Why we need to study representations of matrix groups?

Question

Why we need to study representations of matrix groups? For example, the group $\operatorname{SL}_2(\mathbb F_q)$, where $\mathbb F_q$ is the field with $q$ elements, is studied by Drinfeld. I think that these groups are already given by matrices. The representation theory is to represent elements in an algebra or group (or other algebraic structure) by matrices. Why we still need to study representations of matrix groups? Thank you very much.

Answer 1

I will answer a more general question.

If $X$ is an important mathematical object, why should we study the representation theory of $Aut(X)$?

Maybe algebraic geometers care about $X$, or maybe mathematical physicists, or maybe homotopy theorists, etc. Each discipline has techniques that can build a new object $X'$ out of $X$. These new objects are usually more inspiring (if less fundamental) than $X$ itself.

Maybe $X$ is projective $n$-space and algebraic geometers take $X'$ to be a Hilbert scheme. Or maybe $X$ is a manifold and mathematical physicists build the cotangent bundle of the cotangent bundle of $X$. Or maybe $X$ is a genus 3 surface and homotopy theorists build the loop space of the suspension of the space of maps from a genus 5 surface, or whatever.

The point is, these constructions preserve the $Aut(X)$ symmetry.

For some reason, every branch of mathematics seems to have a bunch of nice functors to the category of vector spaces over $\mathbb{C}$. Let $H$ be one such functor. Now $HX'$ will be a $\mathbb{C}$-vector space carrying information about the complicated object $X'$. But it will also be a representation of the group $Aut(X)$.

If the representation theory or $Aut(X)$ is already known, then we gain access to a powerful set of tools with which to study $HX'$. If it's not already known, then we should try to decompose $HX'$ anyway--this is the way to find irreducibles in the first place.

Knowing the representation theory of $Aut(X)$ lets us prepare for any possible construction of $X'$ and any possible functor $H$.

Answer 2

Representation theory is not necessarily about finding a matrix representation of an abstract group; rather, it is about understanding the abstract properties of the group through their reflections in various matrix representations. Not all features of a group are evident in any one representation; as a really simple example, it's not at all obvious from the standard representation of $\mathrm{GL}_2$ that there exists a determinant: namely, a character $\operatorname{det} \colon \mathrm{GL}_2 \to \mathrm{GL}_1$. Nor is it obvious that no nontrivial such character exists for $\mathrm{SL}_2$. Yet both of these facts are simple consequences of the "highest weight theory" for reductive groups, which is basically a complete classification of their representations.

As another example, the fundamental group of a complex reductive group (such as $\mathrm{GL}_n(\mathbb{C})$) can be obtained from the same theory, namely as the quotient group of the lattice of "coweights" by its "coroot" sublattice (these are both concepts coming from representation theory).

As still another example, the problem of forming the quotient group $G/H$ by a closed subgroup $H$ is basically solved by a representation theory lemma that $H$ is the stabilizer of a line in some representation $V$ of $G$ (then $G/H$ is, effectively, the set of $G$-translates of that line, which is at least intuitively a subset of the projective space of lines in $V$). Of course, one can form $G/H$ as a set (or, if $H$ is normal, an abstract group) without any higher concepts, but if $G$ is considered as an algebraic group, then $G/H$ should have the structure of an algebraic variety to be of any use, and this is the value added by the above argument.

Still another justification for using arbitrary representations is the question: what is so special about matrix groups, anyway? You take for granted that having some matrix representation is desirable for the purposes of describing a group, but what is the advantage of this over, say, giving $G$ as its generators and relations (if finite) or by algebraic equations in some large number of variables (if algebraic)? Clearly, the useful additional data is that of linear algebra, whose thoroughly understood structure and accessible techniques provide a ton of basically free descriptors for anything to which it can be applied. If your group is given by a matrix representation in some $\mathrm{GL}_n$ for a huge $n$, or with extremely complicated matrix entries, that representation may not tell you everything you want to know. By casting the shadow of $G$ in various other directions we can view its shape more completely.

