Jon Cheah | Searcharxiv

arXiv2026

Frayed Demazure weaves for Poisson-compatible cluster structures on Bott--Samelson charts

Demazure weaves are combinatorial representations of maps between Bott--Samelson cells and have been used to construct cluster structures on braid varieties. We show the compatibility of these maps and the resulting cluster structures with the standard Poisson structure on the Bott--Samelson variety. Adding frayed strands to Demazure weaves, we further construct Poisson compatible cluster structures on other affine charts of the Bott--Samelson variety in a manner that transition functions across charts become rational quasi-cluster. The mutation sequences we construct for these quasi-cluster morphisms are closely related to those of M\'enard for open Richardson seeds.

math.CO↗

arXiv2024

On upper bounds of frieze patterns

In this note, we show that the sequence of maximum values in frieze patterns of type $A_n$ is the sequence of Fibonacci numbers, and that of frieze patterns of type $C_n$ is the sequence of odd Fibonacci numbers.

math.CO↗

arXiv2022

Real McKay Correspondence: KR-Theory of Graded Kleinian Groups

This project considers the finite symmetry subgroups of the orthogonal group $\mathrm{O}(3) \subset \mathrm{GL}(3,\mathbb{R})$ and the index $2$ containments $G\lhd \widehat{G}$. The special orthogonal group $\mathrm{SO}(3) \subset \mathrm{SL}(3,\mathbb{R})$ admits a double cover from the spinor group $\mathrm{Spin}(3) \cong \mathrm{SU}(2) \subset \mathrm{SL}(2,\mathbb{C})$, and lifting our subgroups up preserves the network of containments. Those subgroups not contained in $\mathrm{SO}(3) \subset \mathrm{O}(3) $ are lifted to the pinor groups $\mathrm{Pin}_{\pm}(3)$ of which there are two choices. For the index $2$ containments $G\lhd \widehat{G}$, we calculate the Real and complex Frobenius-Schur indicators, and apply Dyson's classification of antilinear block structures to produce decorated McKay graphs for each case. We then explore $KR$-theory as introduced by Atiyah in 1966, which is a variant of topological $K$-theory for working with topological spaces equipped with an involution. The GIT quotient spaces $\mathbb{C}^2 // G$, can be equipped by an involution via the action of $\widehat{G} / G$. In 1983, Gonzalez-Sprinberg and Verdier showed how one can view the McKay correspondence as an isomorphism between the $G$-equivariant $K$-theory $K_G(\mathbb{C}^2)$ and the $K$-theory of the minimal resolution of the singularity $\widetilde{\mathbb{C}^2 // G}$. We use this to conjecture an analogous a form of the McKay correspondence for $C_2$-graded groups and $KR$-theory.

math.RT↗