Exotic Representation Theory
Lead Research Organisation:
Loughborough University
Department Name: Mathematical Sciences
Abstract
This proposal lies at the intersection of Geometry, Topology and Algebra. Through the study of certain generalised finite groups which arise naturally in each of these contexts, I will forge new interdisciplinary connections between them. These connections will serve to sustain and develop links between algebra and related disciplines, enhancing the key underpinning role that algebraic, geometric and topological research plays across the mathematical sciences, as well as the in UK's world-leading position in the area. They will also and offer potential applications to Quantum Mechanics, Topological Data Analysis and Artificial Intelligence.
The study of groups begins with the study of symmetries of shapes such as regular polygons or 3-dimensional polyhedra like the cube, tetrahedron or icosahedron. As for these examples, many interesting symmetry groups arise as reflection groups - those generated by reflections (linear transformations fixing a hyperplane of the underlying vector space). A reflection group is called `real' or `complex' according to whether the vector space is real or complex. Both the real and irreducible complex reflection groups have been classified.
The classification of finite simple groups, the building blocks of symmetry, reveals that `most' simple groups are of Lie type, meaning that they are, in some sense, determined by their Weyl groups which are real reflection groups. For example, the groups of rotations and reflections of higher dimensional tetrahedra (the symmetric groups) give rise to simple groups of matrices defined over some finite field.
This project will study the representation theory of spetses, which are yet undefined objects generalising the simple groups of Lie type in which the associated Weyl group is assumed to be a complex reflection group. Despite the absence of an actual group with which to perform calculations, techniques from Algebra and Topology - specifically the theories of Hecke algebras and $p$-compact groups - will be combined to garner new representation-theoretic information (such as character degrees, decomposition matrices and Brauer trees).
I will show that this information is rich enough to satisfy generalised local-global counting conjectures from the modular representation theory of finite groups. For groups, these conjectures will follow as special cases - when the Weyl group is real - thereby settling questions that have been around for more than half a century.
The study of groups begins with the study of symmetries of shapes such as regular polygons or 3-dimensional polyhedra like the cube, tetrahedron or icosahedron. As for these examples, many interesting symmetry groups arise as reflection groups - those generated by reflections (linear transformations fixing a hyperplane of the underlying vector space). A reflection group is called `real' or `complex' according to whether the vector space is real or complex. Both the real and irreducible complex reflection groups have been classified.
The classification of finite simple groups, the building blocks of symmetry, reveals that `most' simple groups are of Lie type, meaning that they are, in some sense, determined by their Weyl groups which are real reflection groups. For example, the groups of rotations and reflections of higher dimensional tetrahedra (the symmetric groups) give rise to simple groups of matrices defined over some finite field.
This project will study the representation theory of spetses, which are yet undefined objects generalising the simple groups of Lie type in which the associated Weyl group is assumed to be a complex reflection group. Despite the absence of an actual group with which to perform calculations, techniques from Algebra and Topology - specifically the theories of Hecke algebras and $p$-compact groups - will be combined to garner new representation-theoretic information (such as character degrees, decomposition matrices and Brauer trees).
I will show that this information is rich enough to satisfy generalised local-global counting conjectures from the modular representation theory of finite groups. For groups, these conjectures will follow as special cases - when the Weyl group is real - thereby settling questions that have been around for more than half a century.
