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

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}.

71 pages, changed content

References in corpus (2)

• Geometric Complexity Theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry
• Braided Symmetric and Exterior Algebras

Cited by in corpus (5)

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• Geometric Complexity Theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry
• On P vs. NP, Geometric Complexity Theory, and the Riemann Hypothesis
• Representation theory of the nonstandard Hecke algebra
• On P vs. NP, Geometric Complexity Theory, Explicit Proofs and the Complexity Barrier