# Seminário de Análise Funcional, Estruturas Lineares e Aplicações

## Sessões anteriores

### 07/12/2018, 15:45 — 16:45 — Sala 6.2.33, Faculdade de Ciências da Universidade de Lisboa

Jocelyn Lochon, *Universidade de Lisboa*

### The asymptotic behaviour of the Super-Plancherel measure

Let $\mathbb{F}_q$ be a finite field and denote $U_n(\mathbb{F}_q)$ the group of $n \times n$ uppertriangular matrices over $\mathbb{F}_q$ with only ones in the diagonal. In recent years the representation theory of $U_n(\mathbb{F}_q)$ has been approached *via* certain Supercharacter-Theories, not only due to their (non-commutative) combinatorial-analogues to the representation theory of the symmetric group $S_n$, but also as a useful tool to address Harmonic-analysis problems.

We consider a particular Supercharacter-Theory for $U_n(\mathbb{F}_q)$ which yields a natural measure on the set-partitions of $\{1,...,n \}$: the Super-plancherel measure $\textbf{SPl}_n$. The aim of this talk is to understand the asymptotic behaviour of $\textbf{SPl}_n$ as $n \rightarrow \infty$; in particular limit objects are interpreted in a *representation-*theoretical setting.

### 06/07/2018, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática

Miguel Tierz, *Faculdade de Ciências ULisboa*

### Minors of Toeplitz matrices, symmetric functions and random matrix ensembles

We study minors of Toeplitz matrices, using the theory of symmetric functions. The symbols of the Toeplitz matrices can either be in the Szegö class or have Fisher-Hartwig singularities and the results apply both to minors of finite and of large dimension. As an application, also using results on inverses of Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skew Schur polynomials. We also discuss the intimate relationship with random matrix theory and the ensuing applications.

### 25/05/2018, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática

Raquel Simões, *Universidade de Lisboa, CEAFEL*

### A geometric model for the module category of a gentle algebra

One of the problems in representation theory is to understand the structure of the module category of an algebra. Gentle algebras are a class of algebras of tame representation type, meaning it is often possible to get a global understanding of their representation theory.

In this talk, we will show how to encode the module category of a gentle algebra using combinatorics of a surface. This is joint work with Karin Baur (Graz).

### 18/05/2018, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática

Olga Azenhas, *Universidade de Coimbra, CMUC*

### Involutions for symmetries of Littlewood-Richardson (LR) coefficients

The action of the dihedral group $\mathbb Z_2\times \mathfrak{ S}_3$ on LR coefficients is considered with the action of $\mathbb Z_2$ realized by the transposition of a partition. The action of $\mathbb Z_2\times\mathfrak{ S}_3$ carries a linear time action of a subgroup $H$ of index two, and a bijection which goes from $H$ into the other coset is difficult. LR coefficients are preserved in linear time by the action of $H$ whereas the other half symmetries consisting of commutativity and transposition symmetries are hidden. The latter are given by a remaining generator, the reversal involution or Schutzenberger involution, by which one is able to reduce in linear time the LR commuters and LR transposers to each other.

To pass from symmetries of LR (skew) tableaux to symmetries of companion Gelfand-Tsetlin (GT) patterns we build on the crystal action of the longest permutation in the symmetric group on an LR tableau, and Lascoux's double crystal graph structure on biwords. This analysis also affords an explicit bijection between two of the interlocking GT patterns in a hive.

This is based on a joint work with A. Conflitti and R. Mamede.

### 04/05/2018, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática

Mario Moctezuma Salazar, *Instituto Politécnico Nacional, México.*

### Toeplitz matrices and Schur polynomials

Toeplitz matrices are ubiquitous and enjoy attractive computational properties, for which its study has captured the interest of many mathematicians in the last century.

In the first part of this talk, we review some necessary properties of Schur polynomials. We start a well known tool, Vieta’s formulas, from which naturally arise the elementary symmetric polynomials, then we study other families of symmetric polynomials: complete homogeneous symmetric polynomials, Schur polynomials, and skew Schur polynomials. We also recall a formula that expresses the product of Schur polynomials in terms of skew Schur polynomials.

