The general linear group

The general linear group ${GL}_{n} (R)$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 17 October 2011

The general linear group ${GL}_{n} (R)$

Let $n \in ℤ_{> 0}$ , let $R$ be a commutative ring, and let $M_{n} (R)$ be the ring of $n \times n$ matrices with entries in $R$ .

The general linear group with entries in $R$ is the group ${GL}_{n} (R) = {g \in M_{n} (R) | there exists g^{- 1} \in M_{n} (R) such that g g^{- 1} = g^{- 1} g = 1}$ with operation given by matrix multiplication.

HW: Note that $R^{\times} = {GL}_{1} (R)$ is the group of units in the ring $R$ , and ${GL}_{n} (R)$ is the group of units in the ring $M_{n} (R)$

HW: Show that ${GL}_{n} (R) = {g \in M_{n} (R) | \det (g) \in {GL}_{1} (R)}$ .

In this case $W_{0} = S_{n} acts on 𝔥_{ℤ} = \sum_{i = 1}^{n} ℤ ε_{i}$ by permuting the $ε_{i}$ and the reflecting hyperplanes are $𝔥^{ε_{i} - ε_{j}}, with 1 \leq i < j \leq n, where ⟨ ε_{i}, ε_{j} ⟩ = δ_{i j} .$

For $1 \leq i, j \leq n$ and $i \neq j$ , let $E_{i j}$ be the $n \times n$ matrix with a 1 in the $(i, j)$ entry and all other entries 0, $s_{i j} = 1 - E_{i i} - E_{j j} + E_{i j} + E_{j i} and x_{i j} (f) = 1 + f E_{i j}, for f \in R,$ so that the $s_{i j}$ generate the group $W_{0}$ as permutation matrices. Let $ε_{1}, \dots, ε_{n}$ be a $ℤ$ -basis of $𝔥_{ℤ}$ so that $𝔥_{ℤ} = {λ = λ_{1} ε_{1} + \dots + λ_{n} ε_{n} | λ_{1}, \dots, λ_{n} \in ℤ},$ and define $h_{λ} (g) = diag (g^{λ_{1}}, \dots, g^{λ_{n}}), for λ \in 𝔥_{ℤ}, g \in R^{\times} .$

Let $𝔽$ be a field. Then the group ${GL}_{n} (𝔽)$ is presented by generators $x_{i j} (f) and h_{λ} (g),$ for $1 \leq i, j \leq n$ , $i \neq j$ , $f \in 𝔽^{\times}$ and $λ \in 𝔥_{ℤ}$ and $g \in 𝔽^{\times}$ ,
and relations $\begin{array}{l} x_{i j} (f_{1}) x_{i j} (f_{2}) = x_{i j} (f_{1} f_{2}), \\ h_{λ} (g_{1}) h_{λ} (g_{2}) = h_{λ} (g_{1} g_{2}), \\ h_{λ} (g) h_{μ} (g) = h_{λ + μ} (g), \\ x_{i j} (f_{1}) x_{k l} (f_{2}) = x_{k l} (f_{2}) x_{i j} (f_{1}), & if i \neq l and j \neq k, \\ x_{i j} (f_{1}) x_{j l} (f_{2}) = x_{j l} (f_{2}) x_{i j} (f_{1}) x_{i l} (f_{1} f_{2}), & if i \neq l, \\ x_{i j} (f_{1}) x_{k i} (f_{2}) = x_{k i} (f_{2}) x_{i j} (f_{1}) x_{k j} (- f_{1} f_{2}), & if j \neq k, \\ x_{i j} (g) x_{j i} (- g^{- 1}) x_{i j} (g) = h_{ε_{j}} (- 1) h_{ε_{i} - ε_{j}} (g) s_{i j}, \\ h_{λ} (g) x_{i j} (g) {h_{λ} (g)}^{- 1} = x_{i j} (f g^{⟨ λ, ε_{i} - ε_{j} ⟩}), \\ w x_{i j} (f) w^{- 1} = x_{w (i), w (j)} (f), & for w \in S_{n} . \end{array}$

PUT IN THE PICTORIAL VERSION OF THE ELEMENTARY MATRICES Introduce a pictorial notation

(4.1)

For example

=

corresponds to the matrix identity

(\begin{matrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{matrix}) (\begin{matrix} 1 & 0 & c & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) = (\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ c & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) (\begin{matrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{matrix})

Notes and References

This presentation of the general linear group is the motivation for the definition of Chevalley groups. Multiplication by elementary matrices is often called "row reduction". The first half of Theorem ??? says that ${GL}_{n} (𝔽)$ is generated by elementary matrices, a fact which is usually proved (by row reduction) in a first course in linear algebra right after the definition of matrix multiplication (see [Ar, ???]). The pictorial notation for elementary matrices appears in [Th, Sec 4.1.2], where it is very handy for proving identities in unipotent Hecke algebras.

References

[Ar] M. Artin, Algebra, ????, Prentice-Hall ???.

[Th] N. Thiem, Unipotent Hecke algebras: the structure, representation theory and combinatorics, Ph.D. Thesis, University of Wisconsin, Madison 2004.

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