[Paper Review] Elementary derivation of Weingarten functions of classical Lie groups

Marcel Novaes|arXiv (Cornell University)|Jun 9, 2014

Advanced Combinatorial Mathematics2 references3 citations

TL;DR

This paper presents an elementary, direct derivation of Weingarten functions for classical Lie groups—unitary, orthogonal, and symplectic—using power sum symmetric functions, Schur function expansions, and group integrals. The key contribution is a unified, accessible method that avoids advanced representation theory, yielding explicit formulas for Weingarten functions in terms of characters and normalization constants for each group.

ABSTRACT

Integration of polynomials over the classical groups of unitary, orthogonal and symplectic matrices can be reduced to basic building blocks known as Weingarten functions. We present an elementary derivation of these functions.

Motivation & Objective

• To provide a direct, elementary derivation of Weingarten functions for classical compact Lie groups without relying on advanced representation theory.
• To unify the treatment of unitary, orthogonal, and symplectic groups through a common framework based on symmetric functions and power sums.
• To derive explicit formulas for Weingarten functions using only basic calculus and combinatorial identities, accessible to a broader audience in mathematical physics.
• To establish a systematic method for computing polynomial integrals over classical groups via Weingarten calculus.

Proposed method

• Express the integrand as a derivative of a power sum symmetric function in matrix elements.
• Transform the power sum basis into Schur functions using standard symmetric function identities.
• Evaluate the group integral using orthogonality of characters and known integral formulas for Schur functions.
• Revert back to the power sum basis to prepare for differentiation.
• Take the derivative to recover the original polynomial integral, yielding the Weingarten function as a sum over permutations.
• Derive explicit expressions for the Weingarten functions of U(N), O(N), and Sp(2N) in terms of characters, dimensions, and normalization constants.

Experimental results

Research questions

• RQ1 Can Weingarten functions for classical Lie groups be derived without invoking representation theory or Jucys-Murphy elements?
• RQ2 What is the role of symmetric functions—particularly power sums and Schur functions—in simplifying group integrals over classical groups?
• RQ3 How can the structure of cycle types and matchings in the symmetric group be used to classify and compute Weingarten functions?
• RQ4 What are the explicit algebraic forms of the Weingarten functions for orthogonal and symplectic groups, analogous to the unitary case?

Key findings

• The Weingarten function for the unitary group is given by $\text{Wg}^U(\tau^{-1}\sigma) = \frac{1}{n!} \sum_{\substack{\lambda \vdash n \\ \ell(\lambda) \leq N}} \frac{d_\lambda}{J^{(1)}_\lambda(1^N)} \chi_\lambda(\tau^{-1}\sigma)$, where $d_\lambda$ is the dimension of the irreducible representation and $J^{(1)}_\lambda(1^N)$ is a normalization factor.
• For the orthogonal group, the Weingarten function is $\text{Wg}^O(\tau^{-1}\sigma) = \frac{2^n n!}{(2n)!} \sum_{\substack{\lambda \vdash n \\ \ell(\lambda) \leq N}} \frac{d_{2\lambda}}{J^{(2)}_\lambda(1^N)} \omega_\lambda(\tau^{-1}\sigma)$, with $\omega_\lambda$ being the irreducible character of the hyperoctahedral group.
• For the symplectic group, the Weingarten function is $\text{Wg}^{Sp}(\tau^{-1}\sigma) = \frac{n!}{(2n)!} \sum_{\substack{\lambda \vdash n \\ \ell(\lambda) \leq N}} \frac{d_{\lambda \cup \lambda}}{J^{(1/2)}_\lambda(1^N)} \psi_\lambda(\widetilde{\pi})$, where $\psi_\lambda$ is a character related to the symplectic structure.
• The method successfully reduces all three cases to a common sequence of steps: derivative of power sum → Schur function expansion → group integral → reversion to power sums → final derivative, yielding closed-form expressions.
• The derivation avoids heavy machinery such as Schur-Weyl duality or Gelfand pairs, making the results accessible through elementary calculus and symmetric function theory.