Schur Functions and Characters of Lie Algebras and Superalgebras

Abstract.
The relationship between Schur functions and characters of irreducible representations of classical Lie algebras will be discussed, including both determinantal and combinatorial definitions of these characters. Some classical generating functions for sums of Schur functions indexed by partitions of particular type will generalised to situations in which either the size or the number of parts of the relevant partitions are restricted. The relationship of these identies to characters of either finite or infinite-dimensional irreducible representations of classical Lie algebras will be discussed. In this connection the importance of Howe dual pairs will be emphasised as a means of calculating the relevant characters, as well as analogies between spin representations of orthogonal groups and metaplectic representations of symplectic groups. The extension will then be made of some of these ideas to the case of Schur supersymmetric functions and characters of Lie superalgebras, including the orthosymplectic superalgebras. Some applications to random matrix theory will be briefly described.

Tentative content:

Lecture 1

1. Schur functions, Young tableaux and characters of gl(n)
2. Partitions, Frobenius notation, modification rules
3. Infinite series and the Cauchy identity
3. Characters of o(n) and sp(n)
4. Howe dual pairs and the evaluation of characters

Lecture 2

5. Schur function expansions - Weyl, Littlewood, Cauchy
6. Row length restrictions of classical expansions - Macdonald, Stembridge, Okada, Krattenthaler
7. Relationship with Frobenius notation partitions
8. New column length restrictions of classical expansions
9. Connection with characters of o(n) and sp(n)
10. Row and column restricted Cauchy identities

Lecture 3

13. Supersymmetric functions and characters of gl(m|n)
14. Characters of gl(m|n), osp(m|n) and spo(m|n)
15. Howe supersymmetric dual pairs
16. Applications to random matrices