Corrado De Concini

• PhD
• Professor (Full) at Sapienza University of Rome

For a simple complex Lie algebra $\mathfrak g$ we study the space $Hom_\mathfrak g(L,\bigwedge \mathfrak g)$ when $L$ is either the little adjoint representation or, in type $A_{n-1}$, the $n$-th symmetric power of the defining representation. As main result we prove that $Hom_\mathfrak g(L,\bigwedge \mathfrak g)$ is a free module, of rank twice th...

For a simple complex Lie algebra $\mathfrak g$ we study the space of invariants $A=\left( \bigwedge \mathfrak g^*\otimes\mathfrak g^*\right)^{\mathfrak g}$, (which describes the isotypic component of type $\mathfrak g$ in $ \bigwedge \mathfrak g^*$) as a module over the algebra of invariants $\left(\bigwedge \mathfrak g^*\right)^{\mathfrak g}$. As...

For a simple complex Lie algebra $\mathfrak g$ we study the space of invariants $A=\left( \bigwedge \mathfrak g^*\otimes\mathfrak g^*\right)^{\mathfrak g}$, (which describes the isotypic component of type $\mathfrak g$ in $ \bigwedge \mathfrak g^*$) as a module over the algebra of invariants $\left(\bigwedge \mathfrak g^*\right)^{\mathfrak g}$. As...

These proceedings contain the contributions of some of the participants in the “intensive research period” held at the De Giorgi Research Center in Pisa, during the period May-June 2010. The central theme of this research period was the study of configuration spaces from various points of view. This topic originated from the intersection of several...

In this paper we construct a generalization of the classical Steinberg section for the quotient map of a semisimple group with respect to the conjugation action. We then give various applications of our construction including the construction of a sort of Gelfand Zetlin basis for a generic irreducible representation of quantum GL(n) at odd roots of...

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 P...

In this chapter we return to the theory of Chapter 14, using all of its notation, and complete the theory of the partition function.

In this chapter we want to give a taste to the reader of the wide area of approximation theory. This is a very large subject, ranging from analytical to even engineering-oriented topics. We merely point out a few facts more closely related to our main treatment. We refer to [70] for a review of these topics. We start by resuming and expanding the...

In this chapter we want to sketch a theory that might be viewed as an inverse or dual to the spline approximations developed from the Strang-Fix conditions. Here the main issue is to approximate or fit discrete data through continuous or even smooth data. In this setting, an element of the cardinal spline space åaeÙ BX(x-a)g(a)\sum \nolimits_{\alp...

In this chapter we discuss an approximation scheme as in [33] and [51], that gives some insight into the interest in box splines, which we will discuss presently.

In this chapter we compute the cohomology, with complex coefficients, of the complement of a toric arrangement. A different approach is due to Looijenga [74].

In this chapter we begin to study the problem of counting the number of integer points in a convex polytope, or the equivalent problem of computing a partition function. We start with the simplest case of numbers. We continue with the theorems of Brion and Ehrhart and leave the general discussion to the next chapters.

The main purpose of this chapter is to discuss the theory of Dahmen–Micchelli describing the difference equations that are satisfied by the quasipolynomials that describe the partition function TX\mathcal{T}_X on the big cells. These equations allow also us to develop possible recursive algorithms. Most of this chapter follows very closely the pa...

All the modules over Weyl algebras that will appear are built out of some basic irreducible modules, in the sense that they have finite composition series in which only these modules appear. It is thus useful to give a quick description of these modules. Denote by F the base field (of characteristic 0) over which V,U := V* are finite-dimensional ve...

This chapter is devoted to the study of the periodic analogue of a hyperplane arrangement, that we call a toric arrangement. Thus the treatment follows the same strategy as in Chapter 8, although with several technical complications. We shall then link this theory with that of the partition functions in a way similar to the treatment of TXT_X in C...

In this article, we start to recall the inversion formula for the convolution with the Box spline. The equivariant cohomology and the equivariant K-theory with respect to a compact torus G of various spaces associated to a linear action of G in a vector space M can be both described using some vector spaces of distributions, on the dual of the grou...

