Group Representations I, TCD 2014/15
Syllabus
The purpose of this course is to give an introduction to representation theory for the case of finite groups (and demonstrate that most of those approaches work well for infinite compact groups). A really important idea in mathematics is that a proper theory for anything should possess enough symmetries, and that for studying mathematical theories it makes sense to study their symmetries. Representation theory is the main instrument for studying symmetries.The course covers the following topics:
- Representation of a group. Examples of representations. Trivial representation. Regular representation.
- Equivalent representations. Arithmetics of representations. Irreducible representations. Schur's lemma.
- Characters and matrix elements. Orthogonality relations for matrix elements and characters.
- Applications. Representations of a product of two groups. Tensor powers of a faithful representation contain all irreducibles as constituents. Dimensions of irreducibles divide the order of the group. Burnside's paqb-theorem.
- Set representations. Orbits, intertwining numbers etc.
- Representations and character table of A5.
Learning outcomes
On successful completion of this module, students will be able to:- construct complex irreducible representations for various finite groups of small orders
- reproduce proofs of basic results that create theoretical background for dealing with group representations
- apply orthogonality relations for characters of finite groups to find multiplicities of irreducible constituents of a representation
- apply representation theoretic methods to simplify problems from other areas that "admit symmetries"
- identify group theoretic questions arising in representation theoretic problems, and use results in group theory to solve problems on group representations.
Materials
Notes of a similar (but not the same!) course from the academic year 2012/13, taken by Stiofáin Fordham (use at your own risk; I shall try to correct misprints as we progress, - please notify me of misprints you locate) [PDF]
Summary of classes
| Lecture | Topics covered | Lecture notes/slides | Homeworks/Tutorials/Solutions |
|---|---|---|---|
| 1 (12/01) | Motivation and intuition for representation theory. Recollections of background results. Finite groups of small orders. | ||
| 2 (14/01) | Examples of representations. Trivial representation. Left and right regular representations. One-dimensional representations of groups of small orders. | ||
| 3 (15/01) | Homomorpishms and isomoprhisms of representations. Examples. Reducible and irreducible representations. Decomposable and indecomposable representations. Example of an indecomposable reducible representation over a field of 2 elements. | L3 [PDF] | |
| 4 (19/01) | Complete reducibility. Example: complement of the trivial subrepresentation in the regular representation. | L4 [PDF] | HW1 [PDF] |
| 5 (21/01) | Schur's Lemma. Intertwining numbers. Example of the left regular representation. | L5 [PDF] | |
| 6 (22/01) | No class on 22/01. | ||
| The next class is on February 2. The classes we missed will be recovered in the last teaching week of the semester. | |||
| 7 (02/02) | Orthogonality relations for matrix elements of irreducible representations | ||
| 8 (04/02) | Orthogonality relations for characters. Application to multiplicities of irreducibles in the regular representations. Properties of characters. Class functions. | ||
| 9 (05/02) | The number of irreducible representations is equal to the number of conjugacy classes. Examples. The character table of S3. | HW2 [PDF] HW1 solutions [PDF] | |
| 10 (09/02) | Tensor products of vector spaces. Tensor products of representations. Examples. | ||
| 11 (11/02) | Tutorial class: character table of S4 | T1 [PDF] | |
| 12 (12/02) | Finishing the tutorial: representations of S4. The group S4 and rotations of the cube. | ||
| 13 (16/02) | The group S4 and rotations of the cube. Applications to averaging of numbers on faces of the cube. | ||
| 14 (18/02) | Representations of a group vs representations of its index two subgroup. Example of S4 and A4. | L14 [PDF] | HW3 [PDF] HW2 solutions [PDF] |
| 15 (19/02) | Isotypic components of representations. Projection operators via characters. Examples. | ||
| Reading week, no classes. Suggested reading: projection operators on individual summands of an isotypic component via matrix elements (pp.23-24 in lecture notes from 2012/13, or Section 2.7 in Serre's book). | |||
| 16 (02/03) | No class on 02/03. | ||
| 17 (04/03) | Faithful representations. Tensor powers of a faithful representation. | ||
| 18 (05/03) | Burnside's theorem. Decomposition of the Dedekind-Frobenius determinant. | ||
| 19 (09/03) | Permutation representations of finite groups. Intertwining numbers. Constructions of some representations of S5. | ||
| 20 (11/03) | Representations of S5 and of A5. | HW4 [PDF] HW3 solutions [PDF] | |
| 21 (12/03) | Geometric interpretation of three-dimensional representations of A5. Towards representations of Sn for arbitrary n. | ||
| 22 (16/03) | Representations of Sn for arbitrary n. Outline of the approach, and statement of the result. | ||
| 23 (18/03) | Representations of Sn for arbitrary n. Proof of the result. | L22-23 [PDF] | |
| 24 (19/03) | Tutorial class: using general methods to construct representations of S6. | T2 [PDF] | |
| 25 (23/03) | Decomposing tabloid representations of S6. | ||
| 26 (25/03) | Schur-Weyl duality. | HW5 [PDF] HW4 solutions [PDF] | |
| 27 (26/03) | Young symmetrizers. Constructing the Schur functors that give polynomial representations of GL(V). | ||
| 28 (30/03) | Algebraicity results and their applications. | ||
| 29 (01/04) | Burnside's paqb-theorem. | ||
| 30 (02/04) | Summary / revision. | HW5 solutions [PDF] |
Textbooks
Three main recommended textbooks are:- Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina, "Introduction to representation theory" [PDF]
- William Fulton, Joe Harris, "Representation theory: a first course"
- J.-P. Serre, "Linear representations of finite groups"
All three of these books cover (much) more topics than we shall address, but most things we discuss in class is in at least one of them. The most accessible reference might be the second one, the two others are more concise and to the point. There are a couple of copies of the third one which School of Maths bought for students taking this course to use, you can borrow them from me when you need them.