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General Capelli-type identities

Naihuan Jing, Yinlong Liu, Jian Zhang

TL;DR

The paper addresses the problem of unifying and extending Capelli-type identities for matrices with noncommutative entries by developing a deformation framework based on a matrix $H$ and immanants. It introduces a master Capelli identity that leverages Jucys–Murphy elements in the symmetric group algebra to relate products of $X$ and $Y$ to their immanant-type combinations, reducing to Williamson’s and Okounkov’s identities in appropriate limits. The authors then derive generalized Turnbull identities for determinants of symmetric matrices and permanents of antisymmetric matrices, as well as generalized Howe-Umeda-Kostant-Sahi identities for antisymmetric contexts, with Pfaffian techniques and a suite of lemmas underpinning the proofs. Collectively, these results furnish a cohesive, deformation-based framework that encompasses a breadth of classical identities, confirms conjectures on antisymmetric determinants, and broadens the toolkit for invariant-theoretic and representation-theoretic investigations within noncommutative algebra.

Abstract

The classical Capelli identity is an important determinantal identity of a matrix with noncommutative entries that determines the center of the enveloping algebra of the general linear Lie algebra, and was used by Weyl as a main tool to study irreducible representations in his famous book on classical groups. In 1996 Okounkov found higher Capelli identities involving immanants of the generating matrix of $U(gl(n))$ which correspond to arbitrary orthogonal idempotent of the symmetric group. It turns out that Williamson also discovered a general Capelli identity of immanants for $U(gl(n))$ in 1981. In this paper, we use a new method to derive a family of even more general Capelli identities that include the aforementioned Capelli identities as special cases as well as many other Capelli-type identities as corollaries. In particular, we obtain generalized Turnbull's identities for both symmetric and antisymmetric matrices, as well as the generalized Howe-Umeda-Kostant-Sahi identities for antisymmetric matrices which confirm the conjecture of Caracciolo, Sokal, and Sportiello.

General Capelli-type identities

TL;DR

The paper addresses the problem of unifying and extending Capelli-type identities for matrices with noncommutative entries by developing a deformation framework based on a matrix and immanants. It introduces a master Capelli identity that leverages Jucys–Murphy elements in the symmetric group algebra to relate products of and to their immanant-type combinations, reducing to Williamson’s and Okounkov’s identities in appropriate limits. The authors then derive generalized Turnbull identities for determinants of symmetric matrices and permanents of antisymmetric matrices, as well as generalized Howe-Umeda-Kostant-Sahi identities for antisymmetric contexts, with Pfaffian techniques and a suite of lemmas underpinning the proofs. Collectively, these results furnish a cohesive, deformation-based framework that encompasses a breadth of classical identities, confirms conjectures on antisymmetric determinants, and broadens the toolkit for invariant-theoretic and representation-theoretic investigations within noncommutative algebra.

Abstract

The classical Capelli identity is an important determinantal identity of a matrix with noncommutative entries that determines the center of the enveloping algebra of the general linear Lie algebra, and was used by Weyl as a main tool to study irreducible representations in his famous book on classical groups. In 1996 Okounkov found higher Capelli identities involving immanants of the generating matrix of which correspond to arbitrary orthogonal idempotent of the symmetric group. It turns out that Williamson also discovered a general Capelli identity of immanants for in 1981. In this paper, we use a new method to derive a family of even more general Capelli identities that include the aforementioned Capelli identities as special cases as well as many other Capelli-type identities as corollaries. In particular, we obtain generalized Turnbull's identities for both symmetric and antisymmetric matrices, as well as the generalized Howe-Umeda-Kostant-Sahi identities for antisymmetric matrices which confirm the conjecture of Caracciolo, Sokal, and Sportiello.
Paper Structure (7 sections, 13 theorems, 137 equations)

This paper contains 7 sections, 13 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. General Capelli identity
  3. Generalized Turnbull's identities
  4. General Howe-Umeda-Kostant-Sahi identity
  5. Identities for Pfaffians
  6. Proof of Theorem \ref{['huks']}
  7. Proof of Theorem \ref{['odd-anti']}

Key Result

Theorem 2.1

Let $\mathcal{A}$ be an algebra, and let $X=(X_{ij}) \in \mathrm{Mat}_{n\times m}(\mathcal{A})$, $Y=(Y_{ij})\in \mathrm{Mat}_{m \times s}(\mathcal{A})$, where the entries of $X$ are mutually commutative and those of $Y$ are arbitrary. Suppose in $\mathcal{A}\otimes \mathrm{Hom}(\mathbb C^{s}\otimes where $H$ is an $n\times s$ matrix. In terms of entries the relation cape-rela1 is For $r>m=1$, we

Theorems & Definitions (26)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • ...and 16 more