Let G be a torus acting linearly on a complex vector space M and let X be the list of weights of G in M. We determine the topological equivariant K-theory of the open subset M f of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the character lattice [^(G)] \widehat{G} , satisfying the cocircuit di...

This chapter is an introduction to the theory of hyperplane arrangements, which appear in our treatment as the hyperplanes associated to the list of linear equations given by a list X of vectors. We discuss several associated notions, including matroids, Tutte polynomials, and zonotopes. Finally, we expand the example of root systems and compute in...

In this note several computations of equivariant cohomology groups are performed. For the compactly supported equivariant cohomology, the notion of infinitesimal index developed in arXiv:1003.3525, allows to describe these groups in terms of certain spaces of distributions arising in the theory of splines. The new version contains a large number of...

In this note, we study an invariant associated to the zeros of the moment map generated by an action form, the infinitesimal index. This construction will be used to study the compactly supported equivariant cohomology of the zeros of the moment map and to give formulas for the multiplicity index map of a transversally elliptic operator.

We introduce and study a notion of analytic loop group with a Riemann-Hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity with non trivial central charge. We introduce a Poisson structure and study properties of its Poisson dual group. We prove that the Hopf-Poisson structure is isomorphic to th...

We want to explain some formulas appearing in connection with root systems and their zonotopes which are relevant for the theory of the Kostant partition function. In particular, we compute explicitly the Tutte polynomial for all exceptional root systems. A more systematic treatment of these topics will appear in a forthcoming book Topics in Hyperp...

This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory are in [4], while in [5] and [6] we shall investigate the cohomological formulas generated by this...

We study the quotient of a completion of a symmetric variety G/H under the action of H. We prove that this is isomorphic to the closure of the image of an isotropic torus under the action of the restricted Weyl group. In the case the completion is smooth and toroidal we describe the set of semistable points.

Let $G$ be a simply connected semisimple algebraic group and let $K$ be the subgroup of points fixed by an involution of $G$. For certain representations containing a line $r$ preserved by $K$, we study the normality of the closure of the set of vectors which are $G$ conjugate to a vector in $r$. Some applications of our result to the normality of...

Supermanifold theory is used to give a geometric interpretation of the standard constraints which are imposed in superspace formulations of supersymmetric gauge theories. The results obtained are exploited, together with a few facts from supermanifold cohomology, to provide a simple proof of Weil triviality in anomalous supersymmetric gauge theorie...

Without Abstract

The operator formalism for string theory at arbitrary genus is presented in great detail. A Hamiltonian operator is provided. This dictates the time evolution of any operator of the theory and, in particular, allows us to derive the equations of motion of the fundamental fields. Scattering amplitudes are defined as correlation functions of suitable...

We recall some deformation theory of susy-curves and construct the local model of their (compactified) moduli ‘spaces’. We also construct universal deformations “concentrated” at isolated points, which are the mathematical counterparts of the usual choices done in the physical literature. We argue that these cannot give a projected “atlas” for supe...

For the complement of a hyperplane arrangement we construct a dual homology basis to the no-broken-circuit basis of cohomology. This is based on the theory of wonderful embeddings and nested sets developed in [4]. Our result allows us to express the so-called Jeffrey-Kirwan residues in terms of integration on some explicit geometric cycles.

We show a curious identity on root systems which gives the evaluation of the volume of the spherical simpleces cut by the cone generated by simple roots.

In this paper we want to revisit results of Dahmen and Micchelli on box-splines which we reinterpret and make more precise. We compare these ideas with the work of Brion, Szenes, Vergne and others on polytopes and partition functions.

We show a curious identity on root systems which gives the evaluation of the volume of the spherical simpleces cut by the cone generated by simple roots.

We study the restriction of representations of Cayley-Hamilton algebras to subalgebras. This theory is applied to determine tensor products and branching rules for representations of quantum groups at roots of 1.

Motivated by the counting formulas of integral polytopes, as in Brion and Vergne, and Szenes and Vergne, we start to form the foundations of a theory for toric arrangements, which may be considered as the periodic version of the theory of hyperplane arrangements.

We use the results of AG/0406290 to discuss the counting formulas of network flow polytopes and magic squares, i.e. the formula for the corresponding Ehrhart polynomial in terms of residues. We also discuss a description of the big cells using the theory of non broken circuit bases.

