# Seminars Academic Year 2018-2019

- 14 September 2018

Cinzia Bisi (Università Degli Studi Di Ferrera)

The harmonicity of slice regular functions - 15 February 2019

Amedeo Altavilla (University of Rome Tor Vergata)

Implementing zonal harmonics with Fueter theorem - 22 February 2019

Sebbe Stouten (Master student Ghent University)

Kronecker coefficients for symmetric group characters - 1 March 2019

David Eelbode (University of Antwerp)

The Cayley-Dirac operator on oriented Grassmannians - 1 March 2019

Güner Muarem (Master student University of Antwerp)

Symmetries of differential operators and the Fueter theorem - 8 March 2019

Srđan Lazendić (Ghent University)

The Finite Heisenberg-Weyl Groups in Image Processing - 15 March 2019

Sigiswald Barbier (Ghent University)

The blocks of the periplectic Brauer algebra in positive characteristic - 29 March 2019

Brecht Denis (Ghent University)

Shadowing effect of multi-particle catalyst systems in TAP pulse experiments - 5 April 2019

Hadewijch De Clercq (Master student Ghent University)

The higher rank q-Bannai-Ito algebra and multivariate (-q)-Racah polynomials - 5 April 2019

Asmus Bisbo (Ghent University)

Polynomial representations of osp(1|2n) - 26 April 2019

Sam Claerebout (Ghent University)

Segal-Bargmann transformations on superspaces - 3 May 2019

Astrid Massé (Ghent University)

Discrete heat equations - 10 May 2019

Nicolas Vercheval

Modelling point cloud data: an approach from topology - 17 May 2019

Wouter Van de Vijver

Racah problems for the oscillator algebra and sl(n) - 24 May 2019

Ren Hu

Bargmann-Radon transform for axially monogenic functions - 7 June 2019

Alexis Langlois Rémillard

Representation theory and cellular structure of seam algebras - 14 June 2019

Teppo Mertens

Radon-type transforms for holomorphic functions in the Lie ball

## The harmonicity of slice regular functions (Cinzia Bisi, Università Degli Studi di Ferrera)

*Abstract:*

I will start improving the definition of slice regular function over the quaternions given by Gentili-Struppa in 2006-2007, and of monogenic function over a real Clifford algebra introduced by Colombo-Sabadini-Struppa in 2009. Then, bringing new ideas to the theory, I will answer positively to the question: is a slice regular function over the quaternions (analogously to a holomorphic function over the complex) ”harmonic” in some sense, i.e. is it in the kernel of some order-two differential operator over the quaternions? Finally, I will deduce novel integral formulas as applications. This is part of a project with J. Winkelmann.

## Implementing zonal harmonics with Fueter theorem (Amedeo Altavilla, University of Rome Tor Vergata)

*Abstract:*

In this talk I will give an overview of my research in collaboration with Hendrik De Bie and Michael Wutzig. The results of this research are some new implementations of zonal harmonic functions. The main techniques come from a suitable adaptation of the Fueter theorem in this particular context. At the end of the talk I will try to state some open problem emerged from our work.

## Kronecker coefficients for symmetric group characters (Sebbe Stouten, Master student Ghent University)

*Abstract:*

When studying the ring of symmetric functions, Schur functions show up as a useful Z-basis. We can define the Littlewood-Richardson coefficients by multiplying two Schur functions. However, we can also use the connection between Schur functions and characters of the symmetric group to define another multiplication on Schur functions, given by the Kronecker coefficients. We will discuss coproducts using both kinds of coefficients, and will end with a generating function for the Kronecker coefficients.

## The Cayley-Dirac operator on oriented Grassmannians (David Eelbode, University of Antwerp)

*Abstract:*

In his PhD thesis, Tim Janssens has studied a differential operator which we have dubbed the Cayley-Dirac operator, for the simple reason that it factorises the so-called Cayley-Laplace operator (which has been studied in the past). Much to our surprise, these operators nicely generalise some properties from the classical case (harmonic/Clifford analysis in one vector variable) to a new setting, in which one essentially considers a suitable oriented Grassmannian instead of a sphere as the underlying 'basic manifold'. During this talk, we would like to highlight a few of these facts: we will start from the Howe duality for a certain function space, and see how far we can go with that.

This is joint work with Yasushi Homma and Tim Janssens.

