Artículos (Álgebra)

URI permanente para esta colecciónhttps://hdl.handle.net/11441/10804

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Determinant of the distance matrix of a tree
(University of Vienna, 2024) Briand, Emmanuel; Esquivias Quintero, Luis; Gutierrez, Alvaro; Lillo, Adrian; Rosas Celis, Mercedes Helena; Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII); Universidad de Sevilla. Departamento de Álgebra; MICIU/AEI/10.13039/501100011033; Junta de Andalucía, FEDER, PAIDI2020
We present a combinatorial proof of the Graham–Pollak formula for the determinant of the distance matrix of a tree, via sign-reversing involutions and the Lindström–Gessel–Viennot lemma.
Explicit models for perverse sheaves, II
(Springer, 2008) Gudiel Rodríguez, Félix; Narváez Macarro, Luis; Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII); Universidad de Sevilla. Departamento de Álgebra
In this paper we show that any p-perverse sheaf on an arbitrary stratified topological space ( p is a perversity function) is functorially determined by a system of usual sheaves on the open sets Ur (r ≥ 0) and certain gluing data, where Ur is the union of strata of perversity ≤ r.
On ν-quasiordinary surface singularities and their resolution
(Springer, 2024-08-05) Aroca, F.; Tornero Sánchez, José María; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
Quasiordinary power series were introduced by Jung at the beginning of the 20th century, and were not paid much attention until the work of Lipman and, later on, Gao. They have been thoroughly studied since, as they form a very interesting family of singular varieties, whose properties (or at least many of them) can be encoded in a discrete set of integers, much as what happens with curves. Hironaka proposed a generalization of this concept, the so-called ν-quasiordinary power series, which has not been examined in the literature in such detailed way. This paper explores the behavior of these series under the resolution process in the surface case.
A generalisation of the Phase Kick-Back
(Springer, 2023-03-13) Ossorio Castillo, J.; Pastor Díaz, Ulises; Tornero Sánchez, José María; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
In this paper, we present a generalisation of the Phase Kick-Back technique, which is central to some of the classical algorithms in quantum computing. We will begin by recalling the Phase Kick-Back technique to then introduce the new generalised version for f:{0,1}n→{0,1}m functions using the eigenvalues of the oracle function Uf. After that, we will present a new generalised version of the Deutsch–Jozsa problem and how it can be solved using the previously defined technique. We will also deal with a generalised version of the Bernstein–Vazirani problem and solve it using the generalised Phase Kick-Back. Finally, we show how we can use this technique to obtain an algorithm for Simon’s problem that improves the classical one.
Combinatorics and their evolution in resolution of embedded algebroid surfaces
(International Press, 2019-05-30) Cobo Pablos, M. Helena; Soto Prieto, Manuel Jesús; Tornero Sánchez, José María; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
The seminal concept of characteristic polygon of an embedded algebroid surface, first developed by Hironaka, seems well suited for combinatorially (perhaps even effectively) tracking of a resolution process. However, the way this object evolves through the resolution of singularities is not really well understood, as some references had pointed out. The aim of this paper is to study how this object changes as the surface gets resolved. In order to get a precise description of the phenomena involved, we need to use different techniques and ideas. Eventually, some effective results regarding the number of blow-ups needed to decrease the multiplicity are obtained as a side product.
Blurred combinatorics in resolution of singularities: (A little) Beyond the characteristic polytope
(Duke University Press, 2017-10-04) Cobo Pablos, M. Helena; Soto Prieto, Manuel Jesús; Tornero Sánchez, José María; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
We introduce a variation of the well-known Newton–Hironaka polytope for algebroid hypersurfaces. This combinatorial object is a perturbed version of the original one, parameterized by a real number ε∈R≥0ε∈R≥0. For well-chosen values of the parameter, the objects obtained are very close to the original, while at the same time presenting more (hopefully interesting) information in a way that does not depend on the choice of parameter.
On the reduced Bernstein-Sato polynomial of Thom-Sebastiani singularities
(Springer Nature, 2024-09-10) Castaño Domínguez, Alberto; Narváez Macarro, Luis; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
Given two holomorphic functions f and g defined in two respective germs of complex analytic manifolds (X, x) and (Y, y), we know thanks to M. Saito that, as long as one of them is Euler homogeneous, the reduced (or microlocal) Bernstein-Sato polynomial of the Thom-Sebastiani sum f+g can be expressed in terms of those of f and g. In this note we give a purely algebraic proof of a similar relation between the whole functional equations that can be applied to any setting (not necessarily analytic) in which Bernstein-Sato polynomials can be defined.
