34670 - Advanced Algebra 2

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.

(modulo Regonati)

TEORIA ALGEBRICA DEGLI INVARIANTI

Vector invariant theory ; brackets, bitableaux, straightening; First and Second Fundamental Theorems.
Invariants of binary forms; simbolic method;  canonical forms.
Invariants of symmetric and skew-symmetric tensors;   letterplace superalgebras Super[L|P],
biproducts, super-straightening, supersymmetric symbolic method.

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