Organisations
- Loughborough University (Collaboration, Lead Research Organisation)
- University of Milan (Collaboration)
- University of Paris (Collaboration)
- University of Birmingham (Collaboration)
- Manchester University (Collaboration)
- Technologiezentrum Dresden (Collaboration)
- University of Louisiana at Lafayette (Collaboration)
- Technical University Kaiserslautern (Collaboration)
- University of Dresden (Collaboration)
- University of Southampton (Collaboration)
- UNIVERSITY OF MANCHESTER (Collaboration)
- Mimar Sinan Fine Arts University (Collaboration)
People |
ORCID iD |
| Jason Semeraro (Principal Investigator) |
Publications
Kessar R
(2024)
Weights for compact connected Lie groups
in manuscripta mathematica
Kessar R
(2023)
The principal block of a Zl-spets and Yokonuma type algebras
in Algebra & Number Theory
Lynd J
(2023)
Weights in a Benson-Solomon block
in Forum of Mathematics, Sigma
Radha Kessar
(2024)
Weights for compact connected Lie groups
Radha Kessar
(2024)
Weight conjectures for $\ell$-compact groups and spetses
in Annales scientifiques de l'Ćcole normale supĆ©rieure
Semeraro J
(2023)
Weights for l-local compact groups
in Journal of Algebra
Semeraro J
(2024)
Corrigendum to Centralizers of subsystems of fusion systems [J.~Group Theory 18 (2015), 393--405]
in Journal of Group Theory
| Description | Private meeting with Loughborough VC |
| Geographic Reach | Europe |
| Policy Influence Type | Influenced training of practitioners or researchers |
| Title | The theory of p-compact groups in group representation theory |
| Description | My work has led to a new approach in the local representation theory of finite groups. Fusion systems of a unipotent block of a finite group of Lie type control much of the representation theory of that block, yet they are notoriously hard to calculate, and previous methods have largely been ad-hoc. My work shows that these fusion systems are exactly given by the homotopy fixed points of the p-compact group associated to the relative Weyl group of the block. |
| Type Of Material | Improvements to research infrastructure |
| Year Produced | 2024 |
| Provided To Others? | No |
| Impact | These tools will be implemented in next-generation approaches to the local-global conjectures of modular representation theory. |
| Title | Small fusion systems |
| Description | The database consists of a list of fusion systems on small p-groups discovered using the MAGMA package "Fusion Systems" developed by Prof. Chris Parker and myself. |
| Type Of Material | Database/Collection of data |
| Year Produced | 2020 |
| Provided To Others? | Yes |
| Impact | There is a project to implement a GAP version of our code led by David Burell: https://davidburrell.github.io/#projects. The long term aim is to provide databases for fusion systems on groups of order 2^10 and 3^9. It is conjectured that all such fusion systems are already known. |
| URL | https://github.com/chris1961parker/Fusion-Systems |
| Description | Fusion systems with an abelian normal essential subgroup |
| Organisation | Technologiezentrum Dresden |
| Country | Germany |
| Sector | Private |
| PI Contribution | Our project is to classify fusion systems with an abelian normal essential subgroup. Such fusion systems arise in p-local Lie theory, and so this work directly relates to my grant. A two week workshop was held at the university of Birmingham in May 2023, during which this research commenced. The workshop resulted in a long (100 page) draft of results to be used in a research monograph at some later date. One aspect of the work relies on the classification of a particular family of fusion systems - those on the p-Sylow subgroup of the semi-direct product associated to the symmetric power representations of SL_2(q). The list of such systems was thought to be complete, but I was able to locate a new family (on the largest such p-group) in Summer 2023. This will form the basis of a separate paper, which is basically complete. |
| Collaborator Contribution | I am collaborating with Chris Parker and David Craven (Birmingham), Martin van Beek (Manchester), Ellen Henke (Dresden), Bob Oliver (Paris Nord) and Valentina Grazian (Milan). Each collaborator brings a unique skill set to the project. Chris Parker is a seasoned group theorist with an abundance of experience; David Craven has expert knowledge on modular representations of simple groups; Ellen Henke and Justin Lynd has been involved in the revision programme led by Michael Aschbacher to reclassify the finite simple groups using the theory of fusion systems; Martin van Beek and Valentina Grazian bring expert knowledge in the classification of fusion systems using amalgams. |
| Impact | "Fusion systems related to polynomial representations of SL(2,q)", arXiv preprint: https://arxiv.org/abs/2502.20873 |
| Start Year | 2023 |
| Description | Fusion systems with an abelian normal essential subgroup |
