This is the unofficial seminar homepage of the GAPT group at Cardiff University.
The research interests of the GAPT group sweep a broad range of topics, from algebra, geometry, topology, including operator algebras, and non-commutative geometry in pure mathematics, to algebraic and conformal quantum field theory and integrable statistical mechanics in mathematical physics. Usually the seminar takes place every Thursday at 15:10. Speakers give 50 minute talks. The GAPT group has a mailing list as well. You can sign up to it here.Past Events: |

**Upcoming Seminar Talks**

**Thursday, 8th February 2024 15:10 - 16:00**

Taro Sogabe (Kyoto University)

Spanier-Whitehead duality and its application to a duality of extensions of C* algebras

The KK-theory is an interesting invariant in the theory of C*-algebras which gives a nice category of C*-algebras and the Spanier-Whitehead duality is defined in the category.

This duality was investigated in the work of J. Kaminker and I. Putnam by using a special extension of Cuntz-Krieger algebras. Recently, K. Matsumoto discovered an interesting duality between K-theory and KK-theory of the special extension. In this talk, I would like to explain what these two dualities are and I will try to show a picture to understand K. Matsumoto’s result via the Spanier-Whitehead duality.

This is based on a joint work with Ulrich Pennig.

Spanier-Whitehead duality and its application to a duality of extensions of C* algebras

The KK-theory is an interesting invariant in the theory of C*-algebras which gives a nice category of C*-algebras and the Spanier-Whitehead duality is defined in the category.

This duality was investigated in the work of J. Kaminker and I. Putnam by using a special extension of Cuntz-Krieger algebras. Recently, K. Matsumoto discovered an interesting duality between K-theory and KK-theory of the special extension. In this talk, I would like to explain what these two dualities are and I will try to show a picture to understand K. Matsumoto’s result via the Spanier-Whitehead duality.

This is based on a joint work with Ulrich Pennig.

**Thursday, 15th February 2024 15:10 - 16:00**

Juan Villarreal (University of Bath)

Tensor structures arising from affine Lie algebras

First, we will review the construction of the tensor product for affine Lie algebras done by Kazhdan and Lusztig. Then, we will explain the relation of the tensors with the intertwiners that appears on vertex algebras. Finally, we will mention the relation with a lambda bracket formalism as in the work Lambda bracket and intertwiners arXiv:2310.18872

Tensor structures arising from affine Lie algebras

First, we will review the construction of the tensor product for affine Lie algebras done by Kazhdan and Lusztig. Then, we will explain the relation of the tensors with the intertwiners that appears on vertex algebras. Finally, we will mention the relation with a lambda bracket formalism as in the work Lambda bracket and intertwiners arXiv:2310.18872

**Thursday, 22nd February 2024 15:10 - 16:00**

Jamal Shafiq (Cardiff University)

Vector-valued modular forms arising from the affine sl(2) VOA

After a preamble on the necessary background in vertex operator algebras (VOAs) and modular forms, including the notion of vector-valued modular forms, 1-point functions for the affine VOA associated to $\mathfrak{sl}(2)$ are introduced. These are a natural generalisation of characters in conformal field theory by considering any module of the VOA acting rather than simply the vacuum. The connection to modularity is motivated and organising these 1-point functions into vector-valued modular forms, results are presented on the congruence properties of the representations under which these transform. Furthermore, explicit generators for the spaces of vector-valued modular forms up to dimension three are calculated.

Vector-valued modular forms arising from the affine sl(2) VOA

After a preamble on the necessary background in vertex operator algebras (VOAs) and modular forms, including the notion of vector-valued modular forms, 1-point functions for the affine VOA associated to $\mathfrak{sl}(2)$ are introduced. These are a natural generalisation of characters in conformal field theory by considering any module of the VOA acting rather than simply the vacuum. The connection to modularity is motivated and organising these 1-point functions into vector-valued modular forms, results are presented on the congruence properties of the representations under which these transform. Furthermore, explicit generators for the spaces of vector-valued modular forms up to dimension three are calculated.

