[Paper Review] Plethystic identities and mixed Hodge structures of character varieties

Anton Mellit|arXiv (Cornell University)|Mar 1, 2016

Advanced Combinatorial Mathematics2 references3 citations

TL;DR

This paper establishes explicit relations between mixed Hodge polynomials of character varieties using plethystic identities derived from Macdonald polynomial identities and the Hausel–Letellier–Villegas conjecture. By leveraging $λ$-ring structures and plethystic substitution, it proves a duality symmetry in Macdonald polynomials that leads to a new identity connecting Hodge polynomials of different character varieties, resolving a conjecture in wild character variety Laplace transforms.

ABSTRACT

I demonstrate how certain identities for Macdonald's polynomials established by Garsia, Haiman and Tesler, together with the conjecture of Hausel, Letellier and Villegas imply explicit relations between mixed Hodge polynomials of different character varieties.

Motivation & Objective

• To derive explicit relations between mixed Hodge polynomials of character varieties using plethystic identities.
• To connect Macdonald polynomial identities—specifically those of Garsia, Haiman, and Tesler—with mixed Hodge structures in character varieties.
• To verify and apply the Hausel–Letellier–Villegas conjecture to establish new symmetries in Hodge-theoretic invariants.
• To resolve a conjecture from Hausel, Letellier, and Villegas regarding the Laplace transform of wild character varieties.

Proposed method

• Utilizes plethystic substitution in $λ$-rings to manipulate symmetric functions and Macdonald polynomials.
• Applies the Macdonald-Koornwinder duality identity to derive symmetry in generating functions involving $H_{\lambda}[1+uD_{\mu}]$.
• Employs the operator $V^*$, conjugate to $V$, to relate $H_{\lambda}[X]\operatorname{Exp}[-B_{\lambda}Y]$ to $\Omega[X,Y,-1]$ via reproducing kernel properties.
• Uses the $\nabla$-operator and $T$, $T^*$ to conjugate and transform generating functions, preserving degree and symmetry.
• Applies the identity $\operatorname{Exp}[X(1-u)] = \sum_{i,j} (-1)^j u^j h_i e_j$ to interpret $F[1-u]$ in terms of inner products of symmetric functions.
• Derives a bijection between Hodge-theoretic invariants by transforming $\Omega[X,1-y_1,\dots,1-y_k]$ into $\operatorname{Exp}[(X+Y)/Q]\Omega[X+1,Y,-1]$.

Experimental results

Research questions

• RQ1 How do plethystic identities in symmetric functions relate to mixed Hodge polynomials of character varieties?
• RQ2 Can the Macdonald polynomial duality identity be used to derive new symmetries in Hodge-theoretic invariants?
• RQ3 Does the Hausel–Letellier–Villegas conjecture imply explicit relations between mixed Hodge polynomials of different character varieties?
• RQ4 What is the Hodge-theoretic interpretation of the Laplace transform of wild character varieties?
• RQ5 How does the $\lambda$-ring structure facilitate the derivation of identities involving $e_n[(1-u)/(1-q)]$ and $h_n[(1-u)/(1-q)]$?

Key findings

• The identity $\operatorname{Exp}[uB_{\lambda}]H_{\lambda}[1+uD_{\mu}] = \frac{H_{\lambda}[1+uD_{\mu}]}{\prod_{r,c\in\lambda}(1-uq^c t^r)}$ is symmetric in $\lambda$ and $\mu$, establishing a Macdonald-Koornwinder duality.
• Corollary 6.4 shows $H_{\lambda}[1-u] = \prod_{r,c\in\lambda}(1 - u q^c t^r)$, providing a closed-form Hodge-theoretic evaluation.
• Corollary 6.5 gives $H_{\lambda}[-1] = (-1)^{|λ|} q^{n(\lambda')} t^{n(\lambda)}$, linking Hodge polynomials to partition statistics.
• Theorem 7.1 establishes a bijection between Hodge invariants: $\mathbb{H}(h_{\lambda}, h_{|λ|-μ_1}e_{\mu_1}, \dots) = \mathbb{H}(h_{\lambda,|\mu|-|\lambda|}, h_{\mu}, e_{|\mu|})$ for $1 \leq \mu_1 \leq |\lambda| \leq |\mu|$.
• Corollary 7.2 resolves a conjecture from [HMW16] by showing $\mathbb{H}(h_{\lambda}, h_{n-1,1}^{(k)}) = \mathbb{H}(h_{\lambda,k-n}, h_{1^k}, e_k)$ for $k \geq n \geq 1$, $\lambda \vdash n$, linking to the Laplace transform of wild character varieties.
• The transformation $\Omega[X,1-y_1,\dots,1-y_k] = \operatorname{Exp}[(X+Y)/Q]\Omega[X+1,Y,-1]$ enables a direct computation of Hodge invariants via conjugate operators $V^*$ and $\nabla$.