In the second part, we give relations between Schur polynomials and Toeplitz banded matrices, for instance, we express determinants and minors of Toeplitz matrices through Schur polynomials.

The results presented here have been obtained jointly with Egor Maximenko; they are based on some ideas by William F. Trench and Per Alexandersson.

The speaker has been partially supported by IPN-SIP projects.

### 27/04/2018, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática

Pedro Freitas, *Instituto Superior Técnico, Universidade de Lisboa*

### The spectral determinant of the quantum harmonic oscillator in arbitrary dimensions

We show that the spectral determinant of the isotropic quantum harmonic oscillator converges exponentially to one as the space dimension grows to infinity. We determine the precise asymptotic behaviour for large dimension and obtain estimates valid for all cases with the same asymptotic behaviour in the large.

As a consequence, we provide an alternative proof of a conjecture posed by Bar and Schopka concerning the convergence of the determinant of the Dirac operator on $S^{n}$, determining the exact asymptotic behaviour for this case and thus improving the estimate on the rate of convergence given in the proof by Moller.

The seminar is joint with Geometria em Lisboa seminar.

### 02/03/2018, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática

Albrecht Böttcher, *Technische Universität Chemnitz, Germany*

### Four concrete applications of Toeplitz operators

I present four concrete and very different applications of Toeplitz operators.

These applications are (1) a problem in optimal ell-one control, (2) Wiener-Hopf and spectral factorization of polynomials of degree 20000, (3) computation of the volume of the fundamental domains of some high-dimensional lattices, and (4) the determination of the Hausdorff limit of the zero set of polynomials of the Fibonacci type. The talk allows you to switch off four times and to re-enter the same number of times.

### 01/02/2018, 16:00 — 17:00 — Sala 6.2.33, Faculdade de Ciências da Universidade de Lisboa

Inês Rodrigues, *Faculdade de Ciências, Universidade de Lisboa*

### Robinson-Schensted and RSK correspondences for Skew and Skew Shifted Tableaux

The Robinson-Schensted correspondence, introduced by Schensted (1961) in its most well-known form, presents a bijective correspondence between permutations and pairs of standard Young tableaux of the same shape. Knuth (1970) presented a generalization, the RSK correspondence, for semistandard Young tableaux.

Asthe Young tableaux are deeply tied to the study of linear representations of the symmetric group, there is a variant for the projective representations, the shifted tableaux, based on strict partitions. The shifted tableaux also arise in the study of $Q$-functions, an important basis for the subalgebra of the symmetric functions generated by the odd power sums, which resembles the classic Schur functions.

In this seminar, we will show generalizations of both Robinson-Schensted and RSK correspondences for skew tableaux and analogues for shifted skew tableaux (Sagan, Stanley, 1990), based on variants of the insertion algorithm. We will also present a brief introduction on some aspects of the theory of shifted tableaux.

### 05/01/2018, 15:30 — 16:30 — Sala P3.10, Pavilhão de Matemática

Yuri Karlovich, *Universidad Autónoma del Estado de Morelos, México*

### One-sided invertibility of discrete functional operators with bounded coefficients

The one sided invertibility of discrete functional operators with bounded coefficients on the spaces $l^p(\mathbb{Z})$ with $p\in[1,\infty]$ is studied. Criteria of the one sided invertibility of such operators generalize those obtained in the case of slowly oscillating behavior of coefficients. Criteria of the one-sided invertibility of discrete functional operators associated with infinite slant-dominated matrices are established. Applications to studying the two- and one-sided invertibility of functional operators on Lebesgue spaces are also considered.

