- Docente: Andrea Brini
- Credits: 6
- SSD: MAT/02
- Language: Italian
- Moduli: Andrea Brini (Modulo 1) Francesco Regonati (Modulo 2)
- Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Mathematics (cod. 8208)
Course contents
ALGEBRAIC COMBINATORICS AND REPRESENTATION THEORY. (Brini)
Superalgebras: basic definitions and constructions. Associative and Lie superalgebras.
Supersymmetric algebras. Letterplace superalgebras.
Superderivations and superpolarization operators. Actions of Lie superalgebras. Super[L|P] as a bimodule.
Z_2-graded tensor spaces and symmetric groups. The classical action and the Berele-Regev-Sergeev action.
The method of virtual variables. Capelli type operators and their virtualization/devirtualization.
The Grosshans-Rota-Stein biproducts. Virtual presentation and Laplace expansions.
Basic combinatorics ofYoung tableaux. Superstandard Young tableaux and the hook property. Symmetrized bitableaux and Gordan-Capelli series. Young-Capelli symmetrizers and combinatorics. Symmetry coefficients and triangularity theorems. Super[L|P] as a semisimple module. Complete decomposition theorems and double centralizer theorems.
Readings/Bibliography
A. Brini, Combinatorics, Superalgebras, Invariant theory and
Representation theory, Seminaire Lotharingien de Combinatoire 55
(2007), pp. 118
Frank D. Grosshans, The work of Gian-Carlo Rota on invariant theory, Algebra univers. 49 (2003) 213-258
C. Procesi, Lie Groups: An Approach Through Invariants and
Representations, (Universitext) Springer 2006
Assessment methods
The examination consists of an oral examination lasting 45 minutes. Will occur 'the student's Competency both in terms of acquisition of concepts and methods, with application to concrete cases.
Office hours
See the website of Andrea Brini
See the website of Francesco Regonati