q-ANALOGUES OF PROJECTIVE REPRESENTATIONS OF SYMMETRIC GROUPS


Dr Maxim Nazarov, University of York


The classical Young symmetrizers (1901) are certain distinguished idempotents in the group ring of the symmetric group Sn where n is any positive integer. Under the action of the group Sn on the nth tensor power of the complex N-dimensional vector space by permutations of the tensor factors, the images of these idempotents provide explicit realizations of irreducible polynomial representations of the general linear Lie group GLN . A new approach to the Young symmetrizers was proposed by Cherednik (1986), it was motivated by the representation theory of affine quantum groups, and is based on a certain limiting process called fusion procedure. That approach was further developed by Nazarov (1997). In particular, that development unveiled the counterparts of the Young symmetrizers for a non-trivial covering of the group Sn . This covering group was defined by Schur (1911), it linearizes projective representations of the group Sn over the complex field.

The first aim of the present project was to investigate the properties of these "projective" symmetrizers, and of their q-analogues constructed by Jones and Nazarov (1999). In partucular, it is now proved that these q-analogues are actually idempotents, after a suitable normalization. This idempotency property was only conjectured by Jones and Nazarov. The second aim of this project was to construct the counterparts of the Young symmetrizers in the Brauer centralizer algebra (1937). Existence of these counterparts was surmised by Weyl; this aim has been achieved by building on his results (1939). This construction provided new information about the irreducible polynomial representations of the orthogonal and symplectic Lie groups ON and SpN . The third aim of the project was to construct the counterparts of Young symmetrizers in the so called "walled" Brauer algebra. These counterparts provide explicit realizations of the irreducible rational representations of the group GLN , thus extending the classical construction of Young to its natural limits. Research esults of this project are available from the web archive Mathematics, see pages


EPSRC Grant and Duration: 2475 over 36 months from 1.4.2000, reference GR/N34888


For further information please contact: Dr Maxim Nazarov, Department of Mathematics, University of York, York YO10 5DD, England; maxim.nazarov@york.ac.uk