### 06/10/2017, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática

Bernd Silbermann, *Technische Universität Chemnitz, Germany*

### On the spectrum of the Hilbert matrix operator

For each, $\lambda\in\mathbb{C}$, $\lambda\neq 0,-1,-2,...$ the (generalized) Hilbert matrix $\mathcal{H}_{\lambda}$ is given by $$\mathcal{H}_{\lambda}:=\left(\frac{1}{n+m+\lambda}\right)_{n,m\geq0}.$$ If $\lambda=1$ then $\mathcal{H}_{\lambda}$ is the classical Hilbert matrix introduced by D. Hilbert about 125 years ago. These matrices have been the subject of numerous investigations. The talk mainly concerns the description of spectral properties of Hankel operators generated by these matrices on the Hardy spaces $H^{p}$ and $l^{p}$ $(1 < p < \infty$). Special attention will be paid to the description of the essential and point spectra of these operators.

### 29/09/2017, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática

Alexey Karapetyants, *Southern Federal University, Rostov-na-Donu, Russia*

### On a simple proof of the boundedness of Bergman projection in various Banach spaces and some related questions

We give a simple proof of the boundedness of Bergman projection in various Banach spaces of functions on the unit disc in the complex plain. The approach of the paper is based on the idea of V. P. Zaharyuta, V. I. Yudovich (1962) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderon-Zygmund operators. We exploit this approach and treat the cases of variable exponent Lebesgue space, Orlicz space and variable exponent generalized Morrey space. In the case of variable exponent Lebesgue space the boundedness results is known, so in that case we provide a simpler proof. The other two cases are new. The major idea of this paper is to show that the approach can be applied to a wide range of function spaces. This opens a door in a sense for introducing and studying new function spaces of Bergman type in complex analysis.

We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollifying dilations.

This is a joint work with Stefan Samko and Humberto Rafeiro.

### 28/07/2017, 16:00 — 17:00 — Sala 6.2.33, Faculdade de Ciências da Universidade de Lisboa

Ali Mohammadian, *Institute for Research in Fundamental Sciences, Tehran, Iran*

### Integral Graphs

A graph $G$ is called integral if all eigenvalues of its adjacency matrix, $A(G)$, consist entirely of integers. The nullity of $G$ is the nullity of $A(G)$, that is the multiplicity of $0$ as an eigenvalue of $A(G)$. In this talk, we are concerned with integral trees. These objects are extremely rare and very difficult to find. We first present a short survey on integral graphs. We show that for any integer $d \gt 1$, there are infinitely many integral trees of diameter $d$. We will also show that for any integer $k \gt 1$, there are only finitely many integral trees with nullity $k$.

### 14/07/2017, 15:00 — 16:00 — Sala P4.35, Pavilhão de Matemática

Yuri Karlovich, *Universidad Autónoma del Estado de Morelos, Cuernavaca, México*

### Algebras of convolution type operators with PSO data

The Fredholm symbol calculus is constructed for the Banach algebras generated on the weighted Lebesgue spaces with slowly oscillating Muckenhoupt weights by all multiplication operators by piecewise slowly oscillating functions and by all convolution operators with piecewise slowly oscillating presymbols being Fourier multipliers. A new approach of identifying local spectra is presented, which allows one to complete the description of the Fredholm symbol calculus.

### 30/06/2017, 15:00 — 16:00 — Sala 6.2.38, Faculdade de Ciências da Universidade de Lisboa

João Gouveia, *Universidade de Coimbra*

### Polytopes, slack ideals and psd-minimality

The slack ideal is an algebraic object that codifies the geometry of a polytope. This notion was motivated by the study of psd-minimality of polytopes: A $d$-polytope is said to be psd-minimal if it can be written as a projection of a slice of the cone of $d+1$ by $d+1$ positive semidefinite matrices, the smallest possible size for which this may happen. We will show how the slack ideal can be used to extract conditions on psd-minimality, completing the classification of psd-minimal $4$- polytopes, settling some open questions and creating new ones. We will proceed to explore the relation of slack ideals and toric ideals of graphs and present some ongoing work and open questions.

