[Paper Review] Symmetric and non-symmetric quantum Capelli polynomials

Friedrich Knop|ArXiv.org|Mar 29, 1996

Algebraic structures and combinatorial models5 references17 citations

TL;DR

This paper introduces a family of symmetric and non-symmetric quantum Capelli polynomials defined by vanishing conditions at specific points related to parameters $q$ and $t$. Using Cherednik-type difference operators, it establishes that the top-degree components of these polynomials are Macdonald polynomials, and proves that the non-symmetric versions are simultaneous eigenfunctions of these operators, enabling a systematic construction and symmetrization process.

ABSTRACT

We define a family of symmetric and a family of non-symmetric polynomials in terms of vanishing conditions. These families depend on two paramters, q and t. Their main feature is that they consist of non-homogeneous polynomials. The symmetric polynomials form the quantized version of polynomials occuring in the context of generalized Capelli identities. We show that these quantum Capelli polynomials are also characterized by q-difference equations. More precisely, they are eigenfunctions of Cherednik type operators and transform under an affine Hecke algebra. Thus, we are able to identify their top homogeneous component as a Macdonald polynomial (symmetric or non-symmetric, respectively).

Motivation & Objective

• To construct symmetric and non-symmetric quantum Capelli polynomials using a quantized vanishing condition at points $q^\mu w_\mu(\varrho)$.
• To show that the top-degree components of these polynomials are Macdonald polynomials, extending classical Capelli identities to the quantum setting.
• To establish a connection between the polynomials and Cherednik-type difference operators, enabling eigenfunction characterization.
• To prove that the non-symmetric polynomials vanish at more points than required (extra vanishing), governed by a generalized dominance order on $\Lambda$.
• To derive an inversion formula and integrality results for normalized versions of the polynomials in $\mathbb{Z}[r]$.

Proposed method

• Define the vector $\varrho = (1, t^{-1}, \dots, t^{-n+1})$ and associate to each $\lambda \in \Lambda = \mathbb{N}^n$ the point $\overline{\lambda} = w_\lambda(q^{\lambda^+}\varrho)$.
• Construct the polynomial $E_\lambda$ as the unique (up to scalar) degree-$|\lambda|$ polynomial vanishing at $\overline{\mu}$ for all $\mu \in \Lambda$ with $|\mu| \leq |\lambda|$, $\mu \neq \lambda$.
• Introduce Cherednik-type difference operators $H_i$ and $\overline{H}_i$, which act on the polynomial ring and have the $E_\lambda$ as simultaneous eigenfunctions.
• Use the affine Hecke algebra action to relate the non-symmetric polynomials to non-symmetric Macdonald polynomials via eigenfunction properties.
• Derive an inversion formula using operators $Z_i$ and $\tilde{Z}_i$, showing $\Psi(f) = f(\tilde{Z}_1, \dots, \tilde{Z}_n)(1)$ for a linear automorphism $\Psi$.
• Normalize $E_\lambda$ and $P_\lambda$ via products over cells in the Young diagram to obtain integrality results in $\mathbb{Z}[r]$.

Experimental results

Research questions

• RQ1 How can the classical Capelli identity be generalized to a quantum setting using symmetric and non-symmetric polynomials?
• RQ2 What is the structure of the non-symmetric quantum Capelli polynomials defined by vanishing conditions at $q^\mu w_\mu(\varrho)$?
• RQ3 How do the top-degree components of these polynomials relate to Macdonald polynomials?
• RQ4 What is the role of Cherednik-type difference operators in characterizing these polynomials as eigenfunctions?
• RQ5 What is the nature of the extra vanishing property, and how is it governed by an order on $\Lambda$?

Key findings

• The non-symmetric quantum Capelli polynomials $E_\lambda$ are simultaneous eigenfunctions of Cherednik-type difference operators $H_i$, which are constructed from the affine Hecke algebra.
• The top-degree homogeneous part of $E_\lambda$ is the non-symmetric Macdonald polynomial, and the top-degree part of the symmetric version $P_\lambda$ is the symmetric Macdonald polynomial.
• The polynomials $E_\lambda$ vanish not only at the prescribed points $\overline{\mu}$ for $|\mu| \leq |\lambda|$, $\mu \neq \lambda$, but also at additional points, a phenomenon termed 'extra vanishing'.
• In the classical limit $q \to 1$, $t = q^r$, the operators $H_i$ and $\Phi$ converge to $\sigma_i$ and $\tilde{\Phi}$, respectively, yielding a graded Hecke algebra action with $\sigma_i^2 = 1$.
• The normalized polynomials $\tilde{\cal E}_\lambda$ and $\tilde{\cal P}_\lambda$ are shown to have coefficients in $\mathbb{Z}[r]$, extending integrality results from earlier work.
• An inversion formula holds: $\Psi(f) = f(\tilde{Z}_1, \dots, \tilde{Z}_n)(1)$, where $\tilde{Z}_i = \sigma_i \cdots \sigma_{n-1} \tilde{\Phi} \sigma_1 \cdots \sigma_{i-1}$, and $\Psi$ maps the leading term of $\tilde{E}_\lambda$ to itself.