# 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).