Engineering and Physical Sciences Research
Polaris House, North Star Avenue, Swindon, Wiltshire,
United Kingdom SN2 1ET
Telephone +44 (0) 1793 444000. Internet http://www.epsrc.ac.uk
FINAL REPORT FORM (Fellowships)
Grant Reference
GR/R44102/01
Organisation
UMIST
FELLOWSHIP DETAILS
Title
Dr
Forename(s)
Sarah
Surname
Perkins
Organisation
Birkbeck College
Division or Department
Sch of Economics, Mathematics and Stats
PROJECT DETAILS
Title
Coxeter Groups and Length Functions
Funds Awarded (£)
63,435
Fellowship Support Fund (£)
Total Grant Value (£)
-
Start Date:
01/02/2002
End Date:
31/01/2004
63,435
OBJECTIVES
Original objectives as presented in the proposal
N/A
These objectives remained valid throughout the Fellowship (tick box), or
These objectives were changed (tick box).
NX0219FR v1.0
(if so please explain how/why in your report)
Page 1 of 7
COMPLIANCE WITH THE DATA PROTECTION ACT 1998
In accordance with the Data Protection Act 1998, the personal data provided on this form will be processed by EPSRC, and may be held on computerised database and/or manual
files. Further details may be found in the guidance notes.
New objectives
FELLOWSHIP SUMMARY
Fellowship summary at time of proposal
The length function is at the core of what it means to be a Coxeter Group. I shall investigate three main strands of interest in this area.
There are known results for Weyl groups about the way in which the lengths of elements may be altered by conjugation. I will look at
ways of extending such results, for example to include affine Coxeter groups.
I have recently developed a notion of length for subsets of Coxeter groups. I have obtained some fascinating results, in the cases
where the subsets are, for example, parabolic subgroups and conjugacy classes. It is clear from my initial work that this is a fertile area
for research. I intend to delve deeper into the emergent structure and explore more fully some of the patterns that are appearing.
I intend to work on a range of problems, viewing Coxeter groups from a wider perspective. I have been invited to attend a workshop on
open problems relating to the Macdonald polynomials, which will allow me to extend my knowledge of current research relating to the
representation theory of Lie groups. My existing expertised in Coxeter groups can then be applied to problems in this broader context.
Summary of Outcomes: In simple terms describe your work in such a way that it could be publicised to a general audience.
I have been able to carry out an extensive investigation into the notion of length of subsets of Coxeter groups. Certain subsets, known
as parabolic subgroups, play a vital role in Coxeter groups, and for these sets, a comprehensive description of their lengths has been
found. Another type of subset that has been considered is cosets. These have been found to yield interesting partially ordered sets
which can be viewed as a generalisation of the important Bruhat ordering on Coxeter groups.
A further question was that of the action of conjugacy classes on the root system of Coxeter groups. An intriguing pattern had been
observed in crystallographic Coxeter groups, relating to the number of negative roots in the image of a given positive root under the
action of a given conjugacy class. This pattern was explained and was proved to hold for all finite crystallographic Coxeter groups.
The project has provided a solid base for further investigation into lengths of subsets of Coxeter groups.
Web address with further details if you wish:
BENEFICIARIES
Intended beneficiaries at time of proposal
N/A
NX0219FR v1.0
Page 2 of 7
COMPLIANCE WITH THE DATA PROTECTION ACT 1998
In accordance with the Data Protection Act 1998, the personal data provided on this form will be processed by EPSRC, and may be held on computerised database and/or manual
files. Further details may be found in the guidance notes.
OUTCOMES OF THE FELLOWSHIP
Major benefits of EPSRC Fellowships include i) the potential to impact positively on the careers of individual Fellows and
ii) the research results.
FELLOW
Details of career advancement that may be attributed to the Fellowship – please ensure that this aspect is addressed in the narrative
Review Report that must accompany this form. In particular, if your career progression during the Fellowship included one or more
moves into new employment, please provide brief details, listing employer and position held.
NX0219FR v1.0
Page 3 of 7
COMPLIANCE WITH THE DATA PROTECTION ACT 1998
In accordance with the Data Protection Act 1998, the personal data provided on this form will be processed by EPSRC, and may be held on computerised database and/or manual
files. Further details may be found in the guidance notes.
RESEARCH RESULTS
Publications
(1) Please state how many research results arose principally as a result of the research funded through this grant and indicate the type (journal, conference,
book, patent, software, other)
JOURNAL
CONFERENCE
PAPERS
Total Number of Publications
2
1
Number of Refereed Publications (if
different from above)
2
BOOK
PATENT
SOFTWARE
OTHER
5
Number of Publications That
Contain An Industrial Co-Author
Number of Publications That
Contain An International Co-Author
(2) Please list up to five significant publications that arose principally as a result of the research funded through this grant and indicate the type (journal,
TITLE
TYPE
Journal/Conf.
year
vol.
page
Dr SB Perkins; Professor PJ Rowley
Coxeter Length
J. Algebra
2004
273,
no.1
344358
Journal
Dr SB Perkins; Professor PJ Rowley
On Negative Orbits of Finite Coxeter Groups
J. Alg. Comb.
2004
to
appear
to
appear
Journal
NX0219FR v1.0
Page 4 of 7
COMPLIANCE WITH THE DATA PROTECTION ACT 1998
In accordance with the Data Protection Act 1998, the personal data provided on this form will be processed by EPSRC, and may be held on computerised database and/or manual files. Further details may be found in the guidance notes.