All linear symmetries of the SU(3) tensor multiplicities
(IOP Science, 2024) Briand, Emmanuel; Rosas Celis, Mercedes Helena; Trandafir, Stefan; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM333: Algebra Computacional en Anillos no Conmutativos y Aplicaciones
The SU(3) tensor multiplicities are piecewise polynomial of degree 1 in their labels. The pieces are the chambers of a complex of cones. We describe in detail this chamber complex and determine the group of all linear symmetries (of order 144) for these tensor multiplicities. We represent the cells by diagrams showing clearly the inclusions as well as the actions of the group of symmetries and of its remarkable subgroups.
Vector partition functions and Kronecker coefficients
(IOP Science, 2021-04-01) Mishna, Marni; Rosas Celis, Mercedes Helena; Sundaram, Sheila; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM333: Algebra Computacional en Anillos no Conmutativos y Aplicaciones
The Kronecker coefficients are the structure constants for the restriction of irreducible representations of the general linear group GL(nm) into irreducibles for the subgroup GL(n) × GL(m). In this work we study the quasipolynomial nature of the Kronecker function using elementary tools from polyhedral geometry.We write the Kronecker function in terms of coefficients of a vector partition function. This allows us to define a new family of coefficients, the atomic Kronecker coefficients. Our derivation is explicit and self-contained, and gives a new exact formula and an upper bound for the Kronecker coefficients in the first nontrivial case.
Partial symmetries of iterated plethysms
(Springer, 2023-05-03) Gutiérrez Cáceres, Álvaro; Rosas Celis, Mercedes Helena; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM333: Algebra Computacional en Anillos no Conmutativos y Aplicaciones
This work highlights the existence of partial symmetries in large families of iterated plethystic coefficients. The plethystic coefficients involved come from the expansion in the Schur basis of iterated plethysms of Schur functions indexed by one-row partitions.The partial symmetries are described in terms of an involution on partitions, the flip involution, that generalizes the ubiquitous w involution. Schur-positive symmetric functions possessing this partial symmetry are termed flip-symmetric. The operation of taking plethysm with 8λ preserves flip-symmetry, provided that λ is a partition of two. Explicit formulas for the iterated plethysms 82 o 8b and 8c o 82 o 8a, with a, b, and c ≥ 2 allow us to show that these two families of iterated plethysms are flip-symmetric. The article concludes with some observations, remarks, and open questions on the unimodality and asymptotic normality of certain flip-symmetric sequences of iterated plethystic coefficients.
Necessary conditions for the positivity of Littlewood–Richardson and plethystic coefficients
(Académie des Sciences, 2023-01-26) Gutiérrez Cáceres, Álvaro; Rosas Celis, Mercedes Helena; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM333: Álgebra computacional en anillos no conmutativos y aplicaciones.
We give necessary conditions for the positivity of Littlewood–Richardson coefficients and SXP coefficients. We deduce necessary conditions for the positivity of the plethystic coefficients. Explicitly, our main result states that if Sλ(V ) appears as a summand in the decomposition into irreducibles of Sμ(Sν(V )), then ν’s diagram is contained in λ’s diagram.
Rectangular symmetries for coe cients of symmetric functions
(The Electronic Journal of Combinatorics, 2015-07-31) Briand, Emmanuel; Orellana, Rosa; Rosas Celis, Mercedes Helena; Universidad de Sevilla. Departamento de Matemática Aplicada I; Universidad de Sevilla. FQM333: Álgebra computacional en anillos no conmutativos y aplicaciones.
We show that some of the main structural constants for symmetric functions (Littlewood-Richardson coe cients, Kronecker coe cients, plethysm coe cients, and the Kostka{Foulkes polynomials) share symmetries related to the operations of taking complements with respect to rectangles and adding rectangles.
A comment of the combinatorics of the vertex operator Γ(t|X)
(Project euclid, 2019-12-08) Rosas Celis, Mercedes Helena; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM333: Algebra Computacional en Anillos no Conmutativos y Aplicaciones
The Jacobi--Trudi identity associates a symmetric function to any integer sequence. Let Γ(t|X) be the vertex operator defined by Γ(t|X)sα=∑n∈Zs(n,α)[X]tn. We provide a combinatorial proof for the identity Γ(t|X)sα=σ[tX]sα[x−1/t] due to Thibon et al. We include an overview of all the combinatorial ideas behind this beautiful identity, including a combinatorial description for the expansion of s(n,α)[X] in the Schur basis, for any integer value of n.