**Thursday, 29th February 2024 15:10 - 16:00**

Daniel Drimbe (University of Oxford)

Rigidity theory in von Neumann algebras and orbit equivalence

In the early 1940s, Murray and von Neumann found a natural way to associate a von Neumann algebra to any countable group $G$ and to any measure preserving action of it. The classification of von Neumann algebras has since been a central theme in operator algebras driven by the following fundamental question: what aspects of the group $G$ or of a measure preserving action of $G$ are remembered by their associated von Neumann algebras? The goal of this talk is to present major breakthroughs in the theory of von Neumann algebras and orbit equivalence and to survey some of the progress made recently using Popa’s deformation/rigidity theory.

Rigidity theory in von Neumann algebras and orbit equivalence

In the early 1940s, Murray and von Neumann found a natural way to associate a von Neumann algebra to any countable group $G$ and to any measure preserving action of it. The classification of von Neumann algebras has since been a central theme in operator algebras driven by the following fundamental question: what aspects of the group $G$ or of a measure preserving action of $G$ are remembered by their associated von Neumann algebras? The goal of this talk is to present major breakthroughs in the theory of von Neumann algebras and orbit equivalence and to survey some of the progress made recently using Popa’s deformation/rigidity theory.

**Thursday, 7th March 2024 15:10 - 16:00**

Otto Sumray (University of Oxford)

Quiver Laplacians and Feature Selection

Laplacians have been developed for a broad range of discrete structures, ranging from graphs, to simplicial complexes, to cellular sheaves. In this talk we introduce the notion of a Laplacian for quiver representations and present some results on their spectra. We shall then use these Laplacians to solve an important problem in feature selection: how can one combine different features from across a data set? We then apply these methods to a bioinformatics problem, regarding the identification of relevant genomic regions in single-cell chromatin accessibility data.

Quiver Laplacians and Feature Selection

Laplacians have been developed for a broad range of discrete structures, ranging from graphs, to simplicial complexes, to cellular sheaves. In this talk we introduce the notion of a Laplacian for quiver representations and present some results on their spectra. We shall then use these Laplacians to solve an important problem in feature selection: how can one combine different features from across a data set? We then apply these methods to a bioinformatics problem, regarding the identification of relevant genomic regions in single-cell chromatin accessibility data.

**Thursday, 14th March 2024 15:10 - 16:00**

Arthur Pander Maat (Queen Mary University London)

The Morita homotopy theory of $C^*$-categories

$C^*$-categories are a horizontal categorification of $C^*$-algebras, sharing many of their fundamental properties, including an appropriate notion of Hilbert modules. One advantage of generalizing from $C^*$-algebras with $C^*$-categories is that taking the $C^*$-category of Hilbert modules becomes an endofunctor on the category of $C^*$-categories. After giving an introduction to $C^*$-categories, we use an Eilenberg-Watts theorem, to show that this endofunctor is a reflective localization, giving a model for the

The Morita homotopy theory of $C^*$-categories

$C^*$-categories are a horizontal categorification of $C^*$-algebras, sharing many of their fundamental properties, including an appropriate notion of Hilbert modules. One advantage of generalizing from $C^*$-algebras with $C^*$-categories is that taking the $C^*$-category of Hilbert modules becomes an endofunctor on the category of $C^*$-categories. After giving an introduction to $C^*$-categories, we use an Eilenberg-Watts theorem, to show that this endofunctor is a reflective localization, giving a model for the

*Morita homotopy theory*of operator algebraic objects. We give several applications, including a new model for KK-theory.**Thursday, 21st March 2024 15:10 - 16:00**

Natasha Blitvic (Queen Mary University London)

Combinatorial moment sequences

We will look at a number of interesting examples — some proven, others merely conjectured — of Hamburger moment sequences in combinatorics. We will consider ways in which this positivity may be expected: for instance, in different types of combinatorial statistics on perfect matchings that encode moments of noncommutative analogues of the classical Central Limit Theorem. We will also consider situations in which this positivity may be surprising, and where proving it would open up new approaches to a class of very hard open problems in combinatorics.

Combinatorial moment sequences

We will look at a number of interesting examples — some proven, others merely conjectured — of Hamburger moment sequences in combinatorics. We will consider ways in which this positivity may be expected: for instance, in different types of combinatorial statistics on perfect matchings that encode moments of noncommutative analogues of the classical Central Limit Theorem. We will also consider situations in which this positivity may be surprising, and where proving it would open up new approaches to a class of very hard open problems in combinatorics.