9
International
co-author?
AUTHOR(S)
Industrial
co-author?
REFERENCE
Refereed?
conference, book, patent, software, other); whether there was an international or industrial co-author and whether the publication was refereed.
Other Results of the Fellowship
Please indicate the timescale for potential exploitation (e.g. achieved, expected within 3 years, expected within
10 years, none foreseeable in 10 years) and include patent nos., dates, company names and addresses etc. as
appropriate, in the ‘details’ and ‘reference’ columns.
DETAILS
REFERENCE
TIMESCALE
Licences or Patents
Spin Off Company
Direct Consultancy
Industrial Training
Courses
Public Understanding
of Science and
Engineering
I hope to increase my opportunities to speak about my area of
research to the lay public. I have already given several talks to
schoolchildren about mathematics, as well as a poster
presentation at the British Association Festival and recently a
(very!) brief TV slot which was a direct outcome of a talk given to
a group of mathematics schoolteachers.
Expected within
3 years
Other (please specify)
Related Support Awarded / Pending
Please indicate the level of research support that has arisen principally as a result of the Fellowship.
FUNDING SOURCE
DETAILS
SUPPORT (£)
EPSRC (from start of
Fellowship)
Other UK Research
Council
Other UK Government
UK Industrial
UK Other
European Commission
Other Industrial
Other Non-Industrial
Australian Research Council partially funded my visit to Sydney University to
work with Professor RB Howlett
2,000
Pending
I have applied to the Nuffield Foundation for an award for New Lecturers
5,000
NX0219FR v1.0
Page 5 of 7
COMPLIANCE WITH THE DATA PROTECTION ACT 1998
In accordance with the Data Protection Act 1998, the personal data provided on this form will be processed by EPSRC, and may be held on computerised database and/or manual files.
Further details may be found in the guidance notes.
GRANT ADDITIONAL/SPECIFIC CONDITIONS AND INFORMATION
Additional and Specific Conditions/Information
SCHEME CONDITIONS
GC1
General Arrangements
GC10
Use of Equipment
GC11
Expenditure Statements
GC12
Individual Grant Review
GC13
Sanctions
GC2
Use of Funds
GC3
Payment Arrangements
GC5
Extensions
GC6
Claims for Large Capital
GC7
Transfer Between Headings
GC8
Staff and Indirect Costs
GC9
Procurement of equipment
SCHEME INFORMATION
www1
Standard Grant Condition GC19 requires Investigators to publish the results of their
EPSRC funded research. In this context EPSRC considers that the World Wide Web
(WWW) is one of the most effective information dissemination mechanisms available.
Therefore, Investigators are encouraged to place a summary of their Individual Grant
Review (IGR) Report on their own web site at the same time as the IGR is submitted to
EPSRC.
Signatures:
Fellow
Head of Department
Administration Authority
( electronic submission )
( electronic submission )
( electronic submission )
Date:
Date:
Date:
COMPLIANCE WITH THE DATA PROTECTION ACT 1998
In accordance with the Data Protection Act 1998, the personal data provided on this form will be processed by EPSRC, and may be held
on computerised databases and/or manual files.
This information will be used for administration, management purposes and statutory returns, and contact information may be published
on the EPSRC external web site alongside details of grants/awards made. The information may also be made available to organisers of
EPSRC related courses/conferences.
The information held will be retained for no longer than necessary.
If you have any queries regarding this statement please contact the Data Protection Officer for EPSRC, telephone 01793/444100.
NX0219FR v1.0
Page 6 of 7
COMPLIANCE WITH THE DATA PROTECTION ACT 1998
In accordance with the Data Protection Act 1998, the personal data provided on this form will be processed by EPSRC, and may be held on computerised database and/or manual files.
Further details may be found in the guidance notes.
FINAL REPORT ASSESSMENT (Fellowships)
Grant Reference: GR/R44102/01
Project Title: Coxeter Groups and Length Functions
This sheet is an integral part of the Final Report
(Fellowships) Form and should be returned with it.
Fellow: Dr Sarah Perkins
Organisation: Birkbeck College
Please note that your self-assessment form will be forwarded to the external assessors. If, in the context of this particular grant, you
consider one or more of the assessment criteria “not applicable” please tick the relevant box. Please indicate, in the box provided,
your perception of the importance of each assessment criteria in relation to the objectives of the project
(H=High, M=Medium, L=Low).
Assessment
Not
Applicable
Unsatisfactory
Tending to
Í
National
standing
Tending to
Î
Internationally
leading
Relevance to
the Project
H/M/L
Criterion
Research Quality
9
High
Research Planning
9
Medium
and Practice
Potential Scientific
9
Impact
Assessment
Not
Applicable
Unsatisfactory
Tending to
Good
Í
to Society
Public Awareness
Role Fulfilment
Outstanding
9
Prospects / Plans
Potential Benefits
Tending to
Î
Criterion
Impact of the
Fellowship on your
Your Standing in
the Research
Community
Medium
Relevance to
the Project
H/M/L
High
9
High
9
Medium
9
Medium
Please highlight, in your narrative report, what you consider to be the single most valuable outcome of the Fellowship.
NX0219FR v1.0
Page 7 of 7
COMPLIANCE WITH THE DATA PROTECTION ACT 1998
In accordance with the Data Protection Act 1998, the personal data provided on this form will be processed by EPSRC, and may be held on computerised database and/or manual files.
Further details may be found in the guidance notes.