EPSRC Reference: 
GR/M09506/01 
Title: 
CANONICAL BASES,TOTAL POSITIVITY AND THE STRUCTURE OF ALGEBRAIC GROUPS 
Principal Investigator: 
Rietsch, Professor K 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Pure Maths and Mathematical Statistics 
Organisation: 
University of Cambridge 
Scheme: 
Postdoc Res Fellowship PreFEC 
Starts: 
25 April 1999 
Ends: 
24 April 2001 
Value (£): 
61,423

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
My research is in the area of algebraic groups. I propose to study the newly discovered structures intrinsic to reductive groups, coming from the deep theory of positivity for canonical bases of quantized universal enveloping algebras. For flag varieties, for example, these ideas give rise to a remarkable polyhedral subspace (see [4]). Its faces are closely related to the well known Bruhat decompositions. I have conjectured characterisation of these cells in terms of canonical bases, generalising a theorem of Lusztigs. This conjecture would also imply new symmetry properties.One further project I want to pursue concerns intersections of Bruhat cells. In [2, 3] some questions regarding their topology are answered. They give rise to interesting combinatorics, related to canonical bases. I have some ideas for extracting more detailed topological information, thus adding more detail to the combinatorial picture.Returning to canonical bases and positivity, I would like to study one particular subalgebra of a universal enveloping algebra that has a basis coming from a completely different theory. By V. Ginzburg and Dale Peterson it has identified with the cohomology ring of an affine partial flag variety, therefore has a natural Schubert basis. Peterson has recently identified the structure constants as GromovWitten invariants, thus positive integers (via the theory of quantum cohomology for flag varieties). One goal of my research would be to relate this basis with the canonical basis.

Key Findings 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Potential use in nonacademic contexts 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Impacts 
Description 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk 
Summary 

Date Materialised 


Sectors submitted by the Researcher 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Project URL: 

Further Information: 

Organisation Website: 
http://www.cam.ac.uk 