In this paper we study the Schwarz genus for the covering of the space of polynomials with distinct roots by its roots. We show that, for the first unknown case (degree 6), the genus is strictly less than the one predicted by dimension arguments, contrary to what happens in all other reflection groups.

For the complement of a hyperplane arrangement we construct a dual homology basis to the no broken circuit basis of cohomology. This is based on the theory of wonderful embeddings and nested sets.

Given a representation ρ : G → GL(N) of a semisimple group G, we discuss the normality or non normality of the cone over ρ(G) using the wonderful compactification of the adjoint quotient of G and its projective normality [K]. These methods are then used to discuss the normality or non normality of certain other orbit closures including determinanta...

Let F be a field and i < j be integers between 1 and n. A map of Grassmannians f : Gr(i, F^n) --> Gr(j, F^n) is called nesting, if l is contained in f(l) for every l in Gr(i, F^n). We show that there are no continuous nesting maps over C and no algebraic nesting maps over any algebraically closed field F, except for a few obvious ones. The continuo...

Given a finite Coxeter system (W, S), we exhibit an explicit "small" resolution of the trivial Z[W]-module. This is obtained by an explicit combina-torial construction of k(W, 1), which is described as a CW-complex with cells corresponding to flags in S. Some applications are given.

Introduction. Let R := Q [q; q Gamma1 ] be the ring of rational Laurent polynomials in one variable q. Let Br(n) be the Artin braid group with n strings and let R q be the Br(n)- module given by the action over R defined by mapping each standard generator of Br(n) to the multiplication by Gammaq. In [DPS] the first three authors computed the cohomo...

The symmetric varieties considered in this paper are the quotientsG/H, whereG is an adjoint semi-simple group over a fieldk of characteristic 2, andH is the fixed point group of an involutorial automorphism ofG which is defined overk. In the casek=C, De Concini and Procesi (1983) constructed a wonderful compactification ofG/H. We prove the existenc...

this paper we are going to consider cohomology with cofficients in a very natural module over Bm , which always gives non trivial top-cohomology, and such that all cohomology has a very interesting and neat description. Let R q be the Bm+1-module given by the action over the ring

These are the informal notes of two seminars held at the Universita` di Roma "La Sapienza", and at the Scuola Normale Superiore in Pisa in Spring and Autumn 1997. We discuss in detail the content of the parts of Givental's paper dealing with mirror symmetry for projective complete intersections.

Let W be a Coxeter group and let Gw be the associated Artin group. We consider the local system over k(Gw, 1) with coefficients in which associates to the standard generators of Gw the multiplication by q. For the all list of finite irreducible Coxeter groups we calculate the top-cohomology of this local system. It turns out that the ideal which we...

This paper is an expanded version of the talk I gave at the first ECM held in Paris in July 92. Quantum groups, or better quantized enveloping algebras have been defined around 1985 by Drinfeld and Jimbo, [D], [J], as neither commutative nor cocommutative Hopf algebras obtained by suitably deforming the defining relations of the enveloping algebra...

this paper we describe, for any given finite family of subspaces of a vector space or for linear subspaces in affine or projective space, a smooth model, proper over the given space, in which the complement of these subspaces is unchanged but the family of subspaces is replaced by a divisor with normal crossings. This model can be described explici...

This paper stems from our attempt to understand Drinfeld's construction (cf. [Dr2]) of special solutions of the Knizhnik-Zamolodchikov equation (cf. [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 construc...

this paper. In doing this we have been necessarily somewhat sketchy but we refer to the original papers where the topics here recalled are treated in detail. In x2 we introduce and study certain ideals in the algebras F q [G]; F q [B

We introduce a form of the quantum function algebra on a Drinfeld-Jimbo quantum group over the ring Z[q, q-1]. Specializing q to a root of 1, we show that over the cyclotomic field this algebra is a projective module over its central sub-algebra, which is the usual coordinate algebra of the group. We study the induced Poisson-Lie structure of the g...

In this paper we study finite group symmetries of differential behaviors (i.e., kernels of linear constant coefficient partial differential operators). They lead us to study the actions of a finite group on free modules over a polynomial ring. We establish algebraic results which are then used to obtain canonical differential representations of sym...

