Geometric Complexity Theory VII: Nonstandard quantum group for the plethysm problem

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 arbitrary complex semisimple Lie groups. The sequel \cite{GCT8} gives conjecturally correct algorithms to construct canonical bases of the coordinate rings of these nonstandard quantum groups and canonical bases of the dually paired nonstandard deformations of the symmetric group algebra. A positive $#P$-formula for the plethysm constant follows from the conjectural properties of these canonical bases and the duality and reciprocity conjectures herein.

💡 Research Summary

The paper “Geometric Complexity Theory VII: Nonstandard quantum group for the plethysm problem” introduces a novel algebraic framework aimed at providing a positive combinatorial formula for plethysm constants—an outstanding open problem in algebraic combinatorics and representation theory. The authors position this work within the broader Geometric Complexity Theory (GCT) program, which seeks to connect complexity‑theoretic lower bounds with deep structural properties of algebraic varieties and representation‑theoretic objects.

Background and Motivation
In the representation theory of complex semisimple Lie groups, the Littlewood‑Richardson (LR) rule gives a positive integer description of tensor product multiplicities. This rule can be derived from the standard Drinfeld‑Jimbo quantum group via canonical bases (Lusztig, Kashiwara). By contrast, plethysm constants (p_{\lambda}^{\mu,\nu})—the multiplicities occurring when one symmetric function is composed with another—have resisted any known positive combinatorial description. Existing results only establish that computing plethysm is #P‑hard, but they do not provide a constructive, sign‑free formula.

Nonstandard Quantum Group Construction
To bridge this gap, the authors define a “nonstandard quantum group” (G_q^{\text{ns}}). This object differs from the standard quantum group in several crucial ways:

1. R‑matrix deformation – The universal R‑matrix is enriched with extra parameters that encode the plethystic composition rather than a simple braiding.
2. Coordinate algebra – The coordinate ring (\mathcal{O}_q(G^{\text{ns}})) is equipped with a Hopf algebra structure whose comultiplication reflects the plethysm operation.
3. Dual nonstandard symmetric‑group algebra – Simultaneously, a quantum deformation (\mathbb{C}_q^{\text{ns}}