## Symmetries of differential operators and the Fueter theorem (Güner Muarem, University of Antwerp)

*Abstract:*

It is well-known that the quaternions discovered by Hamilton in 1843 form a number system that extends the complex numbers. An obvious first reflex in that time was to mimic the techniques and results from complex analysis in order to obtain a similar function theory. The first step was to find the correct analogue of the notion of holomorphic in the quaternionic setting: the so-called Fueter regularity. In this context the Swiss mathematician Fueter [1935] found a way of producing regular functions from holomorphic ones. One can now extend these notions and results to the general context of Clifford analysis: a function theory studying the so-called Dirac operator and its solutions(called monogenics) that is both a generalization of complex and Harmonic analysis (i.e. the function theory associated to the Laplace operator) but is also a refinement.

In this talk we shall explain how the Fueter theorem can be proven in Clifford algebra context by using a representation theory inspired approach. This approach shall be closely related to (generalized) symmetries of given differential operators and their induced Lie algebraic structure. Furthermore, we shall generalize these results even more for symplectic Clifford algebras, something that was still missing in the existing literature and explore the connection with special functions.

## The Finite Heisenberg-Weyl Groups in Image Processing (Srđan Lazendić, Ghent University)

*Abstract:*

The continuous Heisenberg-Weyl groups have a long history in physics and signal processing. However, their discrete variants haven’t received the deserved attention. When a signal is transformed by an element of the (much larger) discrete symplectic group, autocorrelations, which are mathematical tool for finding repeating patterns in a signal, can be calculated from trace inner products of covariance matrices with signed permutation matrices from the discrete Heisenberg-Weyl group. The mapping between a signal and its autocorrelation coefficients based on the Heisenberg-Weyl group is called the Weyl transform. This instance of the Weyl transform is a special case of a general framework for representation of operators in harmonic analysis. We will make it clear how those beautiful properties of the Weyl transform can be used for particular application in image processing and show the original results for image classification.

## The blocks of the periplectic Brauer algebra in positive characteristic (Sigiswald Barbier, Ghent University)

*Abstract:*

Representation theory of the symmetric group can be related to representation theory of the general linear group via Schur-Weyl duality. Similarly, Schur-Weyl duality also relates the orthogonal Lie group, the symplectic Lie group and the encompassing orthosymplectic Lie supergroup to the Brauer algebra, and it relates the periplectic Lie supergroup to the periplectic Brauer algebra.

In this talk I will introduce this periplectic Brauer algebra and give a classification of its blocks in positive characteristic. This is joint work with Anton Cox and Maud De Visscher.

## Shadowing effect of multi-particle catalyst systems in TAP pulse experiments (Brecht Denis, Ghent University)

*Abstract:*

Catalysts are the workhorses of chemical transformations in the industry. They offer often a green alternative path for reactions, favored by reduced activation energy but following a more complex path. Temporal Analysis of Products (TAP) is a pulsed transient technique able to extract intrinsic kinetic data of this complex path over a catalyst from reaction-transport data. Since the configuration of the catalyst could be shrunk to a single micron-sized catalyst particle, the characteristics of multi-particle systems for TAP are of interest. A shadowing effect has been studied by modeling the TAP reactor with a parabolic partial differential equation (PDE). First results will be shown solving the PDE with a Monte Carlo method and the finite element method.

## The higher rank q-Bannai-Ito algebra and multivariate (-q)-Racah polynomials (Hadewijch De Clercq, Master student Ghent University)

*Abstract: *

The q-Bannai-Ito algebra is a quadratic quantum algebra with remarkable properties. It encodes the bispectrality of the (-q)-Racah polynomials, which are the most fundamental orthogonal polynomials of q-hypergeometric type. In this talk, I will explain how this connection can be generalized to multiple variables. We will exploit the Hopf algebraic structure of quantum groups to build a higher rank extension of the q-Bannai-Ito algebra. We will study its action on the discrete series representation of the corresponding quantum group, and identify a class of canonical bases. Several such bases are in duality, in the sense that their overlap coefficients can be expressed as multivariate (-q)-Racah polynomials. The bispectral operators for these polynomials give rise to a discrete realization of the higher rank q-Bannai-Ito algebra.

## Polynomial representations of osp(1|2n) (Asmus Bisbo, Ghent University)

*Abstract:*

The well-studied boson Fock representation of osp(1|2n) can be easily realized in terms of partial derivatives and variable multiplications on the space of n-variable complex valued polynomials. Generalizing this situation by substituting the partial derivatives with p-dimensional Dirac operators and the variable multiplications with corresponding Clifford operators acting on spinor-valued polynomials, we get realizations of the paraboson representations of osp(1|2n) with lowest weight (p/2,...,p/2). Using a nice description of the characters of these representations in terms of Schur functions, we find natural bases of spinor-valued polynomials indexed by semi-standard Young tableaux.

