Every 2-periodic positive real mesh frieze of finite type A, D or E is constant, or lies in a one- or two-parameter family, with the exact count depending on the type.
Existence and rotatability of the two-colored Jones-Wenzl projector
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abstract
The two-colored Temperley-Lieb algebra $2\mathrm{TL}_R({}_s {n})$ is a generalization of the Temperley-Lieb algebra. The analogous two-colored Jones-Wenzl projector $\mathrm{JW}_R({}_s {n}) \in 2\mathrm{TL}_R({}_s {n})$ plays an important role in the Elias-Williamson construction of the diagrammatic Hecke category. We give conditions for the existence and rotatability of $\mathrm{JW}_R({}_s {n})$ in terms of the invertibility and vanishing of certain two-colored quantum binomial coefficients. As a consequence, we prove that Abe's category of Soergel bimodules is equivalent to the diagrammatic Hecke category in complete generality.
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2-periodic frieze patterns
Every 2-periodic positive real mesh frieze of finite type A, D or E is constant, or lies in a one- or two-parameter family, with the exact count depending on the type.