Elenco seminari del ciclo di seminari
“ALGEBRA E GEOMETRIA”

If G is an algebraic group acting on a variety X, the sheets of X are the irreducible components of subsets of elements of X with equidimensional G-orbits. For G complex connected reductive, the sheets for the adjoint action of G on its Lie algebra g were studied by Borho and Kraft in 1979. More recently, Losev has introduced finitely-many subvarieties of g consisting of equidimensional orbits, called birational sheets: their definition is less immediate than the one of a sheet, but they enjoy better geometric and representation-theoretic properties and are central in Losev's proposal to give an Orbit method for semisimple Lie algebras. In the first part of the seminar we give an historical overview on sheets and recall some basics about algebraic groups and Lusztig-Spaltenstein induction in terms of the so-called Springer generalized map and analyse its interplay with birationality. This will allow us to introduce Losev's birational sheets. The last part is aimed at defining an analogue of birational sheets of conjugacy classes in G, under the hypothesis that the derived subgroup of G is simply connected. We will conclude with an overview of the main features of these varieties, which mirror some of the properties enjoyed by the objects defined by Losev.

In the last ten years, there has been a renewed interest in the theory of quantum symmetric pairs (QSPs). As the name suggests, a QSP is an algebraic datum which quantizes the notion of symmetric space. It consists of a Drinfeld-Jimbo quantum group of a simple Lie algebra and a distinguished coideal subalgebra quantizing the fixed point subalgebra of an involution. Although their representation theory remains quite mysterious, it is becoming more and more evident that QSPs possess an incredibly rich structure with their own theory of canonical basis and braid group actions yet adapted to a particular “boundary behavior”. In the first part of the talk, I will give an overview of this emerging theory, while in the second part I will report on ongoing joint work with B. Vlaar and T. Przezdziecki devoted to the construction of a meromorphic boundary Yang-Baxter operator for quantum affine symmetric pairs and a quantum Schur-Weyl duality which arises from the study of its poles.

For a complex connected semisimple linear algebraic group G of adjoint type and of rank n, De Concini and Procesi constructed its wonderful compactification X, which is a smooth Fano G x G-variety of Picard number n enjoying many interesting properties. In this talk, it is shown that the wonderful compactification X is rigid under Fano deformations. Namely, for any family of smooth Fano varieties over a connected base, if one fiber is isomorphic to X, then so are all other fibers. This is a joint work with Qifeng Li.

Let G be a permutation group acting on a finite set Omega. A subset B of Omega is called a base for G if the pointwise stabilizer of B in G is trivial. In the 19th century, bounding the order of a finite primitive permutation group G was a problem that attracted a lot of attention. Early investigations of bases then arose because such a problem reduces to that of bounding the minimal size of a base of G. Some other far- reaching applications across Pure Mathematics led the study of the base size to be a crucial area of current research in permutation groups. In the first part of the talk, we will investigate some of these applications and review some results about base size. We will present a recent improvement of a famous estimation due to Liebeck that estimates the base size of a primitive permutation group in terms of its degree. In the second part of the talk, we will define the concept of irredundant bases of G and the concept of IBIS groups. Whereas bases of minimal size have been well studied, irredundant bases and IBIS groups have not yet received a similar degree of attention. Indeed, Cameron and Fon-Der-Flaas, already in 1995, defined such groups and proposed to classify some meaningful families. But only this year, a systematic investigation of primitive permutation IBIS groups has been started. We will discuss how we reduced the classification of primitive IBIS groups to the almost simple groups and affine groups. Eventually, we will conclude by mentioning recent advances towards a complete classification.