### 05/06/2017, 15:00 — 16:00 — Sala 6.2.38, Faculdade de Ciências da Universidade de Lisboa

Alicia Roca, *Universitat Politècnica de València*

### Lattices of invariant subspaces

Given $\mathbb{F}$ an arbitrary field and $A\in M_{n}(\mathbb{F})$, the set of $A$-invariant subspaces of $\mathbb{F}^{n}$ is a lattice with inclusion as order, intersection as meet and linear sum as join. We denote this lattice by $\operatorname{Inv}(A)$.

An $A$-invariant subspace $V\subseteq \mathbb{F}^{n}$ is $A$-hyperinvariant ($A$-characteristitc) if it is invariant for every matrix $T\in Z(A)$, i.e. commuting with $A$ ($T\in Z^*(A)$, i.e. commuting with $A$ and $T$ non singular). It is straightforward to see that the set of $A$-hyperinvariant ($A$-characteristic) subspaces is a sublattice of $\operatorname{Inv}(A)$. We denote this sublattice by $\operatorname{Hinv}(A)$ ($\operatorname{Chinv}(A)$). Obviously, \[\operatorname{Hinv}(A)\subseteq \operatorname{Chinv}(A)\subseteq \operatorname{Inv}(A).\] If the characteristic polynomial of $A$ splits over $\mathbb{F}$, the study of these lattices can be reduced to the nilpotent case. Let $J\in M_{n}(\mathbb{F})$ be a nilpotent Jordan matrix.

In this talk we recall the general properties of $\operatorname{Inv}(J)$ and analyze which of those properties are preserved in the sublattice $\operatorname{Hinv}(J)$ and, if $\mathbb{F}=GF(2)$, in $\operatorname{Chinv}(A)$ (the only case where $\operatorname{Hinv}(J)\neq \operatorname{Chinv}(J)$).

In addition we analyze the cardinality of $\operatorname{Hinv}(J)$ and $\operatorname{Chinv}(J)$.

Joint work with David Mingueza, M. Eulàlia Montoro.

### 02/06/2017, 15:00 — 16:00 — Sala 6.2.38, Faculdade de Ciências da Universidade de Lisboa

Ricardo Mamede, *Universidade de Coimbra*

### Gray codes for the hyperoctahedral group

A (cyclic) $n$-bit Gray code is a (cyclic) ordering of all $2^n$ binary words of length $n$ such that consecutive words differ in a single bit. Alternatively, an $n$-bit Gray code can be viewed as a Hamiltonian path of the $n$- dimensional hypercube $Q_n$, and a cyclic Gray code as a Hamiltonian cycle of $Q_n$. This idea has been generalized as follows. A Gray code for any combinatorial family of objects is a listing of all objects in that family such that successive objects differ in some prescribed, usually “small", way. The definition of “small" depends on the particular family, its context, and its applications. In this talk, we construct Gray codes for the finite reflection groups, with a particular focus on signed permutations and some of its restrictions: signed involutions and signed involutions without fixed points.

### 19/05/2017, 15:00 — 16:00 — Sala 6.2.38, Faculdade de Ciências da Universidade de Lisboa

Deividi Pansera, *Universidade do Porto*

### Semisimple Hopf actions and factorization through group actions

Let $H$ be a Hopf algebra over a field $F$ acting on an algebra $A$. Let $I \subseteq \operatorname{Ann}_H(A)$ be a Hopf ideal of $H$, then one says that the action of $H$ on $A$ \textit{factors through} the quotient Hopf algebra $H/I$. If there exists $I \subseteq \operatorname{Ann}_H(A)$ such that $H/I \cong F[G]$, for some group $G$, we say that the action of $H$ on $A$ *factors through a group action*. In 2014, Etingof and Walton have shown that any semisimple Hopf action on a commutative domain factors through a group action [EtingofWalton]. Also in 2014, using their previous result, Cuadra, Etingof and Walton showed that any action of a semisimple Hopf algebra $H$ on the $n$th Weyl algebra $A=A_n(F)$, with $\operatorname{char}(F) = 0$, factors through a group action [CuadraEtingofWalton].