| Organisation | University of Birmingham |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | Our project is to classify fusion systems with an abelian normal essential subgroup. Such fusion systems arise in p-local Lie theory, and so this work directly relates to my grant. A two week workshop was held at the university of Birmingham in May 2023, during which this research commenced. The workshop resulted in a long (100 page) draft of results to be used in a research monograph at some later date. One aspect of the work relies on the classification of a particular family of fusion systems - those on the p-Sylow subgroup of the semi-direct product associated to the symmetric power representations of SL_2(q). The list of such systems was thought to be complete, but I was able to locate a new family (on the largest such p-group) in Summer 2023. This will form the basis of a separate paper, which is basically complete. |
| Collaborator Contribution | I am collaborating with Chris Parker and David Craven (Birmingham), Martin van Beek (Manchester), Ellen Henke (Dresden), Bob Oliver (Paris Nord) and Valentina Grazian (Milan). Each collaborator brings a unique skill set to the project. Chris Parker is a seasoned group theorist with an abundance of experience; David Craven has expert knowledge on modular representations of simple groups; Ellen Henke and Justin Lynd has been involved in the revision programme led by Michael Aschbacher to reclassify the finite simple groups using the theory of fusion systems; Martin van Beek and Valentina Grazian bring expert knowledge in the classification of fusion systems using amalgams. |
| Impact | "Fusion systems related to polynomial representations of SL(2,q)", arXiv preprint: https://arxiv.org/abs/2502.20873 |
| Start Year | 2023 |
| Description | Fusion systems with an abelian normal essential subgroup |
| Organisation | University of Manchester |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | Our project is to classify fusion systems with an abelian normal essential subgroup. Such fusion systems arise in p-local Lie theory, and so this work directly relates to my grant. A two week workshop was held at the university of Birmingham in May 2023, during which this research commenced. The workshop resulted in a long (100 page) draft of results to be used in a research monograph at some later date. One aspect of the work relies on the classification of a particular family of fusion systems - those on the p-Sylow subgroup of the semi-direct product associated to the symmetric power representations of SL_2(q). The list of such systems was thought to be complete, but I was able to locate a new family (on the largest such p-group) in Summer 2023. This will form the basis of a separate paper, which is basically complete. |
| Collaborator Contribution | I am collaborating with Chris Parker and David Craven (Birmingham), Martin van Beek (Manchester), Ellen Henke (Dresden), Bob Oliver (Paris Nord) and Valentina Grazian (Milan). Each collaborator brings a unique skill set to the project. Chris Parker is a seasoned group theorist with an abundance of experience; David Craven has expert knowledge on modular representations of simple groups; Ellen Henke and Justin Lynd has been involved in the revision programme led by Michael Aschbacher to reclassify the finite simple groups using the theory of fusion systems; Martin van Beek and Valentina Grazian bring expert knowledge in the classification of fusion systems using amalgams. |
| Impact | "Fusion systems related to polynomial representations of SL(2,q)", arXiv preprint: https://arxiv.org/abs/2502.20873 |
| Start Year | 2023 |
| Description | Fusion systems with an abelian normal essential subgroup |
| Organisation | University of Milan |
| Country | Italy |
| Sector | Academic/University |
| PI Contribution | Our project is to classify fusion systems with an abelian normal essential subgroup. Such fusion systems arise in p-local Lie theory, and so this work directly relates to my grant. A two week workshop was held at the university of Birmingham in May 2023, during which this research commenced. The workshop resulted in a long (100 page) draft of results to be used in a research monograph at some later date. One aspect of the work relies on the classification of a particular family of fusion systems - those on the p-Sylow subgroup of the semi-direct product associated to the symmetric power representations of SL_2(q). The list of such systems was thought to be complete, but I was able to locate a new family (on the largest such p-group) in Summer 2023. This will form the basis of a separate paper, which is basically complete. |
