Daniil Sarafannikov | Searcharxiv

arXiv2026

Categorification of the genus two DAHA

We construct three derived endofunctors on the derived category of graded integrable representations of the current algebra of a flat degeneration of $D(2|1,α)$, and prove that their classes in the Grothendieck group are the genus two Macdonald operators. Computing the graded $\operatorname{Ext}$ pairing explicitly, we deduce the self-adjointness of the genus two Macdonald operators and the orthogonality of genus $2$ Macdonald polynomials with respect to the certain skew-bilinear form.

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arXiv2022

Skew Howe duality and q-Krawtchouk polynomial ensemble

We consider the decomposition into irreducible components of the exterior algebra $\bigwedge\left(\mathbb{C}^{n}\otimes \left(\mathbb{C}^{k}\right)^{*}\right)$ regarded as a $GL_{n}\times GL_{k}$ module. Irreducible $GL_{n}\times GL_{k}$ representations are parameterized by pairs of Young diagrams $(λ,\barλ')$, where $\barλ'$ is the complement conjugate diagram to $λ$ inside the $n\times k$ rectangle. We set the probability of a diagram as a normalized specialization of the character for the corresponding irreducible component. For the principal specialization we get the probability that is equal to the ratio of the $q$-dimension for the irreducible component over the $q$-dimension of the exterior algebra. We demonstrate that this probability distribution can be described by the q-Krawtchouk polynomial ensemble. We derive the limit shape and prove the central limit theorem for the fluctuations in the limit when $n,k$ tend to infinity and $q$ tends to one at comparable rates.

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