Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 024, 56 pages      arXiv:2501.04605      https://doi.org/10.3842/SIGMA.2026.024

Big Algebra in Type $A$ for the Coordinate Ring of the Matrix Space

Nhok Tkhai Shon Ngo
Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria

Received April 30, 2025, in final form February 22, 2026; Published online March 14, 2026

Abstract
In this paper, we consider the big algebra recently introduced by Hausel for the $\mathrm{GL}_n$-action on the coordinate ring of the matrix space $\mathrm{Mat}(n,r)$. In particular, we obtain explicit formulas for the big algebra generators in terms of differential operators with polynomial coefficients. We show that big algebras in type $A$ are commutative and relate them to the Bethe subalgebra in the Yangian $\operatorname{Y}(\mathfrak{gl}_{n})$. We apply these results to big algebras of symmetric powers of the standard representation of $\mathrm{GL}_n$.

Key words: shift of argument subalgebra; Bethe subalgebras; Capelli identities; equivariant cohomology.

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