| Collaborator Contribution | I am collaborating with Chris Parker and David Craven (Birmingham), Martin van Beek (Manchester), Ellen Henke (Dresden), Bob Oliver (Paris Nord) and Valentina Grazian (Milan). Each collaborator brings a unique skill set to the project. Chris Parker is a seasoned group theorist with an abundance of experience; David Craven has expert knowledge on modular representations of simple groups; Ellen Henke and Justin Lynd has been involved in the revision programme led by Michael Aschbacher to reclassify the finite simple groups using the theory of fusion systems; Martin van Beek and Valentina Grazian bring expert knowledge in the classification of fusion systems using amalgams. |
| Impact | "Fusion systems related to polynomial representations of SL(2,q)", arXiv preprint: https://arxiv.org/abs/2502.20873 |
| Start Year | 2023 |
| Description | Fusion systems with an abelian normal essential subgroup |
| Organisation | University of Paris |
| Country | France |
| Sector | Academic/University |
| PI Contribution | Our project is to classify fusion systems with an abelian normal essential subgroup. Such fusion systems arise in p-local Lie theory, and so this work directly relates to my grant. A two week workshop was held at the university of Birmingham in May 2023, during which this research commenced. The workshop resulted in a long (100 page) draft of results to be used in a research monograph at some later date. One aspect of the work relies on the classification of a particular family of fusion systems - those on the p-Sylow subgroup of the semi-direct product associated to the symmetric power representations of SL_2(q). The list of such systems was thought to be complete, but I was able to locate a new family (on the largest such p-group) in Summer 2023. This will form the basis of a separate paper, which is basically complete. |
| Collaborator Contribution | I am collaborating with Chris Parker and David Craven (Birmingham), Martin van Beek (Manchester), Ellen Henke (Dresden), Bob Oliver (Paris Nord) and Valentina Grazian (Milan). Each collaborator brings a unique skill set to the project. Chris Parker is a seasoned group theorist with an abundance of experience; David Craven has expert knowledge on modular representations of simple groups; Ellen Henke and Justin Lynd has been involved in the revision programme led by Michael Aschbacher to reclassify the finite simple groups using the theory of fusion systems; Martin van Beek and Valentina Grazian bring expert knowledge in the classification of fusion systems using amalgams. |
| Impact | "Fusion systems related to polynomial representations of SL(2,q)", arXiv preprint: https://arxiv.org/abs/2502.20873 |
| Start Year | 2023 |
| Description | On e-local structures for spetses |
| Organisation | Loughborough University |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | The project aims to extend a recent result of Damiano's which is analogous to Quillen's observation that the Brown complex of a Lie type group in characteristic $p$ is homotopy equivalent to the Tits building. His result deals with the case of Lie-type groups in non-defining characteristic. We extend this to the setting of spetses where we invoke some old results of Bob Oliver, Ran Levi and Carles Broto to make sense of the analogous topological structures. My own contribution has been to guide our project especially in the way in which topological and algebraic structures can be expected to interact. He I have drawn on experience I have gained through working on related problems in other projects. |
| Collaborator Contribution | Damiano Rossi brings extensive knowledge of the representation theory of Lie-type groups in non-defining characteristic. |
| Impact | "On e-local structures for spetses", in preparation. |
| Start Year | 2023 |
| Description | Sharpness for Clelland-Parker fusion systems |
| Organisation | Loughborough University |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | Anja Meyer and I are in the process of verifying the Sharpness Conjecture for the class of polynomial fusion systems, using the methods of Diaz-Park. |
| Collaborator Contribution | We observed that the criterion supplied by Diaz-Park fits the above situation very well. We are currently in the process of checking the details. |
| Impact | Ongoing. |
| Start Year | 2025 |
| Description | Spectra of subrings of cohomology generated by characteristic classes for fusion systems |
| Organisation | University of Southampton |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | If $\cF$ is a saturated fusion system on a finite $p$-group $S$, we define the Chern subring $\Ch(\cF)$ of $\cF$ to be the subring of $H^*(S;\FF_p)$ generated by Chern classes of $\cF$-stable representations of $S$. We show that $\Ch(\cF)$ is contained in $H^*(\cF;\FF_p)$ and apply a result of Green and Leary to describe the ideal spectrum of this subring in terms of a certain category of elementary abelian subgroups. We obtain similar results for various related subrings, including those generated by characteristic classes of $\cF$-stable $S$-sets. |
