Artículos (Álgebra)

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

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Mostrando 1 - 20 de 260

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.
A generalisation of the Phase Kick-Back
(Springer, 2023-03-13) Ossorio Castillo, Joaquín; Pastor Díaz, Ulises; Tornero Sánchez, José María; Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII); Universidad de Sevilla. Departamento de Álgebra; Ministerio de Ciencia e Innovación (MICIN). España; Junta de Andalucía; European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER); 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 U f . 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.
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.
Limits of sequences of Pseudo-Anosov maps and of hyperbolic 3–manifolds
(MSP, 2021-08-11) Bonnot, Sylvain; Carvalho, André de; González Meneses López, Juan; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
There are two objects naturally associated with a braid β∈Bn of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism φβ:S2→S2; and the finite-volume complete hyperbolic structure on the 3–manifold Mβ obtained by excising the braid closure of β, together with its braid axis, from S3. We show the disconnect between these objects, by exhibiting a family of braids {βq:q∈Q∩(0,13]} with the properties that, on the one hand, there is a fixed homeomorphism φ0:S2→S2 to which the (suitably normalized) homeomorphisms φβq converge as q→0, while, on the other hand, there are infinitely many distinct hyperbolic 3–manifolds which arise as geometric limits of the form limk→∞Mβqk, for sequences qk→0.
Key polynomials for simple extensions of valued fields
(Worldwide Center of Mathematics, 2022-05-28) Herrera Govantes, Francisco Javier; Mahboub, W.; Olalla Acosta, Miguel Ángel; Universidad de Sevilla. Departamento de Álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
In this paper we present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaqui\'e and reminiscent of approximate roots of Abhyankar and Moh. Given a simple transcendental extension of valued fields, we associate to it a countable well-ordered set of polynomials called key polynomials. We define limit key polynomials and give explicit formulae for them. We give an explicit bound on the order type of the set of key polynomials.
Monomial discrete valuations in k[[X]]
(Cornell University, 2003-07-17) Herrera Govantes, Francisco Javier; Olalla Acosta, Miguel Ángel; Vicente Córdoba, José Luis; Universidad de Sevilla. Departamento de álgebra
Let v be a rank m discrete valuation of k[[X1,...,Xn]] with dimension n-m. We prove that there exists an inmediate extension L of K where the valuation is monomial. Therefore we compute explicitly the residue field of the valuation.
Tautological systems and free divisors
(Elsevier, 2019-08-20) Narváez Macarro, Luis; Sevenheck, Christian; Universidad de Sevilla. Departamento de álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
We introduce tautological systems defined by prehomogeneous actions of reductive algebraic groups. If the complement of the open orbit is a linear free divisor satisfying a certain finiteness condition, we show that these systems underly mixed Hodge modules. A dimensional reduction is considered and gives rise to one-dimensional differential systems generalizing the quantum differential equation of projective spaces.
Rings of differential operators as enveloping algebras of Hasse–Schmidt derivations
(Elsevier, 2020-01) Narváez Macarro, Luis; Universidad de Sevilla. Departamento de álgebra; Universidad de Sevilla. FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
Let k be a commutative ring and A a commutative k-algebra. In this paper we introduce the notion of enveloping algebra of Hasse–Schmidt derivations of A over k and we prove that, under suitable smoothness hypotheses, the canonical map from the above enveloping algebra to the ring of differential operators DA/k is an isomorphism. This result generalizes the characteristic 0 case in which the ring DA/k appears as the enveloping algebra of the Lie-Rinehart algebra of the usual k-derivations of A provided that A is smooth over k.

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