Answer 3

To amplify what user76758 says, from the historical viewpoint much of the interest in group representations does involve a given matrix group (such as a Lie group or algebraic analogue). But the main point was to study the action of that group on related vector spaces including symmetric and exterior powers. This was in part a focus of classical invariant theory but got expanded along with the study of symmetry groups in differential geometry and physics. It's worthwhile to look back for example at Weyl's influential book The Classical Groups.

Similarly, finite simple groups are often studied in connection with their actions on associated combinatorial or geometric objects. For groups of Lie type this may involve their natural realization as matrix groups, or might go in other directions depending on the ground field involved. Representation theory is a natural tool in the study of how groups (whatever their origin) act on other kinds of objects: linear, geometric, combinatorial, etc.

Answer 4

In many "real-life" situations, it's not that one expects to learn things about the literal group $G$ whose representation theory one studies, but, rather, to learn about other natural objects on which $G$ acts by using prior knowledge about the irreducibles of $G$, etc.

At the very least, apart from their intrinsic interest, finite matrix groups are good prototypes for the corresponding matrix groups over $\mathbb R$ or $\mathbb C$ or $\mathbb Q_p$ or $\mathbb A$. As it happens, it is possible, and useful, to look at the finite-field matrix group representation theory in a way that is a good warm-up for those groundfields useful in number theory, for example.

A perhaps-unexpected different extreme example: the circle $S^1=SO(2,\mathbb R)$. This group itself might plausibly be considered not-tricky. The irreducible repns of the circle group $SO(2,\mathbb R)$ are the exponentials $\theta \to e^{in\theta}$ for integers $n$. The representation of the group on $L^2$ of itself by translation amounts to the theory of Fourier series. The most elementary aspects of this are not toooo subtle, but, for example, pointwise convergence of Fourier series (=spectral synthesis after spectral decomposition) was/is subtle enough to motivate Cantor to develop set theory. True, the group is not finite...

Answer 5

This is a slight elaboration on Jim Humphreys' answer. As far as I understand, at least one of the motivations of Hermann Weyl to develop the representations of compact Lie groups (such as $U(n)$, $SO(n)$) was to generalize the classical invariant theory and to apply this to differential geometry (tensor calculus) and closely related questions of general relativity. In these fields representation theory often allows to simplify complicated computations with tensors. For example if you know a priori that some complicated tensor you are interested in must be say $O(n)$-invariant, then if the space of invariant tensors of that type must vanish for representation theoretical reasons, then your complicated expression also vanishes.

To give a concrete example of application of that sort (which in fact is more involved than what I just described), let me remind the so called tube formula due to Weyl himself. Let $(M,g)$ be a Riemannian manifold isometrically imbedded into Euclidean space $\mathbb{R}^N$. Consider the volume in $\mathbb{R}^N$ of the $\varepsilon$-neighborhood of $M$ inside $\mathbb{R}^N$ as a function of $\varepsilon\geq 0$. Weyl proved:

1) This function is a polynomial in $\varepsilon$ for $\varepsilon \geq 0$ small enough.

2) The coefficients of this polynomial, after appropriate normalization, are intrinsic invariants of the Riemannian structure $g$, i.e. are independent of the isometric inmedding into $\mathbb{R}^N$. More precisely they are polynomials in the components of the Riemann curvature tensor of $M$.

The proof of (1) is both elementary and easy. Part (2) is indeed non-trivial. Weyl's proof used the invariant theory of $O(n)$.

