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# Representations and invariants of reductive groups in prime characteristic.

PGR-P-1131

## Key facts

Type of research degree
PhD
Application deadline
Ongoing deadline
Country eligibility
International (open to all nationalities, including the UK)
Funding
Competition funded
Supervisors
Dr Rudolf Tange
Schools
School of Mathematics
Research groups/institutes
Algebra, geometry and integrable systems, Pure Mathematics
<h2 class="heading hide-accessible">Summary</h2>

Representations of reductive groups are of vital importance in Lie Theory.<br /> Below we consider reductive groups G (e.g. GL_n or Sp_n) over a field k of prime characteristic.<br /> <br /> 1) See [4],[5],[6].<br /> The aforementioned representations often arise from reductive group actions on varieties. An important example is the adjoint action of a reductive group on its Lie algebra (for GL_n this is just conjugation). This leads to the adjoint representation on<br /> the algebra of functions k[g] on the Lie algebra g of G. The invariants are well-known and described by the Chevalley Restriction Theorem and the description of the Weyl group invariants on the Cartan subalgebra.<br /> <br /> One problem you could work on is a truncated version of this: You factor out the ideal I generated by the p-th powers of the linear functions and try to describe the invariants on k[g]/I. This is directly related to the centre of the restricted enveloping algebra, a finite dimensional associative algebra whose representations are the same as the restricted representations of g.Even for G=GL_n this is quite a nontrivial problem.<br /> <br /> 2) See [1], [2], [3].<br /> A famous open problem is the description of the decomposition numbers of a reductive group in characteristic p. These are the Jordan-Hoelder multiplicities of the irreducibles in certain standard modules which can be obtained from characteristic 0 by reduction mod p. If you want to work on this problem, you will have to work with a specific reductive group (e.g. GL_n or Sp_n) and make some assumptions on p.<br /> <br />

<h2 class="heading hide-accessible">Full description</h2>

<p>Literature:<br /> [1] R. Tange, Injective and tilting resolutions for induced modules, Preprint.<br /> [2] R. Tange, A combinatorial translation principle and diagram combinatorics for the general linear group, to appear in Transform. Groups.<br /> [3] H. Li and R. Tange, A combinatorial translation principle and diagram combinatorics for the symplectic group, to appear in Transform. Groups.<br /> [4] R. Tange, Highest weight vectors and transmutation, Transform. Groups 24 (2019), no 2, 563-588.<br /> [5] A. Dent and R.Tange, Bases for spaces of highest weight vectors in arbitrary characteristic, Algebr. Represent. Theory 22 (2019), no. 5, 1133-1147.&nbsp;<br /> [6] The Zassenhaus variety of a reductive Lie algebra in positive characteristic, Adv. in Math. 224 (2010), no. 1, 340-354.</p>

<h2 class="heading">How to apply</h2>

<p>Formal applications for research degree study should be made online through the&nbsp;<a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University&#39;s website</a>. Please state clearly in the Planned Course of Study that you are applying for <em><strong>PHD Pure Mathematics FT</strong></em> and in the research information section&nbsp;that the research degree you wish to be considered for is <em><strong>Representations and invariants of reductive groups in prime characteristic</strong></em> as well as <a href="https://eps.leeds.ac.uk/maths/staff/4083/dr-rudolf-tange">Dr Rudolf Tange</a> as your proposed supervisor.</p> <p>If English is not your first language, you must provide evidence that you meet the University&#39;s minimum English language requirements (below).</p> <p style="margin-bottom:11px"><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p> <p class="MsoNoSpacing">Applications will be considered on an ongoing basis. &nbsp;Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application. &nbsp;We also advise that you apply at the earliest opportunity as the application and selection process may close early, should we receive a sufficient number of applications or that a suitable candidate is appointed.</p> <p>Please note that you must provide the following documents at the point you submit your application:</p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> <li>Funding information including any alternative sources of funding that you are applying for or if you are able to pay your own fees and maintenance</li> </ul> <p>&nbsp;</p>

<h2 class="heading heading--sm">Entry requirements</h2>

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

<h2 class="heading heading--sm">English language requirements</h2>

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.

<h2 class="heading">Funding on offer</h2>

<p style="margin-bottom:12px"><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>UK</strong>&nbsp;&ndash;&nbsp;The&nbsp;<a href="https://phd.leeds.ac.uk/funding/209-leeds-doctoral-scholarships-2022">Leeds Doctoral Scholarships</a> and <a href="https://phd.leeds.ac.uk/funding/234-leeds-opportunity-research-scholarship-2022">Leeds Opportunity Research Scholarship</a> are available to UK applicants (open from October 2023). <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Non-UK</strong> &ndash;The&nbsp;<a href="https://phd.leeds.ac.uk/funding/48-china-scholarship-council-university-of-leeds-scholarships-2021">China Scholarship Council - University of Leeds Scholarship</a>&nbsp;is available to nationals of China (open from October 2023). The&nbsp;<a href="https://phd.leeds.ac.uk/funding/73-leeds-marshall-scholarship">Leeds Marshall Scholarship</a>&nbsp;is available to support US citizens. <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Important:</strong>&nbsp; Any costs associated with your arrival at the University of Leeds to start your PhD including flights, immigration health surcharge/medical insurance and Visa costs are <strong>not </strong>covered under these studentships.</p> <p>Please refer to the <a href="https://www.ukcisa.org.uk/">UKCISA</a> website for information regarding Fee Status for Non-UK Nationals.</p>

<h2 class="heading">Contact details</h2>

<p>For general enquiries about applications, contact our Doctoral College Admissions team:<br /> e:&nbsp;<a href="mailto:maps.pgr.admissions@leeds.ac.uk">maps.pgr.admissions@leeds.ac.uk</a></p> <p>For questions about the research project, contact Dr Rudolf Tange: e:&nbsp;<a href="mailto:R.H.Tange@leeds.ac.uk">R.H.Tange@leeds.ac.uk</a></p>

<h3 class="heading heading--sm">Linked research areas</h3>