## Segal-Bargmann transformations on superspaces (Sam Claerebout, Ghent University)

*Abstract: *

The classical Segal-Bargmann is an integral transform between the Schrödinger space of square-integrable functions and the Fock space of holomorphic functions. In recent works the Segal-Bargmann transform was reinterpreted as an intertwining operator between realisations on the Schrödinger and the Fock space of the minimal representation of a Lie algebra. In this talk I will give a generalisation of this approach to superspaces in order to obtain a Segal-Bargmann transform as an integral transform that intertwines the Schrödinger and Fock model for the orthosymplectic Lie superalgebra osp(p,2|2n).

## Discrete heat equations (Astrid Massé, Ghent University)

*Abstract:*

Much research has been devoted to the study of the heat equation, as this partial differential equation finds applications in domains such as physics, probability theory and financial mathematics. Our aim is to discretise this equation in the discrete Clifford framework in one dimension. We will consider both discrete space and discrete time. We determine a fundamental solution of the heat equation in this setting and investigate its properties. Discrete heat polynomials are constructed, i.e. discrete polynomial solutions of the heat equation with initial conditions given by a basic discrete homogeneous polynomial. The analogue problem in the dual space is discussed.

## Modelling point cloud data: an approach from topology (Nicolas Vercheval, Ghent University)

*Abstract:*

Topological data analysis (TDA) is an approach to the analysis of datasets, usually data point cloud, using techniques from topology. When data is high-dimensional, incomplete and noisy, TDA provides a general framework for analysis based on independence from the specific metric, dimensionality reduction and robustness to noise.

Its main tool is persistent homology which computes, studies and encodes efficiently multiscale topological features from a nested families of simplicial complexes and topological spaces and finds applications in biology, chemistry and material science. Contemporary research is focusing on how to adapt its features for deep learning.

## Racah problems for the oscillator algebra and sl(n) (Wouter Van de Vijver, Ghent University)

*Abstract:*

We consider the tensor product of n copies of the oscillator algebra h. Using the Hopf structure and Casimir operator of h, we construct a subalgebra R_n(h) in the same way the higher rank Racah algebra was constructed for su(1; 1) in [1]. One can embed the algebra R_n(h) into sl(n,1) after an affine transformation of the generators by central elements. We study the connection between recoupling coefficients for hand sl(n)-representations. These coefficients turn out to be multivariate Krawtchouck polynomials. The relation with the Wigner-(3nj) symbols for h is explained. Flipping two factors in the tensor product is a symmetry of Rn(h). This leads to an automorphism of sl(n,1). The corresponding group elements of SL(n,1) are constructed.

## Bargmann-Radon transform for axially monogenic functions (Ren Hu, Ghent University)

*Abstract:*

In our work, we study a class of monogenic functions called axially monogenic functions. First we present the explicit form for the general Cauchy-Kowalewski extension for axially monogenic functions. Then we determine the Bargmann-Radon transform for these functions, relying on the Funk-Hecke theorem in the process.

## Representation theory and cellular structure of seam algebras (Alexis Langlois Rémillard, Ghent University)

*Abstract:*

The seam algebras were introduced by Morin-Duchesne, Rasmussen and Ridout (2015) in order to introduce an adequate algebraic framework for the lattice analysis in Kac modules and their continuum limit counterpart, the Virasoro-Kac modules. They were viewed there as quotients of one-boundary Temperley-Lieb algebras and they were proven semisimple for generic value of the loop fugacity were given. It was known that they could also be obtained from classical Temperley-Lieb algebra by acting with a family of idempotents. Using this fact, we constructed a cellular basis for the seam algebra by exploiting the properties of a restriction functor and found complete sets of irreducible and principal modules for all the non-semisimple cases, except for one elusive family. This is joint work with Yvan Saint-Aubin.

## Radon-type transforms for holomorphic functions in the Lie ball (Teppo Mertens, Ghent University)

*Abstract:*

Extending the work of I.Sabadini and F.Sommen, we will discuss certain type of Radon transforms on the Lie Sphere. In order to do this, we will start from the Szego-Radon transform on the unit Sphere and try to find suitable extensions of the monogenic functions used in the classic setting. These extensions will be solutions for powers of the Dirac-operator and, in a way, they are all linked together.