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*q*-ANALOGUES OF PROJECTIVE REPRESENTATIONS OF SYMMETRIC GROUPS

The classical Young symmetrizers (1901) are certain distinguished idempotents
in the group ring of the symmetric group *S*_{n}
where *n* is any positive integer. Under the action
of the group *S*_{n} on the *n*th 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 *GL*_{N} . 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 *S*_{n} . This covering group was defined
by Schur (1911), it linearizes projective representations
of the group *S*_{n} 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 *O*_{N} and *Sp*_{N} .
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 *GL*_{N} ,
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`