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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 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 $S_n$.

We discuss several constructions of swap polynomials, that is 2--tensor valued matrix polynomials which are multiples of the swap or switch operator., Comment: this is a replacement since there was an error in the previous version

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., Comment: this version improves the previous one

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 polynomials in $d^2$ variables., Comment: 14 pages, 1 figure, 3 tables

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, Comment: arXiv admin note: text overlap with arXiv:2008.02222

The aim of this note is to compare work of Formanek \cite{formanek2} on a certain construction of central polynomials with that of Collins \cite{Coll} 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 Weingarten function}. By joining these two approaches we succeed in giving a simplified and {\em very natural} presentation of both Formanek and Collins's Theory., Comment: this version corrects several mistakes and improves some of the presentation

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

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., Comment: 27 pages, minor corrections in the bibliography

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 must replace Formula 8.5 of the book with the next Formula (3). Accordingly one has to change Proposition 9.2 in the obvious way. The remaining parts of the book are not affected but one should remove the first line of 11.3.3 which quotes the incorrect formula. Here we discuss the correct proposition, replacing Proposition 8.5.

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 a KAM algorithm., Comment: 48 pages

We discuss some geometric invariants of polynomial identities of algebras deduced from Kemer's theory and deduce some quantitative information on codimension and co--length

We comment two papers of Domokos and Vaccarino proving that the restriction to diagonal matrices of the scheme of commuting matrices is an isomorphism when restricted to invariants to the symmetric group invariants., Comment: 4 pages

We study the subspace of the exterior algebra of a simple complex Lie algebra linearly spanned by the copies of the little adjoint representation or, in the case of the Lie algebra of traceless matrices, by the copies of the n-th symmetric power of the defining representation. As main result we prove that this subspace is a free module over the subalgebra of the exterior algebra generated by all primitive invariants except the one of highest degree., Comment: Latex file, 11 pages, Final version, appeared in "Rendiconti Lincei - Matematica e Applicazioni"

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 main result we prove that $A$ is a free module, of rank twice the rank of $\mathfrak g$, over the exterior algebra generated by all primitive invariants in $(\bigwedge \mathfrak g^*)^{\mathfrak g}$, with the exception of the one of highest degree., Comment: Final version. More misprints corrected. To appear in Advances in Mathematics

We present a proof of the Amitsur--Levitzki theorem which is a basis for a general theory of equivariant skew--symmetric maps on matrices., Comment: 3pages

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 the answer is affirmative in several special cases, and, moreover, for every such an identity $P$ and every central polynomial $c$ with zero constant term there exists $m\in\mathbb{N}$ such that the affirmative answer holds for $c^mP$. In general, however, the answer is negative. We prove that there exist antisymmetric identities that do not follow from the Cayley-Hamilton identity, and give a complete description of a certain family of such identities., Comment: Version 2: 24 pages. This paper is a replacement of the paper "Quasi-identities and the Cayley-Hamilton quasi-polynomial" by the first and the third author. The changes are substantial. Version 3: 27 pages. The title has been changed slightly, the exposition improved, references added, and some typos removed

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 to prove stability of certain solutions of the non--linear Schr\"odinger (NLS) equation on a torus.

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., Comment: 5 pages

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 improvements.

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., Comment: In this version, some misprints are corrected. An example have been changed of place

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 equivariant K-theory of the open subset of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the lattice of weights of G, satisfying the cocircuit difference equations associated to X, introduced by Dahmen--Micchelli in the context of the theory of splines in order to study vector partition functions. This allows us to determine the range of the index map from G-transversally elliptic operators on M to generalized functions on G and to prove that the index map is an isomorphism on the image. This is a setting studied by Atiyah-Singer which is in a sense universal for index computations.

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., Comment: 10pages