| Collaborator Contribution | Ian Leary (Southampton) began work on this project in the Summer of 2023, following discussions we had at the University of Bristol. His interests were in the more topological aspects of the work. For example, he has introduced alternative definitions of Chern subrings in terms of vector bundles of certain representations. One of these conjectures remains open and has already attracted some interest from other UK instututions. |
| Impact | The paper is now finished and has been accepted by the Bulletin of the London Mathematical Society subject to minor revisions. |
| Start Year | 2023 |
| Description | Unipotent blocks and characters for spetses |
| Organisation | Technical University Kaiserslautern |
| Country | Germany |
| Sector | Academic/University |
| PI Contribution | Prof. Radha Kessar (Manchester) and Gunter Malle (Kaiserslautern) and I have succeeded in generalising the the theory of spetses to the case of non-principal blocks. We have also defined character values for spetses using a formula of Curtis. Using the associated Yokonuma-Hecke algebras, we have shown that these values enjoy orthogonality properties in some cases. The general case of this conjecture is still ongoing. My contribution has been to make conjectures based on computational evidence and worked examples. I have also proved some results. |
| Collaborator Contribution | I have benefited enormously from my collaborators' extensive knowledge in representation theory, They were able to prove several of my conjectures, and streamline some existing proofs. |
| Impact | -Weights for compact connected Lie groups ( Manuscripta Mathematics (to appear) - Weights for $\ell$-local compact groups J. Algebra 636 (2023), 357-372 - The principal block of a $\ZZ_\ell$-spets and Yokonuma type algebras Algebra and Number Theory 17 (2023), 397-433 (joint with Radha Kessar and Gunter Malle) - Weight conjectures for $\ell$-compact groups and spetses (joint with Radha Kessar and Gunter Malle) Annales scientifiques de l'École normale supérieure (to appear) |
| Start Year | 2020 |
| Description | Unipotent blocks and characters for spetses |
| Organisation | University of Manchester |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | Prof. Radha Kessar (Manchester) and Gunter Malle (Kaiserslautern) and I have succeeded in generalising the the theory of spetses to the case of non-principal blocks. We have also defined character values for spetses using a formula of Curtis. Using the associated Yokonuma-Hecke algebras, we have shown that these values enjoy orthogonality properties in some cases. The general case of this conjecture is still ongoing. My contribution has been to make conjectures based on computational evidence and worked examples. I have also proved some results. |
| Collaborator Contribution | I have benefited enormously from my collaborators' extensive knowledge in representation theory, They were able to prove several of my conjectures, and streamline some existing proofs. |
| Impact | -Weights for compact connected Lie groups ( Manuscripta Mathematics (to appear) - Weights for $\ell$-local compact groups J. Algebra 636 (2023), 357-372 - The principal block of a $\ZZ_\ell$-spets and Yokonuma type algebras Algebra and Number Theory 17 (2023), 397-433 (joint with Radha Kessar and Gunter Malle) - Weight conjectures for $\ell$-compact groups and spetses (joint with Radha Kessar and Gunter Malle) Annales scientifiques de l'École normale supérieure (to appear) |
| Start Year | 2020 |
| Description | Weight conjectures for Parker--Semeraro fusion systems |
| Organisation | Manchester University |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | I was partially responsible for generalising the main result to all primes p; initially it was only stated for the prime 7. |
| Collaborator Contribution | We worked together on this project. |
| Impact | A paper "Weight conjectures for Parker--Semeraro fusion systems" was written and submitted to Journal of Pure and Applied Algebra |
| Start Year | 2024 |
| Description | Weight conjectures for Parker--Semeraro fusion systems |
| Organisation | Mimar Sinan Fine Arts University |
| Country | Turkey |
| Sector | Academic/University |
| PI Contribution | I was partially responsible for generalising the main result to all primes p; initially it was only stated for the prime 7. |
| Collaborator Contribution | We worked together on this project. |
| Impact | A paper "Weight conjectures for Parker--Semeraro fusion systems" was written and submitted to Journal of Pure and Applied Algebra |
| Start Year | 2024 |
| Description | Weight conjectures for Parker--Semeraro fusion systems |
| Organisation | University of Dresden |
| Country | Germany |
| Sector | Academic/University |
| PI Contribution | I was partially responsible for generalising the main result to all primes p; initially it was only stated for the prime 7. |