We show that an irreducible representation of a quantized enveloping algebraU ε at a ℓth root of 1 has maximal dimension (=ℓN ) if the corresponding symplectic leaf has maximal dimension (=2N). The method of the proof consists of a construction of a sequence of degenerations ofU ε, the last one being aq-commutative algebraU ε(2N). This allows us to...

In this paper we analyze the structure of some subalgebras of quantized enveloping algebras corresponding to unipotent and solvable subgroups of a simple Lie group G. These algebras have the non--commutative structure of iterated algebras of twisted polynomials with a derivation, an object which has often appeared in the general theory of non-commu...

This paper is a continuation of our earlier papers [ Prog. Math. 92, 471-506 (1990; Zbl 0738.17008), J. Am. Math. Soc. 5, No. 1, 151-189 (1992; Zbl 0747.17018)] on representations of quantum groups at roots of 1. Here we show that an irreducible representation of a quantum group at an odd root of 1 can be uniquely induced from an exceptional repres...

O. INTRODUCTION AND NOTATIONS 0.1. This paper is a continuation of the paper (DCK) on representations of quantum groups at roots of 1. We give a solution to most of the conjectures stated in (DCK, §5) on the center and on the quantum coadjoint action (some of the conjectures needed modification to be correct). As in the case of Lie groups, "simply...

We formulate conformal field theories on the infinite-dimensional grassmannian manifold. Besides recovering the known results for the central charge and correlation functions of the b-c system this formalism immediately lends itself to further generalization.The grassmannian manifold is in fact an ad hoc model for the geometrical interpretation of...

We establish a canonical isomorphism between the second cohomology of the Lie algebra of regular differential operators on Cx of degree ?1, and the second singular cohomology of the moduli space $$\hat F_{g - 1} $$ of quintuples (C, p, z, L, [?]), whereC is a smooth genusg Riemann surface,p a point onC, z a local parameter atp, L a degreeg-1 line...

We characterize the algebraic-geometric potentials for the Schrödinger and AKNS operators using the Weyl m-functions and the Floquet exponent for these operators. The characterization is this: among random ergodic Schrödinger operators, the alebraic-geometric potentials are those for which (i) the spectrum is a union of finitely many intervals (or...

This chapter is independent of the remaining treatment and can be skipped without compromising the understanding of what follows.

Without Abstract

The special basis in spaces of finite dimensional representation ofS N and GL(n) is constructed and its properties are studied.

Quantum fluctuations about a Yang-Mills pseudoparticle solution are studied from the point of view of spontaneous breakdown of conformal symmetry. It is argued that the linearization procedure by which one passes from the nonlinear Yang-Mills equation to linear equations for the small-oscillation modes leads to a group contraction of the conformal...

The purpose of this paper is to generalize the results of Quillen [lo] about the cohomology of (the classifying space of) the general linear groups over a finite field to the orthogonal case. In this paper we will restrict ourselves to the study of the cohomology with mod 2 coefficients of (the classifying space of) the orthogonal groups. While in...

By using projective geometry arguments, it is shown that in a spontaneously broken symmetry theory the linearization procedure by which one passes from non-linear Heisenberg fields equations to the linear free fields equations leads to a group contraction mechanism: the symmetry group that appears in observations is a group contraction of the dynam...

In spontaneously broken symmetry theories, the symmetry group that appears in observations proves to be a group contraction of the dynamical invariance group. Infrared effects play a crucial role in the dynamical rearrangement of symmetry which leads to the group contraction. Many examples are considered. General theorems are given for SU(n) and SO...

This chapter focuses on a portion of classical invariant theory that goes under the name of the first and second fundamental theorem for the classical groups in a characteristic-free way, that is, where the base ring A is any commutative ring, in particular the integers or an arbitrary field. The ring of polynomial functions over A in the entries o...

Sommario. In occasione dei cinquant'anni del C.I.M.E. si espongono alcuni risultati degli ultimi anni che sono direttamente o indirettamente collegati ai risultati contenuti nelle note (11) relative al corso C.I.M.E., Invariant theory, Montecatini, 1982. 1. Introduzione Quando Pietro Zecca, che ringrazio per il grande onore, mi ha chiesto di scrive...