Group character averages via a single Laguerre

Average of exponential ${\rm Tr}R e^X$, i.e. of a group rather than an algebra character, in Gaussian matrix model is known to be an amusing generalization of Schur polynomial, where time variables are substituted by traces of products of non-commuting matrices ${\rm Tr} \left(\prod_i A{k_i}\right)$ and are thus labeled by weak compositions. The entries of matrices $A_k$ are made from extended Laguerre polynomials, what introduces additional difficulties. We describe the generic sum rules, which express arbitrary traces through convolutions of a single Laguerre polynomial $L_{N-1}^1(z_{k_i})$, what is a considerable simplification.

💡 Research Summary

The paper addresses the long‑standing problem of computing averages of group characters Tr_R e^X in the Gaussian Hermitian matrix model Z = ∫ dX e^{−½ Tr X²}. While the algebraic characters Tr X^k are straightforwardly handled via Hermite polynomials, the exponential character involves non‑commuting matrix products and extended Laguerre polynomials, making closed‑form expressions elusive.

The authors introduce a key object: the N × N matrix A(s) with elements
A(s){ij} = ⟨i| e^{s(a + a†)} |j⟩ = s^{i+j} / (i! j!) e^{s²/2} L{j−i}^{(i−j)}(−s²),
where a, a† are harmonic‑oscillator operators and L_{α}^{n}(x) are generalized Laguerre polynomials. This matrix naturally appears when one rewrites the exponential trace Tr e^{sX} as Tr A(s). The one‑point function is already known to be Tr A(s) = e^{s²/2} L_{N−1}^{1}(−s²).

The central achievement of the paper is to show that any higher‑point trace
Tr A(s₁) A(s₂)…A(s_m)
can be expressed solely through convolutions of a single Laguerre polynomial L_{N−1}^{1}. The authors achieve this by exploiting the generating function
∑{n≥0} L{α}^{n}(z) t^{n} = (1−t)^{−α−1} exp