**Friday, 12th April 2024**

If you are interested in the GAPT seminar, then you might also be interested in the following two events happening in Cardiff:

First Meeting of the UK Metric Geometry and Analysis Network

The

The network focuses mainly on topics in differential geometry and geometric analysis which include:

**UK Metric Geometry and Analysis Network**brings together five working groups from Cardiff University, Durham University, University of Oxford, University of Stirling, and the University of Warwick.The network focuses mainly on topics in differential geometry and geometric analysis which include:

- Alexandrov spaces
- Manifolds with lower curvature bounds
- RCD spaces
- Lorentzian geometry
- Geometric PDEs

Topological Physics in Condensed Matter Theory

The aim of

**this workshop**is to bring together mathematicians and physicists interested in topological physics (interpreted in a broad sense), with a view towards condensed matter theory. Apart from invited talks by leading experts, it will feature contributed talks, giving early career researchers the opportunity to present their work to a broad audience. The workshop is co-organised by the Institute of Physics’ Mathematical & Theoretical Physics group, and the Theory of Condensed Matter group.**Thursday, 18th April 2024 15:00 - 15:50 - note the different time**

Oscar Finegan (Cardiff University)

Cohomology of Derived Intersections

Intersections of subobjects in any category with enough structure are given by pullbacks over the diagram of inclusions. In Algebraic Geometry this leads us to the notion of intersection structure on the intersection of subvarieties. Upgrading this logic to the derived category we end up studying derived tensor products of structure sheaves. If the intersection is transverse then there is nothing interesting to say but in other cases there is interesting geometric data encoded in the cohomologies of these objects. I will present a general introduction to these ideas and give a description of my work in this area.

Cohomology of Derived Intersections

Intersections of subobjects in any category with enough structure are given by pullbacks over the diagram of inclusions. In Algebraic Geometry this leads us to the notion of intersection structure on the intersection of subvarieties. Upgrading this logic to the derived category we end up studying derived tensor products of structure sheaves. If the intersection is transverse then there is nothing interesting to say but in other cases there is interesting geometric data encoded in the cohomologies of these objects. I will present a general introduction to these ideas and give a description of my work in this area.

**Thursday, 25th April 2024 15:10 - 16:00**

Nivedita (University of Oxford)

Functorial chiral Conformal Field Theories from Conformal Nets

There are two main axiomatic approaches to study quantum field theory, namely algebraic (AQFT) and functorial (FQFT). We will briefly introduce these and try to relate them in the special, yet important, case of two-dimensional chiral conformal field theory. 2D chiral CFTs have three formulations, VOAs, conformal nets (AQFT) and Segal functorial CFTs (FQFT). We will sketch a partial construction of a

Functorial chiral Conformal Field Theories from Conformal Nets

There are two main axiomatic approaches to study quantum field theory, namely algebraic (AQFT) and functorial (FQFT). We will briefly introduce these and try to relate them in the special, yet important, case of two-dimensional chiral conformal field theory. 2D chiral CFTs have three formulations, VOAs, conformal nets (AQFT) and Segal functorial CFTs (FQFT). We will sketch a partial construction of a

*fully extended*Segal chiral CFT from the input data of a conformal net. More precisely, from a conformal net, we build three equivalent presentations of the category of Solitons (solitonic representations of conformal nets) which the associated fully extended Segal chiral CFT should assign to a point. We also briefly introduce the target category of Bicommutant Categories (a model for 3-Hilb) and show that the category of solitons is a bicommutant category. This target is motivated as a categorification (and delooping) of the Morita category of von Neumann algebras and is work in progress.**Thursday, 2nd May 2024 15:10 - 16:00 - postponed**

Angela Capel (University of Cambridge)

**Seminar Talks - Autumn 2023**

**Thursday, 5th October 2023 15:10 - 16:00**

Ryo Yamagishi (University of Bath)

Moduli spaces of G-constellations

For a finite subgroup G of $SL_n(\mathbb{C})$, a moduli space of G-constellations is a generalization of the G-Hilbert scheme and is important from the viewpoint of the (derived) McKay correspondence. In this talk I will explain its basic properties and give an outline of the proof of the following conjecture of Craw and Ishii: every projective crepant resolution of the quotient variety $\mathbb{C}^3/G$ is isomorphic to a moduli space of G-constellations for some stability condition.

