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Big algebra in type $A$ for the coordinate ring of the matrix space

Nhok Tkhai Shon Ngo

TL;DR

This work develops a concrete, algebraic construction of big algebras for the coordinate ring $\mathbb{C}[\operatorname{Mat}(n,r)]$ under the $\mathrm{GL}_n$-action in type $A$. By exploiting the Kirillov--Wei operator, the authors obtain explicit normal-form generators $M_{p,q}$ and $F_{p,q}$ for the big algebra $\mathscr{B}(\mathcal{P}(n,r))$, relate these to Capelli identities, and connect them to Bethe subalgebras of the Yangian $\mathrm{Y}(\mathfrak{gl}_n)$ to prove commutativity. They show that big algebras are commutative for irreducible polynomial $\mathrm{GL}_n$-modules and establish a precise isomorphism $\mathscr{B}(m\varpi_1)\cong S^m(\mathscr{B}(\varpi_1))$ for the symmetric powers of the vector representation, along with detailed presentations in terms of generators and relations. The paper further develops the Kirillov algebra framework in the weight-multiplicity-free setting, providing explicit descriptions of big algebras for symmetric powers and new formulas for the Kirillov--Wei operator, with connections to diagonal invariants and plethystic substitutions. Overall, the results give a direct, elementary route to commutativity and a rich algebraic structure linking invariant theory, Capelli-type identities, and integrable-system constructions.

Abstract

In this note we consider the big algebra recently introduced by Hausel for the $\mathrm{GL}_n$-action on the coordinate ring of the matrix space $\operatorname{Mat}(n,r)$. 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 $\mathrm{Y}(\mathfrak{gl}_{n})$. We apply these results to big algebras of symmetric powers of the standard representation of $\mathrm{GL}_n$.

Big algebra in type $A$ for the coordinate ring of the matrix space

TL;DR

This work develops a concrete, algebraic construction of big algebras for the coordinate ring under the -action in type . By exploiting the Kirillov--Wei operator, the authors obtain explicit normal-form generators and for the big algebra , relate these to Capelli identities, and connect them to Bethe subalgebras of the Yangian to prove commutativity. They show that big algebras are commutative for irreducible polynomial -modules and establish a precise isomorphism for the symmetric powers of the vector representation, along with detailed presentations in terms of generators and relations. The paper further develops the Kirillov algebra framework in the weight-multiplicity-free setting, providing explicit descriptions of big algebras for symmetric powers and new formulas for the Kirillov--Wei operator, with connections to diagonal invariants and plethystic substitutions. Overall, the results give a direct, elementary route to commutativity and a rich algebraic structure linking invariant theory, Capelli-type identities, and integrable-system constructions.

Abstract

In this note we consider the big algebra recently introduced by Hausel for the -action on the coordinate ring of the matrix space . 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 are commutative and relate them to the Bethe subalgebra in the Yangian . We apply these results to big algebras of symmetric powers of the standard representation of .
Paper Structure (51 sections, 63 theorems, 256 equations)

This paper contains 51 sections, 63 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Big algebras.
  3. Main results.
  4. Contents.
  5. Acknowledgements.
  6. Notation and preliminaries
  7. Operations with tuples.
  8. Action of the symmetric group.
  9. Sign functions for tuples.
  10. Non-commutative matrices.
  11. The Lie algebra $\mathfrak{gl}_n$.
  12. The coordinate ring of the matrix space.
  13. The action of $\mathfrak{gl}_n$.
  14. The algebra $\mathcal{PD}(n,r)$ of differential operators on $\mathcal{P}(n,r)$.
  15. Kirillov algebra, big algebra and medium algebra on $\mathcal{P}(n,r)$
  16. ...and 36 more sections

Key Result

Theorem 1.1

The big algebra of any polynomial finite-dimensional irreducible representation of $\mathrm{GL}_{n}$ is commutative.

Theorems & Definitions (149)

  • Theorem 1.1: Corollary \ref{['cor:big_alg_commut_irrep']}
  • Theorem 1.2: Theorem \ref{['thm:big_alg_sym_pow_iso']}
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 3.1
  • Definition 3.2
  • Proposition 3.3
  • ...and 139 more