| Collaborator Contribution | We worked together on this project. |
| Impact | A paper "Weight conjectures for Parker--Semeraro fusion systems" was written and submitted to Journal of Pure and Applied Algebra |
| Start Year | 2024 |
| Description | Weights in a Benson Solomon block. |
| Organisation | University of Louisiana at Lafayette |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | I provided computational assistance to verify many of the statements claimed in the paper for small examples. This helped us prove several results and allowed us to locate several errors in the original manuscript of Aschbacher and Chermak. |
| Collaborator Contribution | Having worked with this fusion system in several other projects, Justin was able to provide key insights and invaluable expert knowledge. |
| Impact | Weights in a Benson-Solomon block, Forum of Mathematics, Sigma , Volume 11 , 2023 |
| Start Year | 2017 |
| Title | Fusion systems MAGMA package |
| Description | Chris Parker and I continue to maintain this package which is used by several academics working in the area. |
| Type Of Technology | Webtool/Application |
| Year Produced | 2024 |
| Open Source License? | Yes |
| Impact | The package has led to the discovery of several new and important families of saturated fusion systems. |
| Description | International workshop on Categorical and Geometric Representation Theory |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This conference brought together leading experts in geometric representation theory. It provided an excellent opportunity to learn about the subject of "categorification" which offers an alternative perspective on the theory of Hecke algebras, which take centre stage in this research project. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://sites.google.com/view/geometric-and-categorical-lms/home |
| Description | Invited talk: London Algebra Colloquium |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | Regional |
| Primary Audience | Professional Practitioners |
| Results and Impact | I was invited to speak at the London Algebra Colloquium - a prestigious 60 year old colloquium held as a collaborative venture among several London universities. My talk was entitled "The spectrum of the Chern subring". I spoke on recent work with Ian Leary (Southampton). |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://londonalgebra.wordpress.com/ |
| Description | National workshop |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | I gave a short talk at Victor Snaith's memorial conference entitled "Explicit Brauer induction for spetses". This resulted in several interesting questions from audience members, mostly number theorists and topologists. This broadened the reach of my work. I started a collaboration with Ian Leary, one of the conference organisers, after this collaboration. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://heilbronn.ac.uk/2022/07/25/remembering-vic-snaith/ |
| Description | Private dinner with University Executive Board and Fields medalist Prof. James Maynard |
| Form Of Engagement Activity | A formal working group, expert panel or dialogue |
| Part Of Official Scheme? | No |
| Geographic Reach | Local |
| Primary Audience | Professional Practitioners |
| Results and Impact | I was invited to attend a small gathering with Prof. Maynard following his address at the David Wallace Lecture in 2023. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://www.lboro.ac.uk/schools/science/events/2023/sir-david-wallace-lecture/ |
| Description | Research Talk at the Algebra Seminar, University of Birmingham |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | Anja Meyer gave a talk on her work in March 2025 |
| Year(s) Of Engagement Activity | 2025 |
| URL | https://www.birmingham.ac.uk/research/activity/mathematics/algebra/algebra-seminar |
| Description | Research Talk at the Algebra Seminar, University of Bristol |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | Anja Meyer gave a talk entitled "Finite matrix groups: Cohomology and stable elements" in November 2024. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://www.bristolmathsresearch.org/events/algebra/ |
| Description | Research Talk at the Algebra Seminar, University of Lincoln |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | Anja Meyer gave a talk entitled "Finite matrix groups: Cohomology and stable elements" in December 2024. |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://algebra-lincoln.org/2024/11/09/algebra-seminar-academic-year-2024-25/ |
| Description | Research Talk at the Transpennine Topology Conference |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Professional Practitioners |
| Results and Impact | Anja Meyer gave a talk entitled "Finite matrix groups: Cohomology and stable elements" in December 2024 |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://sarah-whitehouse.sites.sheffield.ac.uk/transalpine-topology-tetrahedron/ttt121-sheffield |