Moduli spaces of G-constellations

For a finite subgroup G of $SL_n(\mathbb{C})$, a moduli space of G-constellations is a generalization of the G-Hilbert scheme and is important from the viewpoint of the (derived) McKay correspondence. In this talk I will explain its basic properties and give an outline of the proof of the following conjecture of Craw and Ishii: every projective crepant resolution of the quotient variety $\mathbb{C}^3/G$ is isomorphic to a moduli space of G-constellations for some stability condition.

**Thursday, 12th October 2023 15:10 - 16:00**

Emanuele Dotto (University of Warwick)

Characteristic polynomials of self-adjoint endomorphisms

The algebraic properties of the characteristic polynomial of a matrix can be efficiently packaged by expressing the characteristic polynomial as a ring homomorphism from the cyclic K-theory ring to the ring of Witt vectors. This homomorphism can moreover be interpreted as the effect in $\pi_0$ of a homotopy theoretic trace map. The talk will introduce these ideas and explain how, by extending them to an equivariant context, they give rise to a refined version of the characteristic polynomial for self-adjoint endomorphisms.

Characteristic polynomials of self-adjoint endomorphisms

The algebraic properties of the characteristic polynomial of a matrix can be efficiently packaged by expressing the characteristic polynomial as a ring homomorphism from the cyclic K-theory ring to the ring of Witt vectors. This homomorphism can moreover be interpreted as the effect in $\pi_0$ of a homotopy theoretic trace map. The talk will introduce these ideas and explain how, by extending them to an equivariant context, they give rise to a refined version of the characteristic polynomial for self-adjoint endomorphisms.

**Thursday, 19th October 2023 15:10 - 16:00**

Makoto Yamashita (University of Oslo)

Homology and K-theory of dynamical systems

I will give an overview on homological invariants for dynamical systems through étale groupoids motivated by the theory of aperiodic tilings, hyperbolic dynamics (Smale spaces), and beyond. Such groupoids appear from discrete group actions on topological spaces and reduction of continuous dynamics to transversal subspaces on spaces which have both continuum flavor (manifolds) and combinatorial flavor (totally disconnected spaces). Étale groupoids admit two kinds of homology theories, one being the Crainic-Moerdijk homology, and the other being the K-theory of associated C*-algebras. The latter is a receptacle for invariants motivated by mathematical physics, while the former is computable through toolkits of algebraic topology in many interesting examples. These are related by categorical structures behind KK-theory of C*-algebras. Based on joint works with Valerio Proietti.

Homology and K-theory of dynamical systems

I will give an overview on homological invariants for dynamical systems through étale groupoids motivated by the theory of aperiodic tilings, hyperbolic dynamics (Smale spaces), and beyond. Such groupoids appear from discrete group actions on topological spaces and reduction of continuous dynamics to transversal subspaces on spaces which have both continuum flavor (manifolds) and combinatorial flavor (totally disconnected spaces). Étale groupoids admit two kinds of homology theories, one being the Crainic-Moerdijk homology, and the other being the K-theory of associated C*-algebras. The latter is a receptacle for invariants motivated by mathematical physics, while the former is computable through toolkits of algebraic topology in many interesting examples. These are related by categorical structures behind KK-theory of C*-algebras. Based on joint works with Valerio Proietti.

**Thursday, 26th October 2023 15:10 - 16:00**

Diego Corro Tapia (Cardiff University)

Singular Riemannian foliations as generalized notion of curvature.

In this talk I will introduce singular Riemannian foliations as a generalized notion of curvature. These foliations where first studied by Élie Cartan in the 30s when considering isoperimetric submanifolds in spheres, then in generality by Molino in the 80s, extending the theory of regular foliations. Recently they have resurged as generalization of group actions by isometries in differential geometry and differential topology.

In this talk I will briefly present connections of these generalized symmetries to finding solutions to geometric PDEs, topological rigidity results, and focus on ongoing work on how to deform a Riemannian manifold using these foliations. At the end we will briefly discuss an open direction of research in differential geometry.

Singular Riemannian foliations as generalized notion of curvature.

In this talk I will introduce singular Riemannian foliations as a generalized notion of curvature. These foliations where first studied by Élie Cartan in the 30s when considering isoperimetric submanifolds in spheres, then in generality by Molino in the 80s, extending the theory of regular foliations. Recently they have resurged as generalization of group actions by isometries in differential geometry and differential topology.

