arXiv subjects

Carlos A. A. Florentino

Publications and source records attributed to Carlos A. A. Florentino.

4 recordsLinked to original sources

arXiv2021

Plethystic exponential calculus and characteristic polynomials of permutations

We prove a family of identities, expressing generating functions of powers of characteristic polynomials of permutations, as finite or infinite products. These generalize formulae first obtained in a study of the geometry/topology of symmetric products of real/algebraic tori. The proof uses formal power series expansions of plethystic exponentials, and has been motivated by some recent applications of these combinatorial tools in supersymmetric gauge and string theories. Since the methods are elementary, we tried to be self-contained, and relate to other topics such as the q-binoomial theorem, and the cycle index and Molien series for the symmetric group.

math.CO↗

arXiv2018

Daniel Augusto da Silva, Poet of Mathematics

Daniel da Silva was a remarkable Scientist and Mathematician of the mid 19th century. Working in Portugal, isolated from the main scientific centres of the time, his investigations in pure mathematics had almost no impact. Apart from giving a short biography of his life and work, this article makes the case for considering him an unavoidable character in the History of Science and one of the founders of Discrete Mathematics, through his introduction of a key method in Enumerative Combinatorics: the Principle of Inclusion-Exclusion.

math.HO↗

arXiv2008

Simultaneous similarity and triangularization of sets of 2 by 2 matrices

Let $\mathcal{A}=(A_{1},...,A_{n},...)$ be a finite or infinite sequence of $2\times2$ matrices with entries in an integral domain. We show that, except for a very special case, $\mathcal{A}$ is (simultaneously) triangularizable if and only if all pairs $(A_{j},A_{k})$ are triangularizable, for $1\leq j,k\leq\infty$. We also provide a simple numerical criterion for triangularization. Using constructive methods in invariant theory, we define a map (with the minimal number of invariants) that distinguishes simultaneous similarity classes for non-commutative sequences over a field of characteristic $\neq2$. We also describe canonical forms for sequences of $2\times2$ matrices over algebraically closed fields, and give a method for finding sequences with a given set of invariants.

math.RA↗

arXiv2006

Invariants of 2 by 2 matrices, irreducible SL(2,C) characters and the Magnus trace map

We obtain an explicit characterization of the stable points of the action of G=SL(2,C) on the cartesian product G^n by simultaneous conjugation on each factor, in terms of the corresponding invariant functions, and derive from it a simple criterion for irreducibility of representations of finitely generated groups into G. We also obtain analogous results for the action of SL(2,C) on the vector space of n-tuples of 2 by 2 complex matrices. For a free group F_n of rank n, we show how to generically reconstruct the 2^{n-2} conjugacy classes of representations F_n -> G from their values under the map T_n : G^n = Hom(F_n,G) -> C^{3n-3} considered in [M], defined by certain 3n-3 traces of words of length one and two.

math.GT↗