| Description | Research Talk at the University of Sheffield |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | Regional |
| Primary Audience | Professional Practitioners |
| Results and Impact | Below is an abstract for a talk I gave on 20th November 2024 in Sheffield. It generated fervent discussion among the attendees, and may lead to future collaboration. Abstract: For a prime p, the p-decomposition matrix D of a finite group G records the way each irreducible ordinary representation of G breaks up into irreducible p-Brauer characters under reduction modulo p. Multiplying D by its transpose yields the Cartan matrix, whose determinant is well-known to be a power of p. A representation of a Sylow p-subgroup S of G is fusion-stable if it is left invariant by the conjugation action of G. After first fixing a basis B of fusion-stable representations of S one can consider an analogue of D for fusion-stable representations which records how each irreducible ordinary representation of G breaks up in B under restriction to S. It turns out this matrix has many properties analogous to those of the classical decomposition matrix, and using them one can show that the modulus square of the determinant of the fusion-stable character table (columns indexed by G-classes of p-elements, rows by elements of B) is always a particular power of p independent of the choice of B. I conjectured that the same result holds for any saturated fusion system on S and I'll provide some evidence for this by explicitly computing with some infinite families of exotic examples. If time permits I will also explain how this project fits within the larger framework of "exotic representation theory" whose aim to exten |
| Year(s) Of Engagement Activity | 2024 |
| URL | https://researchseminars.org/seminar/SheffieldPureMaths |
| Description | Research visit (Birmingham) |
| Form Of Engagement Activity | A formal working group, expert panel or dialogue |
| Part Of Official Scheme? | No |
| Geographic Reach | Regional |
| Primary Audience | Professional Practitioners |
| Results and Impact | In April 2023, I spent 2 weeks working with 8 colleagues in Birmingham at a workshop entitled "Patterns in exotic fusion systems". We created a blueprint for a large research monograph which will classify fusion systems with a normal abelian essential subgroup. Some of these fusion systems are of Chevalley type, and therefore this research relates directly to the subject of the grant. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://heilbronn.ac.uk/2023/04/24/frg-fusion-systems/ |
| Description | Research visit (Kaiserslautern) |
| Form Of Engagement Activity | A formal working group, expert panel or dialogue |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | I spent 1 week in Kaiserslautern in January 2024, to work with Prof. Dr. Gunter Malle and his research group. |
| Year(s) Of Engagement Activity | 2024 |
| Description | Research visit (Oberwolfach) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | I visited and took residence in Oberwolfach in April 2023 (shortly after my grant started). This prestigious invite-only conference brought together leading experts from across Europe and beyond. My talk was entitled "Spetses from the p-local perspective" and stimulated much disucssion. |
| Year(s) Of Engagement Activity | 2023 |
| Description | School Visit (Loughborough) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | Local |
| Primary Audience | Professional Practitioners |
| Results and Impact | I gave a talk entitled "Sullivan spheres, Ariki-Koike algebras and the Drinfeld curve" at Loughborough shortly after I first arrived here. At the time, I knew little about others' work, and chose to focus the talk on the algebro-geometric aspects of my research. It provided me with an opportunity to showcase my work whilst isolating themes and opening up doors to future local collaborations. |
| Year(s) Of Engagement Activity | 2023 |
| Description | School Visit (Manchester) |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | Regional |
| Primary Audience | Postgraduate students |
| Results and Impact | I was invited to talk about my work at the Manchester Algebra Seminar. The talk, which was entitled "The spectrum of the Chern subring", was both well-attended and well-received. |
| Year(s) Of Engagement Activity | 2023 |
| Description | Women in (Co)homology conference |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | This is a conference organised by Anja Meyer (Loughborough) and Baylee Schutte (Aberdeen) in celebration of the 100 year anniversary of Emmy Noether's profound observations on homology. It features women and nonbinary mathematicians whose research builds on her foundational ideas. |
| Year(s) Of Engagement Activity | 2025 |
| URL | https://sites.google.com/view/whaconference/home |