In this talk I will briefly present connections of these generalized symmetries to finding solutions to geometric PDEs, topological rigidity results, and focus on ongoing work on how to deform a Riemannian manifold using these foliations. At the end we will briefly discuss an open direction of research in differential geometry.

**Thursday, 2nd November 2023 15:10 - 16:00**

Veronika Pedić (University of Zagreb)

Vertex algebras and their orbifolds

Orbifold constructions are one of the basic ways to construct new vertex algebras from given ones. The idea of orbifold theory of vertex algebras is to take a vertex algebra $V$ and some group of its automorphisms $G$, and study the representation theory of the fixed point subalgebra $V^G$. This theory was initiated in the 90s by C. Dong and G. Mason, however, it was applicable only to rational vertex algebras. In this talk we present various extensions of this work to much larger categories of weak modules. Furthermore, we present applications to the case of Weyl vertex algebra and Whittaker modules.

This is joint work with D. Adamović, C.-H. Lam and N. Yu.

Vertex algebras and their orbifolds

Orbifold constructions are one of the basic ways to construct new vertex algebras from given ones. The idea of orbifold theory of vertex algebras is to take a vertex algebra $V$ and some group of its automorphisms $G$, and study the representation theory of the fixed point subalgebra $V^G$. This theory was initiated in the 90s by C. Dong and G. Mason, however, it was applicable only to rational vertex algebras. In this talk we present various extensions of this work to much larger categories of weak modules. Furthermore, we present applications to the case of Weyl vertex algebra and Whittaker modules.

This is joint work with D. Adamović, C.-H. Lam and N. Yu.

**Thursday, 9th November 2023 15:10 - 16:00**

Naoki Koseki (University of Liverpool)

Curve counting via categorification

A central approach in modern enumerative geometry is to assign numbers called "virtual invariants" to a space parametrizing geometric objects we want to count. More recently, people are trying to upgrade (categorify) virtual invariants to vector spaces or even categories to obtain finer information about the space.

In this talk, I will review some recent developments in categorified enumerative geometry, including Gopakumar-Vafa invariants and categorical blow-up formula.

Curve counting via categorification

A central approach in modern enumerative geometry is to assign numbers called "virtual invariants" to a space parametrizing geometric objects we want to count. More recently, people are trying to upgrade (categorify) virtual invariants to vector spaces or even categories to obtain finer information about the space.

In this talk, I will review some recent developments in categorified enumerative geometry, including Gopakumar-Vafa invariants and categorical blow-up formula.

**Thursday, 16th November 2023 15:10 - 16:00**

Chris Bruce (Newcastle University)

From rings to C*-algebras and back again

Each ring gives rise to a concrete operator algebra—called the reduced ring C*-algebra of the ring—acting on the Hilbert space of square-summable complex-valued functions on the ring. The study of such C*-algebras was pioneered by J. Cuntz and X. Li, who were interested in structural properties such as simplicity. I will give a gentle introduction to ring C*-algebras and then present new results on rigidity of the C*-algebra construction: Two ring C*-algebras arising from rings of algebraic integers in number fields are *-isomorphic in a Cartan-preserving way if and only if the rings are isomorphic. This is joint work with Xin Li.

From rings to C*-algebras and back again

Each ring gives rise to a concrete operator algebra—called the reduced ring C*-algebra of the ring—acting on the Hilbert space of square-summable complex-valued functions on the ring. The study of such C*-algebras was pioneered by J. Cuntz and X. Li, who were interested in structural properties such as simplicity. I will give a gentle introduction to ring C*-algebras and then present new results on rigidity of the C*-algebra construction: Two ring C*-algebras arising from rings of algebraic integers in number fields are *-isomorphic in a Cartan-preserving way if and only if the rings are isomorphic. This is joint work with Xin Li.

**Thursday, 23rd November 2023 15:10 - 16:00**

Ambrose Yim (Cardiff University)

Topological Inference

Given finitely many samples on a manifold, can we infer its topological invariants? We discuss some state of the art methods and theoretical bounds for inferring the homology of a manifold from a sufficiently dense point sample. We discuss how these methods are motivated by dynamical systems. Conversely, we also show how these methods can help infer topological invariants associated to gradient dynamics from finite data.

