Università degli Studi di Padova

Facoltà di SSMMFFN

Laurea Specialistica in Matematica ed ALGANT

Some information concerning the course "Representation Theory of Groups"

Teacher: Giovanna Carnovale
Schedule: We will begin on april 16th at 11.30 in room 2AB40, with next lecture on april 17th at 11.30 in room 2AB40. It will continue on mondays and wednesdays: 9.30-11.00, room 2AB40 and on fridays 11.30-13.15, room 2BC60.
Schedule for the last week: monday, tuesday, wednesday 9.30-11.30, room 2AB/40.

Reference texts:
C. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras.
P. Etingof et al, Introduction to Representation Theory, arXiv:0901.0827v3, downloadable from: http://xxx.arxiv.org/abs/0901.0827v3
W. Fulton and J. Harris, Representation Theory, an Introduction;
I. M. Isaacs,Character theory of finite groups. Corrected reprint of the 1976 original;
S. Lang, Algebra;
C. Procesi, Lie Groups, An Approach through Invariants and Representations, Springer UTX, 2007.
J.P. Serre, Répresentations Linéaires des Groupes Finis; (there exists also an English version);
R. Goodman and N.R. Wallach. Representations and invariants of the classical groups. Encyclopedia of Mathematics and its Applications, 68. Cambridge University Press, Cambridge, 1998

We will mainly follow P. Etingof et al. notes, and use the above literature for the remaning parts of the program such as extension of scalars and representations in positive characteristic (Isaacs, Serre), compact groups (Goodman-Wallach, Procesi).

Exam: A written examination: here is the schedule
june 23 2009, h.10:00 - room 2BC/30. You may dowload the exam from here. This is a corrected version.
july 10 2009, h.10:00 - room 2BC/30
september 16 2009, h.10:00 - room 1AD/50
september 24 2009, h.10:00 - room 2AB/45

Here is a copy of a take-home exam of the past year. Notice that the program has been changed (there were no quivers nor Lie algebras last year) so it is not a completely faithful model for this year's examination.

Office hours:
On appointment, in room 630, Torre Archimede, via Trieste 63.

New Program of the course and references for the different subjects
1) Representations of algebras, subrepresentations, morphisms, direct sums, quotient representations, irreducible representations, indecomposable representations, completely reducible representations. Schur's lemma and its direct consequences (when k is algebraically closed). Irreducible representations of commutative algebras.

2) Example of irreducible representations of a commutative algebra which are not 1-dimensional when k is not algebraically closed. Regular representation. Irreducible and indecomposable representations of k[x]. Central character. Cyclic representations.

3) Central character for indecomposable representations. Representations of the Weyl algebra.

4) Quivers, their representations and the path algebra.

5) Representations of a quiver: subrepresentations, irreducible representations, direct sums, indecomposable representations, morphisms of representations. Lie algebras: example, subalgebras, ideals, the centre, morphisms, representations, subrepresentations, direct sums, morphisms of representations, indecomposable and irreducible representations. The universal enveloping algebra of a Lie algebra.

6) Tensor products, symmetric and exterior products.

7) Universal property of the universal enveloping algebra, tensor product of representations of a Lie algebra, dual of a representation of a Lie algebra.

8) Irreducible representations of sl(2).

9) Finite dimensional complex representations of sl(2) are completely reducible. Casimir operator, decomposition of tensor products. Lie Theorem (exercise).

10) Subrepresentations of completely reducible representations. Representations of direct sums of matrix algebras. Filtrations. The radical. Finite dimensional algebras and semisimple algebras. The character of a representation.

11) Linear independence of irreducible characters. The Jordan-Hoelder theorem (proof for characteristic zero). The Krull-Remak-Schmidt theorem (without proof). Irreducible representations of tensor products of algebras. Maschke's theorem. The number of irreducible characters of a fintie group equals the number of conjugacy classes.

12) Examples of irreducible representations of finite groups.

13) Frobenius-Schur indicator. Representations of complex type, real type and quaternionic type.

14) The number of involutions in a finite group. Algebraic numbers and algebraic integers. Frobenius divisibility theorem.

15) Burnside theorem (solvability criterion). Representations of direct products. Virtual representations. Induced representations and their characters.

16) Frobenius reciprocity. Different definitions of induced representation. Decomposition of induced representations.

17) Irreducible representations of the symmetric group.

18) Schur functors and decomposition, as a representation of GL(V) of the n-fold tensor product of V.

19) Artin's Theorem.

20) Simply-laced Dynkin diagrams. The underlying graph of a connected quiver of finite representation type is a simply-laced Dynkin diagram.

21) Roots, simple roots, positive and negative roots. Reflection with respect to a root and the Weyl group. Reflection functors for quiver representations.

22) The indecoposable representations of a Dynkin quiver are in bijection with the positive roots.

Old Program of the course and references for the different subjects:
Algebras modules and representations (Isaacs Chapter I)
Group representations and characters (Isaacs, Chapter II)
Degree 1 representations (Curtis-Reiner)
Dual (contragredient) representation and its character (Serre, 2.1 exercise 3)
Products of characters and tensor product representations, symmetric square and its character (Isaacs Chapter IV, Serre 2.1)
Representation structure on Hom(V,V)=End(V) (Serre, 2.1 exercise 4)
Characters and integrality, Burnside's criterion (Isaacs Chapter III)
Induced representations and induced characters(Serre, partie II, 7 and I, 3.3)
Frobenius reciprocity formula and Mackey's irreducibility criterion (Serre 7.2-7.4)
Examples of induced representations (Serre 8.1)
Representations of direct products and semidirect products (Serre 3.2 Part I, 8.2 Part II)
Irreducible representations of the symmetric groups (Fulton-Harris)
Compact groups and their representation theory (Serre Chapter 4, Part I)
Complete reducibility for complex general linear groups (Fulton-Harris, unitary trick in Goodman-Wallach)
Schur functors and irreducible representations of the complex general linear group (Fulton-Harris)
Composition series of modules, Jordan-Holder's theorem (Lang, Algebra)
Changing the field: absolute irreducibility, splitting fields (Isaacs, Chapter IX, Curtis-Reiner section 29)
Schur index: definition and properties (Isaacs, Chapter X), definition 2 (Serre, 12.2) and comparison between the two approaches.
Grothendieck groups, Artin's and Brauer's Theorem (Isaacs Chapter V and VIII; Serre 9 and 10)
Irreducible characters are Z-linear combinations of monomial characters (Serre 8.5, 10.5)
Characterization of characters (Serre, 11.1)
The extension of Q by (exp G)-th roots of 1 is a splitting field (Isaacs, Chapter X)


Possible subjects for a seminar:
Coxeter groups and their standard representation;
Monodromy representation and Gauss hypergeometric functions;
Proof of the Theorem on absolute reducibility;
The Tits representation of Coxeter groups is faithful;
The Lie algebra sl_2;
Hopf algebras and their representations;
The Grothendieck group of the category of representations of the symmetric group and its Hopf algebra structure;
Representations of GL(2) over a finite field;
Projective representations and the Schur multiplier;
Modular characters;
The Schur multiplier for the symmetric group;
Representations of SU(2) and Jacobi polynomials;
Representations of the Alternating groups.