Big algebra in type A for the coordinate ring of the matrix space
Ngo NT. 2026. Big algebra in type A for the coordinate ring of the matrix space. Symmetry, Integrability and Geometry: Methods and Applications. 22, 024.
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Abstract
In this paper, we consider the big algebra recently introduced by Hausel for the GLn-action on the coordinate ring of the matrix space 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 Y(gln). We apply these results to big algebras of symmetric powers of the standard representation of GLn.
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Publishing Year
Date Published
2026-03-14
Journal Title
Symmetry, Integrability and Geometry: Methods and Applications
Publisher
National Academy of Science of Ukraine
Acknowledgement
I would like to express my gratitude to Tam´as Hausel for introducing me to the subject and
for his constant guidance throughout this work. I would also like to thank Tam´as Hausel,
Mischa Elkner, Jakub L¨owit, Anton Mellit, Marino Romero, Leonid Rybnikov for many fruitful
discussions and feedback on earlier drafts of this paper. We are grateful to the anonymous
referees for many useful comments and suggestions that improved the manuscript. This work was done during the author’s PhD studies at the Institute of Science and Technology Austria (ISTA). The author was supported by the Austrian Science Fund (FWF) grant
“Geometry of the tip of the global nilpotent cone” no. 10.55776/P35847 and the DOC Fellowship of the Austrian Academy of Sciences. The author also acknowledges the long-term program
of support of the Ukrainian research teams at the Polish Academy of Sciences carried out in
collaboration with the U.S. National Academy of Sciences with the financial support of external
partners. For open access purposes, the author has applied a CC BY public copyright license
to any author-accepted manuscript version arising from this submission.
Volume
22
Article Number
024
eISSN
IST-REx-ID
Cite this
Ngo NT. Big algebra in type A for the coordinate ring of the matrix space. Symmetry, Integrability and Geometry: Methods and Applications. 2026;22. doi:10.3842/SIGMA.2026.024
Ngo, N. T. (2026). Big algebra in type A for the coordinate ring of the matrix space. Symmetry, Integrability and Geometry: Methods and Applications. National Academy of Science of Ukraine. https://doi.org/10.3842/SIGMA.2026.024
Ngo, Nhok T. “Big Algebra in Type A for the Coordinate Ring of the Matrix Space.” Symmetry, Integrability and Geometry: Methods and Applications. National Academy of Science of Ukraine, 2026. https://doi.org/10.3842/SIGMA.2026.024.
N. T. Ngo, “Big algebra in type A for the coordinate ring of the matrix space,” Symmetry, Integrability and Geometry: Methods and Applications, vol. 22. National Academy of Science of Ukraine, 2026.
Ngo NT. 2026. Big algebra in type A for the coordinate ring of the matrix space. Symmetry, Integrability and Geometry: Methods and Applications. 22, 024.
Ngo, Nhok T. “Big Algebra in Type A for the Coordinate Ring of the Matrix Space.” Symmetry, Integrability and Geometry: Methods and Applications, vol. 22, 024, National Academy of Science of Ukraine, 2026, doi:10.3842/SIGMA.2026.024.
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