Topological Inference

Given finitely many samples on a manifold, can we infer its topological invariants? We discuss some state of the art methods and theoretical bounds for inferring the homology of a manifold from a sufficiently dense point sample. We discuss how these methods are motivated by dynamical systems. Conversely, we also show how these methods can help infer topological invariants associated to gradient dynamics from finite data.

**Thursday, 30th November 2023 15:10 - 16:00**

Matt Booth (Lancaster University)

Deformation theory and Koszul duality

Deformation theory studies the infinitesimal deformations of geometric objects; it can be thought of as an algebraic version of perturbation theory. Deligne, in a 1987 letter to Millson, conjectured that every deformation problem is controlled in some sense by a dg (differential graded) Lie algebra. This conjecture was solved independently in 2007 by Pridham and in 2010 by Lurie in the setting of derived algebraic geometry, where spaces are modelled on commutative dg algebras (the higher degree cohomology of which records information about `higher data' like non-transverse intersections). One view of this correspondence is as a souped-up version of calculus: a deformation problem has a tangent space, which is a dg Lie algebra, and the infinitesimal nature of deformation theory ensures that one can always integrate. Lurie's proof indicates that one should think of the correspondence as an instance of the Koszul duality between commutative and Lie algebras, and he also proves the analogous statement that noncommutative algebras control noncommutative deformation problems. I'll give an outline of the above ideas, and - time permitting - will briefly talk about some recent work of mine joint with Andrey Lazarev, where we develop model structures for global Koszul duality and investigate the resulting notion of deformation problem (which we call MC stacks).

Deformation theory and Koszul duality

Deformation theory studies the infinitesimal deformations of geometric objects; it can be thought of as an algebraic version of perturbation theory. Deligne, in a 1987 letter to Millson, conjectured that every deformation problem is controlled in some sense by a dg (differential graded) Lie algebra. This conjecture was solved independently in 2007 by Pridham and in 2010 by Lurie in the setting of derived algebraic geometry, where spaces are modelled on commutative dg algebras (the higher degree cohomology of which records information about `higher data' like non-transverse intersections). One view of this correspondence is as a souped-up version of calculus: a deformation problem has a tangent space, which is a dg Lie algebra, and the infinitesimal nature of deformation theory ensures that one can always integrate. Lurie's proof indicates that one should think of the correspondence as an instance of the Koszul duality between commutative and Lie algebras, and he also proves the analogous statement that noncommutative algebras control noncommutative deformation problems. I'll give an outline of the above ideas, and - time permitting - will briefly talk about some recent work of mine joint with Andrey Lazarev, where we develop model structures for global Koszul duality and investigate the resulting notion of deformation problem (which we call MC stacks).

**Thursday, 7th December 2023 15:10 - 16:00**

Daniel Drimbe (University of Oxford) -

**postponed****Thursday, 14th December 2023 15:10 - 16:00**

Christian Korff (University of Glasgow)

Exactly solvable lattice models, symmetric functions and vertex operators

The ring of symmetric functions plays a central role in representation theory. It connects with exactly solvable lattice models of statistical mechanics and quantum many-body systems by observing that the eigenfunctions of the transfer matrices or Hamiltonian (the Bethe wave functions) are symmetric polynomials. For infinite lattices with suitable boundary conditions, one can use the transfer matrices of exactly solvable lattice models to obtain combinatorial formulae for vertex operators of symmetric functions. This links the area of statistical lattice models and quantum spin-chains (via the boson-fermion correspondence) with integrable hierarchies of PDEs such as the Kadomtsev-Petiashvili equation where it is known that particular solutions, tau-functions, are given by symmetric functions.

Exactly solvable lattice models, symmetric functions and vertex operators

The ring of symmetric functions plays a central role in representation theory. It connects with exactly solvable lattice models of statistical mechanics and quantum many-body systems by observing that the eigenfunctions of the transfer matrices or Hamiltonian (the Bethe wave functions) are symmetric polynomials. For infinite lattices with suitable boundary conditions, one can use the transfer matrices of exactly solvable lattice models to obtain combinatorial formulae for vertex operators of symmetric functions. This links the area of statistical lattice models and quantum spin-chains (via the boson-fermion correspondence) with integrable hierarchies of PDEs such as the Kadomtsev-Petiashvili equation where it is known that particular solutions, tau-functions, are given by symmetric functions.