# TR-2007-15

## Geometric Complexity Theory: On canonical bases for the nonstandard quantum groups

Ketan D. Mulmuley. 4 September, 2007.
Communicated by Ketan Mulmuley.
Obsolete: Yes (updated 12/19/12)

### Abstract

This article gives conjecturally correct algorithms to construct canonical bases of the irreducible polynomial representations and the matrix coordinate rings of the nonstandard quantum groups in GCT4 and GCT7, and canonical bases of the dually paired nonstandard deformations of the symmetric group algebra therein. These are generalizations of the canonical bases of the irreducible polynomial representations and the matrix coordinate ring of the standard quantum group, as constructed by Kashiwara and Lusztig, and the Kazhdan-Lusztig basis of the Hecke algebra. A positive (\$\#P\$-) formula for the well-known plethysm constants follows from their conjectural properties and the duality and reciprocity conjectures in \cite{GCT7}.

### Original Document

The original document is available in Postscript (uploaded 4 September, 2007 by Ketan Mulmuley).