Combinatorics on a family of reduced Kronecker coefficients
(Académie des Sciences, 2015-08-28) Colmenarejo Hernando, Laura; Rosas Celis, Mercedes Helena; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM333: Álgebra computacional en anillos no conmutativos y aplicaciones.
The reduced Kronecker coefficients are particular instances of Kronecker coefficients, that nevertheless contain enough information to compute all Kronecker coefficients from them. In this note, we compute the generating function of a family of reduced Kronecker coefficients. We show that these reduced Kronecker coefficients count plane partitions. This allows us to check that these coefficients satisfy the saturation conjecture, and that they are weakly increasing. Thanks to its generating function, we can describe our family by a quasipolynomial, specifying its degree and period.
Schur generating functions and the asymptotics of structural constants from combinatorial representation theory
(2020-02-29) Rosas Celis, Mercedes Helena; Briand, Emmanuel; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM333: Algebra Computacional en Anillos no Conmutativos y Aplicaciones
Extreme Khovanov spectra
(EUROPEAN MATHEMATICAL SOC, 2019-09-28) Cantero Morán, Federico; Silvero Casanova, Marithania; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
We prove that the spectrum constructed by González-Meneses, Manchón and the second author is stably homotopy equivalent to the Khovanov spectrum of Lipshitz and Sarkar at its extreme quantum grading.
The conjugacy stability problem for parabolic subgroups in artin groups
(Springer, 2022-09-14) Cumplido Cabello, María; Universidad de Sevilla. Departamento de Álgebra
Given an Artin group A and a parabolic subgroup P, we study if every two elements of P that are conjugate in A, are also conjugate in P. We provide an algorithm to solve this decision problem if A satisfies three properties that are conjectured to be true for every Artin group. This allows to solve the problem for new families of Artin groups. We also partially solve the problem if A has FC-type, and we totally solve it if A is isomorphic to a free product of Artin groups of spherical type. In particular, we show that in this latter case, every element of A is contained in a unique minimal (by inclusion) parabolic subgroup.
Commensurability in Artin groups of spherical type
(EMS Press, 2021-06-29) Cumplido Cabello, María; Paris, Luis; Universidad de Sevilla. Departamento de Álgebra
We give an almost complete classification of Artin groups of spherical type up to commensurability. Let A and A0 be two Artin groups of spherical type, and let A1; : : : ; Ap (respectively, A0 1; : : : ; A0 q) be the irreducible components of A (respectively, A0). We show that A and A0 are commensurable if and only if p D q and, up to permutation of the indices, Ai and A0 i are commensurable for every i . We prove that, if two Artin groups of spherical type are commensurable, then they have the same rank. For a fixed n, we give a complete classification of the irreducible Artin groups of rank n that are commensurable with the group of type An. Note that there are six remaining comparisons of pairs of groups to get the complete classification of Artin groups of spherical type up to commensurability, two of which have been done by Ignat Soroko after the first version of the present paper.
Parabolic subgroups of large-type Artin groups
(Cambridge University Press, 2022-09-12) Cumplido Cabello, María; Martin, Alexandre; Vaskou, Nicolás; Universidad de Sevilla. Departamento de Álgebra
We show that the geometric realisation of the poset of proper parabolic subgroups of a large-type Artin group has a systolic geometry. We use this geometry to show that the set of parabolic subgroups of a large-type Artin group is stable under arbitrary intersections and forms a lattice for the inclusion. As an application, we show that parabolic subgroups of large-type Artin groups are stable under taking roots and we completely characterise the parabolic subgroups that are conjugacy stable. We also use this geometric perspective to recover and unify results describing the normalisers of parabolic subgroups of large-type Artin groups.
An Effective Criterion for Finite Monodromy of ℓ-Adic Sheaves
(Springer, 2023-02-01) Rojas León, Antonio; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
We provide an effective version of Katz’ criterion for finiteness of the monodromy group of a lisse, pure of weight zero, -adic sheaf on a normal variety over a finite field, depending on the numerical complexity of the sheaf.

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