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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}.
This article describes a {em nonstandard} quantum group that may be used to derive a positive formula for the plethysm problem, just as the standard (Drinfeld-Jimbo) quantum group can be used to derive the positive Littlewood-Richardson rule for arbi
These are lectures notes for the introductory graduate courses on geometric complexity theory (GCT) in the computer science department, the university of Chicago. Part I consists of the lecture notes for the course given by the first author in the sp
Geometric complexity theory (GCT) is an approach to the P vs. NP and related problems. This article gives its complexity theoretic overview without assuming any background in algebraic geometry or representation theory.
Geometric complexity theory (GCT) is an approach to the $P$ vs. $NP$ and related problems. A high level overview of this research plan and the results obtained so far was presented in a series of three lectures in the Institute of Advanced study, Pri
We study a basic algorithmic problem in algebraic geometry, which we call NNL, of constructing a normalizing map as per Noethers Normalization Lemma. For general explicit varieties, as formally defined in this paper, we give a randomized polynomial-t