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Let R be a commutative ring and let n, m be two positive integers. Let A R (n, m) be the polynomial ring in the commuting independent variables x i (j) the ring of invariants for this action: its elements are usually called multisymmetric functions and they are the usual symmetric functions when m = 1. In this paper we give a presentation of A R (n, m) Sn… (More)

The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally defined quantities of nodes and edges, such as node degrees, edge weights and -more recently- correlations between… (More)

We show that the symmetric product of a flat affine scheme over a commutative ring can be embedded into the quotient by the general linear group of the scheme of commuting matrices. We also prove that the symmetric product of the affine space is isomorphic to the above quotient when the base ring is a characteristic zero field. Over an infinite field of… (More)

We show that the ring of multisymmetric functions over a commutative ring is isomorphic to the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices. As a consequence we give a surjection from the ring of invariants of several matrices to the ring of multisymmetric functions generalizing a classical… (More)

- F. Vaccarino, F. VACCARINO
- 2004

Let K be an infinite field and let R be a K-algebra endowed with a homogeneous polynomial norm N of degree n. We will show that R is a quotient of a ring of invariants of n × n matrices, if N satisfies a formal analogue of the Cayley-Hamilton Theorem. To achieve this we use recent results on invariants of matrices in positive characteristic, due to S.… (More)

It is well known that over an infinite field the ring of symmetric functions in a finite number of variables is isomorphic to the one of polynomial functions on a single matrix that are invariants by the action of conjugation by general linear group. We generalize this result showing that the abelianization of the algebra of the symmetric tensors of fixed… (More)

- Federica Galluzzi, Francesco Vaccarino
- 2008

Let k be a commutative ring and let R be a commutative k−algebra. The aim of this paper is to define and discuss some connection morphisms between schemes associated to the representation theory of a (non necessarily commutative) R−algebra A. We focus on the scheme Rep n A //GLn of the n−dimensional representations of A, on the Hilbert scheme Hilb n A… (More)

A multifiltration is a functor indexed by N r that maps any mor-phism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural N r-graded R[x 1 ,. .. , xr]-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and vector spaces. We prove in… (More)

Multipersistence homology modules were introduced by G.Carlsson and A.Zomorodian [1] which gave, together with G.Singh [4], an algorithm to compute their Gröbner bases. Although their algorithm has polynomial complexity when the chain modules are free, i.e. in the one-critical case, it might be exponential in general. We give a new presentation of… (More)