Claudio Procesi

Young diagrams and determinantal varieties

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Wonderful models of subspace arrangements

The motivation stems from our attempt to understand Drinfelds construction (el. [Dr2]) of special solutions of the Khniznik-Zamolodchikov equation (of. [K-Z]) with some prescribed asymptotic behavior and its consequences for some universal constructions associated to braiding: universal unipotent monodromy representations of braid groups, the construction of a universal Vassiliev invariant for knots, braided categories etc. The K-Z connection is a special flat meromorphic connection on C ~ with simple poles on a family of hyperplanes. It turns out that the prescription of the asymptotic behavior for such connections is controlled by the geometry of a suitable modification of C ~ in which the union of the polar hyperplanes is replaced by a divisor with normal crossings. In the process of developing this geometry we realized that our constructions could be developed more generally for subspace arrangements and became aware of the paper of Fulton-MaePherson [F-M] in which a Hironaka model is described for the complement of the big diagonal in the power of a smooth variety X. It became clear to us that our techniques were quite similar to theirs and so we adopted their notation of nested set in the appropriate general form. Although we work in a linear subspaces setting it is clear that the methods are essentially local and one can recover their results from our analysis applied to certain special configurations of subspaces. In fact the theory can be applied whenever we have a subvariety of a smooth variety which locally (in the gtale topology) appears as a union of subspaces.

Semisimple representations of quivers

We discuss the invariant theory of the variety of representations of a quiver and present generators and relations. We connect this theory of algebras with a trace satisfying a formal CayleyHamilton identity

On the geometry of conjugacy classes in classical groups

SummaryWe study closures of conjugacy classes in the Lie algebras of the orthogonal and symplectic groups and determine which ones are normal varieties. Furthermore we give a complete classification of the minimal singularities which arise in this context, i.e. the singularities which occur in the open classes in the boundary of a given conjugacy class. In contrast to the results for the general linear group ([KP1], [KP2]) there are classes with non normal closure; they are branched in a class of codimension two and give rise to normal minimal singularities. The methods used are (classical) invariant theory and algebraic geometry.

Topics in hyperplane arrangements, polytopes and box-splines

Preliminaries.- Polytopes.- Hyperplane Arrangements.- Fourier and Laplace Transforms.- Modules over the Weyl Algebra.- Differential and Difference Equations.- Approximation Theory I.- The Di?erentiable Case.- Splines.- RX as a D-Module.- The Function TX.- Cohomology.- Differential Equations.- The Discrete Case.- Integral Points in Polytopes.- The Partition Functions.- Toric Arrangements.- Cohomology of Toric Arrangements.- Polar Parts.- Approximation Theory.- Convolution by B(X).- Approximation by Splines.- Stationary Subdivisions.- The Wonderful Model.- Minimal Models.

The automorphism group of a free group is not linear

A linear group has ascending chain condition on centralizers. (Malcev [S, p. 511). Finitely generated linear groups are residually finite. (Malcev [S, p. 453). A solvable linear group is nilpotent-by-(abelian-byfinite). (Tits [5, pp. 14551461). A linear group either contains a free group of rank two, or is solvable-by-(locally finite). Lubotzky [3] has characterized finitely generated linear groups over a field of characteristic zero by purely group theoretic conditions. These conditions appear difficult to check, although they have been used to show linearity for certain groups. However, we do not use his theorem. The main result of this paper (Theorem 5) is that for n 2 3, the automorphism group of a free group of rank n is not a linear group. The proof uses the representation theory of algebraic groups to show that a kind of “diophantine equation” between the irreducible representations of

A q-analog for the virasoro algebra

(1994). A q-analog for the virasoro algebra. Communications in Algebra: Vol. 22, No. 10, pp. 3755-3774.

Inequalities defining orbit spaces

SummaryThe orbit space of a representation of a compact Lie group has a natural semialgebraic structure. We describe explicit ways of finding the inequalities defining this structure, and we give some applications.

Homology of the zero-set of a nilpotent vector field on a flag manifold

0.1. Let X be a linear transformation of a finite-dimensional vector space V. The configuration of flags in V which are fixed by X has rather remarkable properties when X is unipotent. Though this case is especially interesting, the proper generality in which to study such configurations is in the theory of reductive algebraic groups, where their definition can be reformulated in the language of Borel subalgebras as follows. Let G be a connected reductive group over C, with Lie algebra g, and let N E g be a nilpotent element. Let q be the variety of all Borel subalgebras of g and let

Cohomology of regular embeddings

We give in this paper an explicit description of the rational cohomology ring of a “complete symmetric variety” or regular compactification of an algebraic symmetric variety. The motivation for this computation comes from the desire to describe an explicit Schubert calculus for these varieties, which have been considered since the work of Chasles and Schubert in several special cases. The technique we use is the result of our attempts to understand better the work of Jurkiewicz and Danilov in the case of torus embeddings. The combinatorial presentation of the cohomology ring of a smooth torus embedding finds a natural explanation and proof using the language of equivariant cohomology. The key point is that the Reisner-Stanley algebra of the torus embedding coincides with its equivariant cohomology ring. We consider a class of spaces, regular embeddings, that contains both “complete symmetric varieties” and smooth torus embeddings. By suitably generalizing the notion of Reisner-Stanley algebra, we are able to compute the rational equivariant cohomology ring of these spaces in an explicit way.