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

Orthogonality is a cornerstone of algebra, geometry, and complexity theory—yet classical bounds, such as the Hurwitz–Radon (HR) theorem, impose rigid limits on the number of mutually orthogonal structures in real vector spaces. These constraints permeate fundamental problems; for example, they cap the number of pairwise orthogonal subspaces that can be used to construct block-diagonal tensor decompositions, thereby limiting the design of fast matrix multiplication algorithms and hindering progress in bounding tensor rank, border rank, and related complexity measures.

In this talk, I will present a new paradigm grounded in Clifford algebra, the Clifford group, and Clifford multigrade harmonic analysis—offering a unified algebraic and spectral framework that transcends classical orthogonality constraints such as the HR bound. By exploiting the multigrade decomposition inherent to Clifford algebras, we introduce a notion of subspace orthogonality that extends beyond vector-level independence, enabling the construction of high-dimensional mutually orthogonal systems not accessible through classical means. Combined with orbit structures arising from Clifford group actions, this framework yields explicit families of tensors and operator systems with enhanced algebraic expressiveness and richer spectral profiles. Drawing on methods from Clifford representation theory, Fourier analysis, coding theory, and geometric complexity, the approach aims to provide a unifying lens for problems spanning different domains.

These constructions culminate in the Clifford Harmonic Spectrum—a spectral framework that generalizes the asymptotic spectrum of tensors. In contrast to classical spectra rooted in vector rank and semiring monotonicity, the Clifford harmonic spectrum captures grade-separated entropy, noncommutative alignment, and subspace orthogonality, with potential applications to matrix multiplication and fine-grained reductions, Shannon and Sperner capacity in communication theory, algebraic complexity and circuit lower bounds, quantum error-correcting codes, and beyond.

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