General Linear Group

The general linear group, often denoted as GL _n(F) or GL(n, F), is the grand assembly of all n × n invertible matrices that operate under the familiar rules of matrix multiplication . It’s a group, you see, because the product of any two matrices that can be inverted will itself be invertible, and the inverse of an invertible matrix is, naturally, also invertible. The identity matrix sits at the head of this operation as the neutral element. The name “general linear” isn’t just for show; it arises from the fact that the columns (and rows, for that matter) of such matrices are linearly independent . This means they transform points in general linear position to other points that remain in that same, well, general linear position.

Now, the specifics of what populates these matrices matter. For instance, the general linear group over the real numbers , denoted GL_n(ℝ) or GL(n, ℝ), is comprised of n × n invertible matrices with real entries. More broadly, for any field F – and yes, that includes the complex numbers – or any ring R, like the humble integers , the general linear group consists of n × n invertible matrices whose entries hail from F or R, respectively. The notation GL(n, F) or GL_n(F) is standard, often simplified to GL(n) if the field is implicitly understood.

We can also speak of the general linear group of a vector space, GL(V), which is the automorphism group – the collection of all bijective linear transformations from V to itself, composed under function composition. If V has a finite dimension n, then GL(V) is isomorphic to GL(n, F), though this isomorphism is not absolute; it hinges on the choice of a basis for V. When a basis {e₁, …, e_n} is selected, an automorphism T in GL(V) can be represented by a matrix whose entries a_ji are determined by how T acts on each basis vector: T(e_i) = Σ_j=1ⁿ a_jie_j.

For a commutative ring R, GL(n, R) can be seen as the group of automorphisms of a free R-module of rank n. One can define GL(M) for any R-module M, but it won’t always be isomorphic to GL(n, R).

Determinants and Invertibility

The connection to determinants is crucial. Over a field F, a matrix is invertible precisely when its determinant is non-zero. So, GL(n, F) can be viewed as the set of n × n matrices with non-zero determinants. When dealing with a commutative ring R, invertibility of a matrix is tied to its determinant being a unit in R – meaning, it has a multiplicative inverse within R. Thus, GL(n, R) comprises matrices whose determinants are units. For non-commutative rings, determinants are less straightforward, and GL(n, R) is defined as the unit group of the matrix ring M(n, R).

Lie Group Structure

Real Case

The general linear group GL(n, ℝ) over the real numbers is a real Lie group with a dimension of n². This stems from the fact that the space of all n × n real matrices, M_n(ℝ), is a real vector space of dimension n². GL(n, ℝ) is the subset of these matrices with non-zero determinants. Since the determinant is a polynomial function, GL(n, ℝ) forms an open affine subvariety of M_n(ℝ), and thus a smooth manifold of the same dimension.

The Lie algebra associated with GL(n, ℝ), denoted 𝔤𝔩_n, is the set of all n × n real matrices, with the commutator acting as the Lie bracket.

As a manifold, GL(n, ℝ) isn’t a single piece; it’s split into two connected components : one for matrices with positive determinants and one for those with negative determinants. The identity component , GL⁺(n, ℝ), comprises matrices with positive determinants. This component is also a Lie group of dimension n² and shares the same Lie algebra as GL(n, ℝ).

The polar decomposition reveals a homeomorphism between GL(n, ℝ) and the product of the orthogonal group O(n) with the set of positive-definite symmetric matrices. Similarly, GL⁺(n, ℝ) is homeomorphic to the product of the special orthogonal group SO(n) and the same set of positive-definite symmetric matrices. Because this latter set is contractible, the fundamental group of GL⁺(n, ℝ) is isomorphic to that of SO(n).

This homeomorphism also confirms that GL(n, ℝ) is noncompact . Its maximal compact subgroup is the orthogonal group O(n), while for GL⁺(n, ℝ), it’s the special orthogonal group SO(n). Much like SO(n), GL⁺(n, ℝ) isn’t simply connected (unless n=1); its fundamental group is isomorphic to ℤ for n=2 and ℤ₂ for n>2.

Complex Case

The general linear group over the complex numbers , GL(n, ℂ), is a complex Lie group with complex dimension n². As a real Lie group, its dimension doubles to 2n². The real matrix group GL(n, ℝ) embeds within it, forming a real Lie subgroup. These inclusions, GL(n, ℝ) < GL(n, ℂ) < GL(2n, ℝ), correspond to real dimensions n², 2n², and 4n², respectively. Complex n-dimensional matrices can be identified with real 2n-dimensional matrices that preserve a linear complex structure – that is, they commute with a matrix J such that J² = -I, where J represents multiplication by the imaginary unit i.

The Lie algebra of GL(n, ℂ) consists of all n × n complex matrices, with the Lie bracket defined by the commutator .

Unlike its real counterpart, GL(n, ℂ) is connected . This is partly due to the multiplicative group of complex numbers, ℂ^×, being connected. GL(n, ℂ) is not compact; its maximal compact subgroup is the unitary group U(n). And, similar to U(n), GL(n, ℂ) is not simply connected , possessing a fundamental group isomorphic to ℤ.

Over Finite Fields

When F is a finite field with q elements, the notation GL(n, q) is often used. For a prime p, GL(n, p) represents the outer automorphism group of the group ℤ_pⁿ, and also its automorphism group, as ℤ_pⁿ is abelian, rendering the inner automorphism group trivial.

The order of GL(n, q) is given by:

$$ \prod_{k=0}^{n-1}(q^{n}-q^{k})=(q^{n}-1)(q^{n}-q)(q^{n}-q^{2})\ \cdots \ (q^{n}-q^{n-1}). $$

This can be derived by counting the possible columns: the first column can be any non-zero vector; the second, any vector not in the span of the first; and so on. In q-analog notation, this is [n]_q! (q-1)ⁿ q^{n choose 2}.

