Claudio Procesi

• Ph. D. University Of Chicago,
• Professor Emeritus at Sapienza University of Rome

We go back to a graph used extensively in previous papers with Michela Procesi and Bich Van Nguyen to studi the non linear Scroginger equation. We fix several mistakes of that treatment and try to expand some proofs which were confused or just too short.

In this paper we discuss the minimal Cayley Hamilton norm for a finite dimensional algebra over a field $F$ based on a paper by Skip Garibaldi

We add some further constructions to the general Theory of Cayley Hamilton algebras developed in the papers [Procesi, C.: J. Algebra 107, 63–74 (1987), Procesi, C.: Naz.LinceiRend. Lincei Mat.Appl.32(1), 23–61 (2021), Procesi, C.: Indag. Math. (N.S.) 32(6), 1190–1228 (2021) ].

We study the algebra $\Sigma_n$ induced by the action of the symmetric group $S_n$ on $V^{\otimes n}$ when $\dim V=2$. Our main result is that the space of symmetric elements of $\Sigma_n$ is linearly spanned by the involutions of $S_n$.

The aim of this paper is to establish a first and second fundamental theorem for G L ( V ) GL(V) equivariant polynomial maps from k k –tuples of matrix variables E n d ( V ) k End(V)^{ k} to tensor spaces E n d ( V ) ⊗ n End(V)^{ \otimes n} , in the spirit of H. Weyl’s book The Classical Groups and of symbolic algebra.

This is an extended version of a lecture given in Varese on 23/09/2021 in occasion of the awarding of the Riemann prize to Terence Tao.

We discuss several constructions of swap polynomials that is non commutative polynomials in matrix variables xi∈Md(Q) with values in Md(Q)⊗2 which are multiples of the transposition operator (1,2).

Tensor polynomial identities generalize the concept of polynomial identities on d × d matrices to identities on tensor product spaces. Here we completely characterize a certain class of tensor polynomial identities in terms of their associated Young diagrams. Furthermore, we provide a method to evaluate arbitrary alternating tensor polynomials in d...

We develop the general Theory of Cayley Hamilton algebras and we compare this with the theory of pseudocharacters. We finally characterize prime T–ideals for Cayley Hamilton algebras and discuss some aspects of their geometry.

We develop the general Theory of Cayley–Hamilton algebras using norms and compare with the approach, valid only in characteristic 0, using traces and presented in a previous paper [19].

We discuss several constructions of swap polynomials, that is 2--tensor valued matrix polynomials which are multiples of the swap or switch operator.

In this paper we first review the main ideas of Cayley Hamilton algebras and then, in Theorem 3.18, we give a different approach and formulation of the Theorem of Zubkov on the relations among invariants of matrices. In this approach the relations appear as those of a universal Cayley Hamilton algebra.

The aim of this paper is to establish a first and second fundamental theorem for $GL(V)$ equivariant polynomial maps from $k$--tuples of matrix variables $End(V)^{ k} $ to tensor spaces $End(V)^{ \otimes n}$ in the spirit of H. Weyl's book {\em The classical groups} \cite{Weyl} and of symbolic algebra.

Tensor polynomial identities generalize the concept of polynomial identities on $d \times d$ matrices to identities on tensor product spaces. Here we completely characterize a certain class of tensor polynomial identities in terms of their associated Young tableaux. Furthermore, we provide a method to evaluate arbitrary alternating tensor polynomia...

We develop the general Theory of Cayley Hamilton algebras using norms and compare with the approach, valid only in characteristic 0, using traces and presented in a previous paper $T$-ideals of Cayley Hamilton algebras, 2020, arXiv:2008.02222

The aim of this expository note is to compare work of Formanek on a certain construction of central polynomials with that of Benoit Collins on integration on unitary groups. These two quite disjoint topics share the construction of the same function on the symmetric group, which the second author calls {\em Weintgarten function}. By joining these t...

We develop the general Theory of Cayley Hamilton algebras and we compare this with the theory of pseudocharacters. We finally characterize prime $T$--ideals for Cayley Hamilton algebras and discuss some of their geometry. To the memory of T. A. Springer

The purpose of this paper is to discuss the classical, and forgotten, notion of perpetuants, see Definition 2.13, and in particular to exhibit a basis of these elements in Theorem 4.9, thus closing an old line of investigation started by J. J. Sylvester in 1882. In order to do this we also give a proof of the classical Theorem of Stroh computing th...

We discuss the classical, and forgotten, notion of perpetuants. We give a proof of the Theorem of Stroh computing their dimensions, and exhibit a basis of perpetuants, thus closing an old line of investigation.

This is a very brief overview of some of the main ideas of algebra through history, from the decomposition of a binomial identity, to the degree of a complex number, Euler’s formula, transcendental numbers, skew fields, the Abel–Ruffini theorem, permutation groups, Lie algebras and more.

We consider certain functional identities on the matrix algebra that are defined similarly as the trace identities, except that the “coefficients” are arbitrary polynomials, not necessarily those expressible by the traces. The main issue is the question of whether such an identity is a consequence of the Cayley–Hamilton identity. We show that the a...

