S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 8: Presentations and the Modular Group

30 The Modular Group and Representations of Abelian Groups

· Denis Auroux

Original Professor Auroux’s handwritten notes (PDF), behind a password for now:

Last time, we discussed presentations of groups by generators and relations, and the Cayley graph.

and

One more example:

Proposition 30.1. is generated by and .

Proof. Given , we want to express it in terms of and .

  • If , then , and is either or .
  • Now assume , and repeatedly apply the following algorithm to modify :
    • if , use Euclidean division to write , . Then This decreases .
    • if , brings us back to .

After finitely many steps we find that the product of with some word in and has and , hence is or . ∎

  • There is a different, geometric proof, based on the fact that acts on the upper half plane by Now acts by and by .

The region is a fundamental domain of this action, in the sense that and its images under exactly tile .

Tiling of the upper half plane by images of the fundamental domain of PSL2(Z): the shaded region I above the unit circle between Re z = -1/2 and 1/2, neighbouring vertical strips T, T^2, T^-1, T^-2, the regions S, TS, T^2S, T^-1S, T^-2S under the unit circles centred at the integers, and smaller regions such as ST, STS, ST^-1, TST bounded by circles of radius 1/3; the real axis is marked from -2 to 2.

Since the regions immediately adjacent to are and , the regions immediately next to are and . The structure of the tiling reproduces exactly the Cayley graph of with generators , and the fact that all regions can be reached from in finitely many steps is equivalent to: generate .

  • Other generators: instead of and , we could use
    • and . These have finite order! .
    • and . These are conjugates! .
    • The images of these matrices in also generate .

Theorem 30.2.

Proof. and , so have orders 2 and 3 in . These relations reduce any word in to the form (the word can start and end with either or ).

Mapping to by , induces a surjective homomorphism and the kernel consists of all elements (words ) such that the corresponding expression in and equals in , i.e. in .

Observe: , . Assume some longer word in and ’s (alternating between these) simplifies to .

  • If it starts and ends with , we get a shorter word by conjugating by : since , .
  • If it starts with , conjugate it to get another word that doesn’t: .

Iterating, we eventually get some word .

But and . So some product of the matrices and equals .

Observe: when multiplying a matrix with entries by or , its entries remain , and the sum of all entries increases.

So we can’t get . Hence no word in simplifies to in , and . ∎

  • This presentation can be rewritten in terms of other generators: with in the first presentation and in the second; and

(And to connect two strands of our discussion: the center of the braid group is generated by and is , so mapping , gives

Representation theory

Next topic (through the end of the semester): representation theory ( Artin chapter 10). Recommended text: Fulton and Harris, Representation Theory: A First Course, part I.

Representation theory = the study of group actions on vector spaces, i.e. homomorphisms (for us, mostly over ).

Historically, groups first arose as geometric symmetries, so in the 19th century groups were mostly thought of as subgroups of , rather than abstract groups! The more modern viewpoint, rather, splits this into the study of groups on their own (what we’ve studied) + how to think of an abstract group as a subgroup of (what we’ll see now).

We’ll focus on representations of finite groups, but the problem is also interesting for discrete infinite groups (e.g. , braid groups, …), or continuous ones (Lie groups: , , …).

Notes not reached in this lecture, covered at the start of {{LINK:L31}}

Definition 30.3. A representation of a group is a vector space (usually finite-dimensional; we mostly consider ) + an action of on by linear operators, i.e. such that for all , is a linear map.

Equivalently: a homomorphism , the group of invertible linear operators .

Definition 30.4.

  • A subrepresentation is a subspace which is invariant under , i.e. for all .
  • A representation is irreducible if it has no nontrivial subrepresentations.

Example 30.5. If is a cyclic group, then a representation of is a vector space together with such that .

Return briefly to linear algebra:

Lemma 30.6. Let be a finite-dimensional -vector space and of finite order, . Then is diagonalizable.

Proof. This is because the minimal polynomial of divides , hence has simple roots.

Explicitly: over , factors as , where . So the eigenvalues of are -th roots of unity (), and the generalized eigenspaces () give , a decomposition of into invariant subspaces of .

Since is invertible on , we have , i.e. . Hence is diagonalizable. ∎

Returning to : invariant subspaces of are subrepresentations, and splits into a direct sum of 1-dimensional (irreducible) representations , for a basis of eigenvectors of , each given by a homomorphism

  • Now, if is a -representation of a finite abelian group , , with , the -action is equivalent to the data of such that and which pairwise commute, (then , with the generators of the factors ).

    By the lemma each is diagonalizable, and by the homework, commuting diagonalizable operators are simultaneously diagonalizable. In fact: the eigenspaces of are invariant under all , and the restriction of to an eigenspace of is of finite order, hence diagonalizable by the lemma. Proceed by induction on .

    This shows that splits into a of 1-dimensional subrepresentations. Those now correspond to homomorphisms .

Definition 30.7. For a finite abelian group, define its dual .

This is an abelian group using pointwise multiplication: if are homomorphisms, then so is (⚠ This uses the fact that is abelian .)

Concretely, for , as well, via , though there is no canonical map .

Similarly, the same ( is determined by the images of the generators of , which are roots of 1 in ).

This completes the classification of (complex) representations of finite abelian groups!