- Docente: Andrea Brini
- Credits: 6
- SSD: MAT/02
- Language: Italian
- Moduli: Andrea Brini (Modulo 1) Marilena Barnabei (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 ENUMERATION. (Barnabei)
The inclusion-exclusion principle: Da Silva and Sylvester
formulas. Derangements. Euler function.
Partially ordered sets and lattices: Partially ordered set.
Covering and Hasse diagram. Duality. Rank. Order morphism. Lattice.
Completeness. Sublattice; lattice morphism. Modular lattice.
Distributive lattice. Birkhoff's Theorems. Atom; atomic lattice.
Boolean algebra.
The Moebius function of number theory : prime numbers. The lattice
of integers ordered by divisibility. Moebius function. An
application: the necklace problem.
The Moebius function of a poset: The incidence algebra and some of
its notable functions. Moebius function. Moebius inversion formula.
Computing the Moebius function of a poset. Closure operators and
Galois connections.
The Moebius function of a lattice: General results. The lattice of
subspaces of V(n; q). The lattice of partitions of a finite set. An
application: the number of connected graphs. The characteristic
polynomial. Cross-cut theorems. Complemented lattices.
Geometric lattices: Semimodular lattices. Geometric lattices.
Matroids. Applications: 1. The chromatic polynomial of a graph. 2.
The critical exponent of a set of points in PG(n; q). 3. Hyperplane
arrangements.
The inclusion-exclusion principle: Da Silva and Sylvester
formulas. Derangements. Euler function.
Partially ordered sets and lattices: Partially ordered set.
Covering and Hasse diagram. Duality. Rank. Order morphism. Lattice.
Completeness. Sublattice; lattice morphism. Modular lattice.
Distributive lattice. Birkhoff's Theorems. Atom; atomic lattice.
Boolean algebra.
The Moebius function of number theory : prime numbers. The lattice
of integers ordered by divisibility. Moebius function. An
application: the necklace problem.
The Moebius function of a poset: The incidence algebra and some of
its notable functions. Moebius function. Moebius inversion formula.
Computing the Moebius function of a poset. Closure operators and
Galois connections.
The Moebius function of a lattice: General results. The lattice of
subspaces of V(n; q). The lattice of partitions of a finite set. An
application: the number of connected graphs. The characteristic
polynomial. Cross-cut theorems. Complemented lattices.
Geometric lattices: Semimodular lattices. Geometric lattices.
Matroids. Applications: 1. The chromatic polynomial of a graph. 2.
The critical exponent of a set of points in PG(n; q). 3. Hyperplane
arrangements.
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.
Office hours
See the website of Andrea Brini
See the website of Marilena Barnabei