For instance, GL(3, 2) has an order of (8 − 1)(8 − 2)(8 − 4) = 168. This group is the automorphism group of the Fano plane and of the group ℤ₂³. It is also isomorphic to PSL(2, 7) . These formulas relate to the counting of points on Grassmannians over F, essentially counting subspaces of a given dimension. This connection to Schubert decomposition and q-analogs of Betti numbers was a hint towards the Weil conjectures .

Interestingly, in the limit as q approaches 1, the order of GL(n, q) appears to go to zero. However, with proper normalization (dividing by (q-1)ⁿ), it converges to the order of the symmetric group S_n. This has led to the philosophical interpretation in the context of the field with one element where S_n is seen as GL(n, 1).

History

The general linear group over a prime field, GL(ν, p), and its order were first described by Évariste Galois in 1832, in his final letter. He used this in his work on the Galois group of the general equation of degree p^ν.

Special Linear Group

The special linear group , SL(n, F), consists of matrices with a determinant of precisely 1. These matrices are distinguished by satisfying a polynomial equation, det(A) = 1, placing them on an algebraic variety . The determinant property ensures that the product of two matrices with determinant 1 also has determinant 1, forming a group.

The determinant function, det: GL(n, F) → F^× (where F^× is the multiplicative group of F), is a group homomorphism . It is surjective, and its kernel is the special linear group SL(n, F). Consequently, SL(n, F) is a normal subgroup of GL(n, F). By the first isomorphism theorem , the quotient group GL(n, F) / SL(n, F) is isomorphic to F^×. In fact, GL(n, F) can be expressed as a semidirect product : GL(n, F) = SL(n, F) ⋉ F^×.

For a field or division ring F, the special linear group is the derived group (or commutator subgroup) of GL(n, F), provided n ≠ 2 or F is not the field with two elements .

When F is ℝ or ℂ, SL(n, F) is a Lie subgroup of GL(n, F) with dimension n² - 1. Its Lie algebra comprises n × n matrices over F with a trace of zero, the Lie bracket being the commutator .

SL(n, ℝ) can be characterized as the group of volume - and orientation-preserving linear transformations of ℝⁿ.

SL(n, ℂ) is simply connected, whereas SL(n, ℝ) is not. The fundamental group of SL(n, ℝ) mirrors that of GL⁺(n, ℝ), being ℤ for n=2 and ℤ₂ for n>2.

Other Subgroups

Diagonal Subgroups

The set of invertible diagonal matrices forms a subgroup of GL(n, F) isomorphic to (F^×)ⁿ. In fields like ℝ and ℂ, these represent scaling operations. Scalar matrices, which are constant multiples of the identity matrix, form a subgroup isomorphic to F^×. This subgroup constitutes the center of GL(n, F), making it a normal, abelian subgroup. The center of SL(n, F) consists of scalar matrices with determinant 1, isomorphic to the group of n^th roots of unity in F.

Classical Groups

The classical groups are subgroups of GL(V) that preserve specific bilinear forms on a vector space V. These include:

• The orthogonal group , O(V), which preserves a non-degenerate quadratic form on V.
• The symplectic group , Sp(V), which preserves a symplectic form on V (a non-degenerate alternating form ).
• The unitary group , U(V), which, over ℂ, preserves a non-degenerate hermitian form on V.

These groups are significant examples of Lie groups.

Related Groups and Monoids

Projective Linear Group

The projective linear group , PGL(n, F), and the projective special linear group , PSL(n, F), are formed by taking the quotients of GL(n, F) and SL(n, F), respectively, by their centers . These groups represent the induced action on the associated projective space .

Affine Group

The affine group , Aff(n, F), is an extension of GL(n, F) by the group of translations in Fⁿ. It can be expressed as a semidirect product : Aff(n, F) = GL(n, F) ⋉ Fⁿ, where GL(n, F) acts on Fⁿ in the standard way. The affine group represents all affine transformations of the affine space underlying the vector space Fⁿ. Analogous constructions exist for other subgroups, such as the special affine group (SL(n, F) ⋉ Fⁿ) and the Poincaré group , which is the affine group associated with the Lorentz group .

General Semilinear Group

The general semilinear group , ΓL(n, F), encompasses all invertible semilinear transformations and contains GL(n, F). A semilinear transformation is akin to a linear transformation but allows for a “twist” via a field automorphism in scalar multiplication. It can be represented as a semidirect product: ΓL(n, F) = Gal(F) ⋉ GL(n, F), where Gal(F) is the Galois group of F, acting on the matrix entries. The projective semilinear group PΓL(n, F) is particularly important as it forms the collineation group of projective space for n > 2, extending PGL(n, F).

Full Linear Monoid

By relaxing the non-zero determinant condition, we arrive at the full linear monoid, a structure sometimes called the full linear semigroup or general linear monoid. It is, in fact, a regular semigroup .

Infinite General Linear Group

The infinite general linear group, or stable general linear group, is the direct limit of the inclusions GL(n, F) → GL(n+1, F) (using block matrix embedding). It’s denoted GL(F) or GL(∞, F) and can also be viewed as the group of invertible infinite matrices that differ from the identity in only a finite number of positions. This group plays a role in algebraic K-theory for defining K₁. Over the reals, it possesses a well-understood topology due to Bott periodicity . It’s important not to confuse this with the group of bounded invertible operators on a Hilbert space , which is a larger, contractible group (by Kuiper’s theorem ).