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

This paper continues the Geometric Complexity Theory (GCT) program by addressing a fundamental gap left open in GCT4 and GCT7: the construction of canonical bases for the “nonstandard” quantum groups that arise in the representation‑theoretic formulation of complexity‑theoretic problems. The authors propose conjectural, algorithmic procedures for three intertwined algebraic objects: (1) the irreducible polynomial representations of the nonstandard quantum group (G_q^{\text{ns}}), (2) the matrix coordinate ring (\mathcal{O}_q^{\text{ns}}(G)) of the same group, and (3) the dually paired nonstandard deformation (\mathcal{H}_q^{\text{ns}}(S_n)) of the symmetric‑group algebra (a nonstandard Hecke algebra).

The first algorithm generalises Kashiwara’s crystal‑basis construction. Starting from a highest‑weight vector, a “nonstandard bar involution” (\tilde{B}) is defined using the nonstandard (R)‑matrix. Repeated application of the quantum root operators together with (\tilde{B}) yields a triangular transition matrix whose diagonal entries are 1. The resulting basis ({G_\lambda}) is conjectured to be canonical: it is invariant under (\tilde{B}), stable under the action of the quantum group, and its structure constants are positive integer polynomials.

The second algorithm builds a canonical basis for the coordinate ring. The nonstandard quantum matrix entries (X_{ij}) are ordered according to a nonstandard PBW (Poincaré–Birkhoff–Witt) rule. Multiplication of these generators is expressed recursively, and at each step a normalisation factor is introduced. The conjecture is that every such factor is a (#P)-computable positive integer polynomial, guaranteeing that the basis elements have a triangular expansion in the PBW monomials.

The third algorithm constructs a Kazhdan–Lusztig–type basis for the nonstandard Hecke algebra. By exploiting the duality between (G_q^{\text{ns}}) and (\mathcal{H}q^{\text{ns}}(S_n)), the authors define nonstandard KL polynomials (P^{\text{ns}}{x,y}(q)) that mirror the classical ones. These polynomials appear as the transition coefficients between the canonical bases of the coordinate ring and the Hecke algebra. The authors verify, by extensive computer experiments for small (n) and low degree, that the three algorithms produce mutually compatible bases and that the transition matrices are indeed triangular with positive integer entries.

A central theme of the paper is the positivity (or (#P)) hypothesis: all structure constants arising in the three constructions are assumed to be positive integer polynomials that can be evaluated in (#P). Under this hypothesis, together with the duality and reciprocity conjectures formulated in GCT7, the canonical bases satisfy a remarkable compatibility: the transition matrix from the canonical basis of the coordinate ring to the canonical basis of the Hecke algebra coincides with the matrix of nonstandard KL polynomials evaluated at (q=1).

This compatibility yields a (#P)-formula for plethysm constants. Plethysm constants (a^{\mu,\nu}\lambda) describe the multiplicity of an irreducible representation (V\lambda) in the composition (V_\mu\circ V_\nu). By expressing these constants as a sum over the symmetric group of products of canonical‑basis transition coefficients and nonstandard KL evaluations, the authors obtain an explicit expression that is a sum of (#P)-computable non‑negative integers. Consequently, the plethysm problem, already known to be (#P)-hard, is shown to admit a concrete (#P)-formula derived from the canonical‑basis framework.

The paper concludes with a discussion of open problems. The most pressing is a proof of the positivity hypothesis, which would likely require a deeper understanding of the representation theory of the nonstandard (R)-matrix and its categorification. Further work is needed to extend the computational verification to larger ranks and higher degrees, to analyse the complexity of the proposed algorithms, and to explore applications such as non‑standard quantum‑group based zero‑knowledge proof systems and potential connections to quantum circuit lower bounds. The authors provide GAP/Sage scripts and data tables in an appendix, facilitating reproducibility and encouraging the community to test and refine the conjectures.

In summary, the article offers the first systematic proposal for canonical bases in the nonstandard quantum‑group setting, ties these bases to a nonstandard Kazhdan–Lusztig theory, and leverages their conjectural properties to produce a (#P)-formula for plethysm constants—thereby deepening the bridge between algebraic representation theory and computational complexity.