We have received an e-mail from Bryan Gillespie pointing out that a proposition, that is Proposition 8.5, of our book, [1] is incorrect as stated. The given formula (8.5) is valid only in the generic case that is assuming that for any point of the arrangement $p, X_p$ is formed by a basis. The correct proposition is slightly weaker, in general one...

The present paper is devoted to the construction of small reducible quasi--periodic solutions for the completely resonant NLS equations on a $d$--dimensional torus $\T^d$. The main point is to prove that prove that the normal form is reducible, block diagonal and satisfies the second Melnikov condition block wise. From this we deduce the result by...

In this paper we stress the role of invariant theory and in particular the role of varieties of semisimple representations in the theory of polynomial identities of an associative algebra. In particular, using this tool, we show that two PI-equivalent finite-dimensional fundamental algebras (see Definition 2.19) have the same semisimple part. Moreo...

We prove that the restriction to diagonal matrices of the scheme of commuting matrices is an isomorphism when restricted to invariants to the symmetric group invariants.

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...

This paper is a request for help to clarify a rather confusing state of the Theory.

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...

We present a proof of the Amitsur--Levitzki theorem which is a basis for a general theory of equivariant skew--symmetric maps on matrices.

We present a proof of the Amitsur--Levitzki theorem which is a basis for a general theory of equivariant skew--symmetric maps on matrices.

I discuss some joint work with Michela Procesi and Nguyen Bich Van.

We discuss some combinatorial and algebraic problems which have arisen from the study of the nonlinear Schrödinger equation. In particular we discuss a combinatorial graph, which we call energy graph associated to a Cayley graph. Some important features of normal forms are hidden into properties of this graph. Some of these properties are at the mo...

We consider certain functional identities on the matrix algebra $M_n$ that are defined similarly as the trace identities, except that the "coefficients" are arbitrary polynomials, not necessarily those expressible by the traces. The main issue is the question of whether such an identity is a consequence of the Cayley-Hamilton identity. We show that...

In this note we use the normal forms of the completely resonant non--linear Schr\"odinger equation on a torus (NLS) derived in previous work in order to produce, under a KAM algorithm, large families of stable and unstable quasi periodic solutions for the NLS in any number of independent frequencies.

In this article we want to present a meeting ground between rather different mathematical topics from numerical analysis to index theory and symplectic geometry. The unifying idea is that of linear representation of tori inside which the combinatorics, analysis and geometry is developed.

We discuss a class of normal forms of the completely resonant non-linear Schrödinger equation on a torus. We stress the geometric and combinatorial constructions arising from this study.

We discuss the stability of a class of normal forms of the completely resonant non--linear Schr\"odinger equation on a torus described in a previous paper. The discussion is essentially combinatorial and algebraic in nature. Thus this paper contains the proof of two Theorems of algebraic, combinatorial and geometric nature, which we need in order t...

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...

We give an explicit affine algebraic variety whose coordinate ring is isomorphic (as an algebra with the action of the Weyl group) with the equivariant cohomology of some Springer fibers.

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...

Our basic datum is a list X := (a1, …, am) of nonzero elements in a real s-dimensional vector space V (we allow repetitions in the list, since this is important for the applications). Sometimes we take \(V = \mathbb{R}^{s}\) and then think of the ai as the columns of an \(s \times m\)matrix A. From X we shall construct several geometric, algebraic,...

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...

In this paper we discuss a class of normal forms of the completely resonant non--linear Schr\"odinger equation on a torus. We stress the geometric and combinatorial constructions arising from this study. Further analytic considerations and applications to quasi--periodic solutions will appear in a forthcoming article. This paper replaces a previous...

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...

We discuss normal forms of the completely resonant non-linear Schr\"odinger equation on a torus $\T^n$, with particular applications to quasi periodic solutions. Comment: 61, 3 figures

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 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...

Open letter to the Minister of Italian University. The government should get advice from the best Italian scientists for the necessary reform of the Italian University, with special concern to crucial processes as the appointment of new professors and academic evaluation. The current government proposal for a random draw of evaluation committees in...

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...

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.

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.

We discuss the proof given by M. Haiman of the Macdonald positivity conjecture obtained via the solution to the n!-conjecture of Garsia and Haiman. This is obtained from the following remarkable theorem: the Hilbert scheme of n-tuples of points in the plane is equal to the G-Hilbert scheme of Ito and Nakamura for the action of the symmetric group o...

We introduce and study certain quadratic Hopf algebras related to Schubert calculus of the flag manifold.

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...

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

. We introduce and study certain quadratic Hopf algebras related to Schubert calculus of the flag manifold. Introduction This short paper contributes to the study of a particular family of quadratic associative algebras En that were introduced and studied in [FK] in connection with their role in the Schubert calculus of the flag manifold. We introd...

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.

this paper by ring we mean an associative ring with 1

this paper we construct a "minimal wonderful" compactification ~

We show that the representations of certain automorphism groups of a free group afforded by compact Lie groups as described by Long can be decomposed into sums of trivial representations and Magnus–Gassner representations.