The Dual Canonical Basis in the Spin Representation via the Temperley-Lieb Algebra

Overview

The spin representation is a mathematical object parametrizing the spin states of multiple-electron systems, introduced by Cartan in 1913 and applied to physics by Dirac in 1928. It has a dual canonical basis introduced by Lusztig that is important in different areas of mathematics, including algebraic geometry, cluster algebras, total positivity, and categorification. This basis was previously defined in an axiomatic way, with no insight on its structure or computation. We prove that the dual canonical basis of the spin representation can be visualized diagrammatically through the Temperley-Lieb algebra, so each element of the basis can be viewed explicitly as a diagram. We demonstrate that our visualization is compatible by reproving some classical results through our new perspective. We then utilize our visualization to develop the first explicit formula to compute the dual canonical basis. We prove other new results on the spherical and aspherical modules of the Hecke algebra, a related object, and also come up with a new axiomatic definition of the canonical basis, allowing us to view the canonical basis diagrammatically. Finally, we compute the explicit decomposition of the spin representation into the irreducible Specht modules, which is only possible due to our diagrammatic visualization, allowing us to view multiple-electron systems explicitly as single-particle systems. Our perspective allows us to work with the spin representation in a new way, through a diagrammatic basis, and our new results give a better understanding of the spin representation, accelerating the quantum computation of spin systems.

Competition history

• CSEF 2026 Mathematical Sciences (Senior Division) · Entry S-14-08

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