In this talk we will briefly present a generalization of Cuadra, Etingof and Walton's result. Namely, that any action of a semisimple Hopf algebra $H$ on an iterated Ore extension of derivation type in characteristic zero factors through a group action [LompPansera]. We also present a work in progress on semisimple Hopf algebra actions on the quantum polynomial algebras which do not factor through a group actions.

This talk is all based on my upcoming Ph.D. Thesis under the supervision of Christian Lomp.

#### References

[EtingofWalton] Etingof, P., Walton, C. (2014). Semisimple Hopf actions on commutative domains. *Adv. Math.* 251: 47–61. doi:10.1016/j.aim.2013.10.008.

[CuadraEtingofWalton] Cuadra, J., Etingof, P., Walton, C. (2015). Semisimple Hopf actions on Weyl algebras *Adv. Math.* 282:47–55. doi:10.1016/j.aim.2015.05.014.

[LompPansera] Lomp, C., Pansera, D. A note on a paper by Cuadra, Etingof and Walton. *Communications in Algebra* 1532-4125. doi:10.1080/00927872.2016.1236933

### 05/05/2017, 15:00 — 16:00 — Sala 6.2.38, Faculdade de Ciências da Universidade de Lisboa

Tara Brough, *Centro de Matemática e Aplicações, UNL*

### Word problems of free inverse monoids

In semigroups, $y$ is an inverse of $x$ if $xyx = x$ and $yxy = y$. An inverse monoid is a monoid (semigroup with identity) in which every element has a unique inverse. I will describe the free objects in the category of inverse monoids: for any set $X$, the free inverse monoid on $X$ is denoted $\operatorname{FIM}(X)$. The word problem of a monoid is, informally, the problem of deciding whether two words over a given generating set represent the same element of the monoid. I will explain how this can be considered as a formal language, and discuss the language type (e.g. context-free, context-sensitive) of the word problem of $\operatorname{FIM}(X)$ for a finite set $X$.

The talk will focus primarily on the rank 1 case, in which words over the standard generating set can be viewed as walks in one dimension.

### 21/04/2017, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática

Teresa Sousa, *Escola Naval*

### Monochromatic edge decompositions of graphs

Let $H=(H_1, ..., H_k)$ be a fixed $k$-tuple of graphs and let $G$ be a graph on $n$ vertices whose edges are colored with $k$ colors. A monochromatic $H$-decomposition of $G$ is a partition of its edge set such that each part is either a single edge or a copy of $H_i$ monochromatic in color $i$. The aim is to find the smallest number, denoted by $f(n,H,k)$, such that, any $k$-edge colored graph with $n$ vertices admits a monochromatic $H$-decomposition with at most $f(n,H,k)$ parts. We will consider the problem when $H$ is a fixed $k$-tuple of cliques (a clique is a complete graph) for all values of $k$. The results presented will involve both the Turán numbers and the Ramsey numbers. This is joint work with Henry Liu and Oleg Pikhurko.

### 24/03/2017, 15:00 — 16:00 — Sala 6.1.27, Faculdade de Ciências da Universidade de Lisboa

Raquel Simões, *Faculdade de Ciências, Universidade de Lisboa*

### Negative Calabi-Yau triangulated categories

Calabi-Yau (CY) triangulated categories are those satisfying a useful and important duality, characterised by a number called the CY dimension. Much work has been carried out on understanding positive CY triangulated categories, especially in the context of cluster-tilting theory. Even though CY dimension is usually considered to be a positive (or fractional) number, there are natural examples of CY triangulated categories where this “dimension” or parameter is negative, for example, stable module categories of self-injective algebras. Therefore, negative CY triangulated categories constitute a class of categories that warrant further systematic study. In this talk, we will consider an important class of generating objects of negative CY triangulated categories, namely simple-minded systems and study their mutation behaviour. We will focus on an example given by triangulated categories generated by spherical objects, whose combinatorics plays a useful role in the study of its representation theory.

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