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

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}.

💡 Research Summary

The paper “Geometric Complexity Theory VIII: On canonical bases for the nonstandard quantum groups” proposes conjectural algorithms for constructing canonical bases of two intertwined objects that arise in the Geometric Complexity Theory (GCT) program: the irreducible polynomial representations and the matrix coordinate ring of the nonstandard quantum groups introduced in GCT 4 and GCT 7, and the dually paired nonstandard deformations of the symmetric group algebra. The motivation is the long‑standing problem of finding a positive #P‑type formula for plethysm constants a_{π,λ,μ}, the multiplicities of an H‑module V_π inside a G‑module V_λ when H embeds into G via a representation ρ.

Standard background. In the classical (standard) setting, Kashiwara and Lusztig constructed canonical bases for the coordinate ring O(M_q(V)) of the quantum matrix space and for the quantum group GL_q(V) by means of a “balanced triple” (V_Q, L_0, L_∞) consisting of a Q