Orthogonality, Orbits, and Spectra: Breaking Classical Barriers via Clifford Harmonic Structures

tarting from examples in graph theory and quantum computing, we show that seemingly natural and intuitive constructions can hide subtle pitfalls, and we address them by introducing a hybrid multigrade-and-rotor representation of Clifford algebra grounded in representation theory, with wide-ranging applications. As one illustrative consequence, the celebrated Hurwitz–Radon number—long regarded as an immovable barrier in the design of orthogonal space–time block codes—emerges as an artifact of vector-grade confinement rather than a fundamental limit; relaxing this confinement enlarges the feasible design space and suggests principled mechanisms for escaping classical orthogonality constraints. Building on this Clifford algebra representation foundation, we introduce the Clifford Harmonic Spectrum—a multigrade and geometric generalization of the asymptotic spectrum of Strassen. Finally, we define the Clifford Harmonic Entropy, which extends Shannon entropy and von Neumann entropy. The gradient and Hessian of the Clifford harmonic entropy functional yield a Clifford Fisher information metric and an entropy curvature tensor, and it leads to a general Entropy-Spectrum Duality principle.

This seminar is part of the Problems in Algebraic Geometry Coming from Complexity Theory series.