:

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$\frac{1}{4}(q^{2}-1)(q^{3}-1)(q^{5}-1)\varepsilon_{1}^{5ia}+\frac{1}{5}(q-1)(q^{2}-1)(q^{3}-1)(q^{4}-1)\varepsilon_{1}^{5ia}$

which when simplified reduces to $q^{10}\varepsilon_{1}^{51a}$. The degree of each of these characters (as computed by the degree formula mentioned earlier) is $q^{10}$ 2. 2.

Let us determine the value of the character of type $C_{1}^{{}^{\prime}}$ on the classes of type $E_{2}$. In this case there are two modes of substitutions for $\rho=\rho_{2}=2.1^{3}$ into the dual class of type $C_{1}^{{}^{\prime}}$ as well as into the class of type $E^{{}^{\prime}}$. Also corresponding to $\rho=\rho_{6},\rho_{7}$ there are no modes of substitutions of $Y^{\rho}$ into $C_{1}^{1}$, and corresponding to $\rho=\rho_{4},\rho_{5},\rho_{6},\rho_{7}$ there are no modes of substitutions of $X^{\rho}$ into the class $E_{2}$ . Thus corresponding to the partitions $\rho_{1}$ and $\rho_{3}$ the contribution to the value of character is:

[TABLE]

$-\frac{1}{4}(q-1)[\varepsilon_{1}^{i(2a+c)+2jb}+\varepsilon_{1}^{i(2b+c)+2ja}]$

To find the contribution corresponding to the partitions $\rho_{2}$ we have to take into consideration both the modes of substitutions. This contributions comes out to be

$-\frac{1}{4}(q+1)\varepsilon_{1}^{i(2a+c)+2jb}+\frac{1}{4}(q-1)\varepsilon_{1}^{i(2b+c)+2ja}-\frac{1}{4}(q+1)[2\varepsilon_{1}^{i(a+2b)+j(a+c)}+\varepsilon_{1}^{i(2b+c)+2ja}]$

$+\frac{1}{4}(q-1)[2\varepsilon_{1}^{i(2a+b)+j(b+c)}+\varepsilon_{1}^{i(2a+c)+2jb}]$

Adding the two contributions we get the value of the characters of type $C_{1}^{{}^{\prime}}$ on the classse of type $E_{2}$ (as can be seen there is no contribution from the partitions $\rho_{4},\rho_{5},\rho_{6},\rho_{7}$). This sum comes out to be equal to

[TABLE]

The degree of each of these characters as computed by the degree formula is $q^{4}(q^{2}+1)(q^{4}+q^{3}+q^{2}+q+1)$

A complete listing of characters of $GL(5,q)$ is given in table 7.

Appendix

List of Tables

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TABLE # 1

REPRESENTATIVES OF CONJUGATE CLASSES OF GL(5,q)

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Table 2. CONJUGATE CLASSES.

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Table 2. CONJUGATE CLASSES (Cont’d).

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Table 2. CONJUGATE CLASSES (Cont’d).

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Table 2. CONJUGATE CLASSES (cont’d).

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Table 3. THE FUNCTIONS $Q(m,c)$.

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Table 3. THE FUNCTIONS $Q(m,c)$ (cont’d).

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Table 3. THE FUNCTIONS $Q(m,c)$ (cont’d).

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Table 4. THE FUNCTIONS $\chi(m,e)$.

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Table 5. DUAL CLASSES (TYPES OF CHARACTERS)

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Table 5. DUAL CLASSES (TYPES OF CHARACTERS) (Cont’d).

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Table 5. DUAL CLASSES (TYPES OF CHARACTERS) (Cont’d).

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Table 5. DUAL CLASSES (TYPES OF CHARACTERS) (Cont’d).

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Note: The contribution from the basic character corresponding to the missing partitions is zero. The classes for which there is no contribution for any of the partitions have been omitted.

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1. J. A. Green, “The characters of general linear groups.” Trans. Amer. Math. Soc. 80(1955), 402-447.

2. S. K. Gupta, Ph.D. thesis, Case Institute of Technology 1971.

3. D. Littlewood, The theory of groups characters and matrix representation of groups. Oxford. 1958.

4. D. Littlewood, “On certain symmetric functions.” $Proc.LondonMath.Soc.$(3) 11 (1961), 485-498.

5. I.G. Macdonald, Symmetric functions and Hall polynomials. Clarendon Press. Oxford 1979.

6. Alun O. Morriss, “The characters of the groups $GL(n,q)$ ”. Math. Z. 81(1963), 733-742.

7. Alun O. Morriss, The multiplication of Hall Functions.” Proc. London Math. Soc. 3(13) 1963, 733-742.

8. W. Specht, “Die Characters der Symmetrischen Gruppe.” Math. Z. 73(1960) 312-329.

9. T. Springer, Characters of special groups. Lecture Notes in Mathematics. No. 131, Springer Verlag, 1970.

10. R. Steinberg, Ph.D. Thesis, University of Toronto, 1948.

11. R. Steinberg, “The representations of $GL(3,q),GL(4,q),PGL(3,q),PGL(4,q)$.” Canad. Jour. of Math. 3 (1951),225-235.

12. R. Steinberg, “A geometric approach to the representations of the full linear group over a Galois field.” Trans. Amer. Math. Soc. 71(1951), 274-282.