Rachel Chen, 18, wins $100,000 for diagrammatic extension of a 1997 quantum-math idea

Chen's award and project scope

Rachel Chen, 18, a student at Marlborough School in Los Angeles, California, placed fourth in the 2026 Regeneron Science Talent Search and received a $100,000 award for her mathematics project, titled "The Dual Canonical Basis in the Spin Representation via the Temperley-Lieb Algebra." The project extends a 1997 mathematical finding involving Temperley-Lieb diagrams, simple point-and-curve illustrations that encode aspects of quantum systems, to describe how a full collection of quantum particles behaves under a magnetic field, potentially giving researchers a more intuitive way to examine relationships between different quantum-mechanical states.

Extending Temperley-Lieb diagrams to spin representations

Chen focuses on the spin representation, a mathematical framework for describing the spin states of multiple particles. As she explains in her project presentation, a single electron has two possible spin states, up and down, and quantum superposition means its state can be represented as a combination of those possibilities rather than simply one or the other. For a system of n electrons, the spin representation is built from repeated tensor products of the complex plane, a structure that becomes increasingly hard to visualize as particle count grows. Chen asked whether that structure could be expressed more visually, and answered by demonstrating that Temperley-Lieb diagrams can serve as a basis for the spin representation, effectively providing building blocks through which the quantum system can be viewed, and by linking the diagrams to Lusztig's dual canonical basis.

A new explicit formula for the dual canonical basis

A central contribution of Chen's project is an explicit, non-inductive formula for the dual canonical basis of the spin representation, which her presentation notes had previously lacked such an expression. Using her diagrammatic approach, she developed what she described as the first explicit formula for that basis and showed consistency with known classical results before using the framework to derive new ones. Her research also examined the related Hecke algebra and supplied a new axiomatic description of the canonical basis of the spin representation.

Follow-up signals

The Regeneron Science Talent Search results, which placed Chen fourth among the 2026 finalists, are the most recent verifiable milestone in the published coverage; no further filings, publications, or hearings tied to this project are identified in the available reporting.

tags - Rachel Chen - Regeneron Science Talent Search - Temperley-Lieb algebra - Spin representation - Marlborough School