Answer 6

In algebra many algebraic groups $G$ of finite type over a field $k$ may be realized as closed subgroups of $\operatorname{GL}_k(V)$, where $V$ is a finite dimensional vector space. Hence there is a set of polynomials $I:=\{f_1,..,f_l\}$ with the property that the zero set $Z(I)\subseteq \operatorname{GL}_k(V)$ defines $G$ as a closed subgroup of $\operatorname{GL}_k(V)$ - the general linear group on $V$. Hence we may view $G$ as a "group of matrices" with coefficients in the field $k$. In fact any affine algebraic group $G$ over a field $k$ may be realized as a closed subgroup of $\operatorname{GL}_k(V)$ for some finite dimensional $k$-vector space $V$. There are non-affine algebraic groups: Abelian varieties. If $E\subseteq \mathbb{P}^2_k$ is an elliptic curve over $k$, it follows $E$ has a group structure $m:E \times E \rightarrow E$, making $(E,m)$ into an abelian algebraic group. The mulitiplication map $m$ is a map of algebraic varieties. Since any affine algebraic group is an affine algebraic variety and an elliptic curve $E$ is a projective variety, we cannot embed $E$ as a closed subgroup of $\operatorname{GL}_k(V)$.

Example: If $G=\operatorname{SL}_k(V)$ where $V$ is a finite dimensional $k$-vector space and $H\subseteq G$ is a closed sub group we may construct the "quotient" $G/H$ and $G/H$ is a smooth quasi projective algebraic variety of finite type over $k$. If $H$ is the subgroup of $G$ fixing a $d$-dimensional sub space ($d< dim(V)$) it follows $G/H\cong \mathbb{G}(d,V)$ is the grassmannian variety parametrizing $d$-dimensional vector subspaces of $V$. The group $H$ is a matrix group - it is a closed sub-group of $G$. Any linear representation $\rho:H \rightarrow \operatorname{GL}_k(W)$ gives rise to a finite rank vector bundle $\pi:E(\rho)\rightarrow \mathbb{G}(d,V)$. Hence in geometry such $k$-linear representations of matrix groups arise in the study of vector bundles on the grassmannian (and other flag varieties). In fact the vector bundle $E(\rho)$ has a canonical left $G$-action, the map $\pi$ is invariant with respect to this action and there is an "equivalence of categories" between the category of "$G$-linearized" finite rank algebraic vector bundles on $\mathbb{G}(d,V)$ and finite dimensional $k$-linear representations of $H$. This type of correspondence between geometry and linear representations of algebraic groups is much studied.

Example: Let $V:=k\{e_0,e_1\}$ and $V^*:=k\{x_0,x_1\}$ with $x_0,x_1$ coordinate functions on $V$. It follows $\operatorname{Sym}_k(V^*)=k[x_0,x_1]$ is the polynomial ring in two variables. It follows $\operatorname{Proj}(\operatorname{Sym}_k(V^*)):=\mathbb{P}^1_k$ is the projective line. Let $\mathcal{O}(l)$ be the tautological bundle with $l\geq 1$. It follows

F1. $\Gamma(\mathbb{P}^1_k, \mathcal{O}(l))=\operatorname{Sym}_k^l(V^*)$

is the $l$'th symmetric product of $V^*$. In this case if we choose a line $L$ in $V$ and let $H$ be the subgroup of $\operatorname{SL}(V)$ fixing $L$ it follows $\operatorname{SL}(V)/H \cong \mathbb{P}^1_k$. Formula F1 gives a geometric construction of all irreducible finite dimensional $\operatorname{SL}(V)$-modules. They are all on the form $Sym_k^l(V^*)$ for some $l\geq 1$. This generalize to higher dimension: For any $V$ we may realize all irreducible finite dimensional $\operatorname{SL}(V)$-modules as global sections of invertibel sheaves on flag varieties $\operatorname{SL}(V)/P$. (The Borel-Weil-Bott formula). Hence global sections of invertible sheaves and higher rank vector bundles on grassmannians and flag varieties carries the structure of a linear representation of a matrix group.

There are classification results on connected algebraic groups over a field $k$:

Thm. Every connected algebraic group $G$ may be realized as an extension of an abelian variety $A$ by an affine algebraic group $H$.

And $H$ may be realized as a matrix group. Hence any connected algebraic group is the extension of an abelian variety by a "matrix group". Hence in algebra they appear "everywhere".