Ennola duality for decomposition of tensor products
Emmanuel Letellier, Fernando Rodriguez-Villegas
TL;DR
The work addresses the problem of decomposing tensor products of irreducible characters for finite general linear and unitary groups via Ennola duality. It shows that generic multiplicities are governed by polynomials $V_{\bm\omega}(q)$ and $V'_{\bm\omega}(q)$ with $V'_{\bm\omega}(q)=\pm V_{\bm\omega}(-q)$ and provides a geometric interpretation using the cohomology of quiver varieties, realized through an $\mathbf{S}'_n$-module $\mathbb{M}^\bullet_n$. For unipotent (non-generic) cases, the naive $q\mapsto -q$ substitution fails, motivating the interpolation polynomials $\mathcal{T}_{\bm\mu}(u,q)$ which recover unipotent multiplicities via $U_{\bm\mu}(q)=\mathcal{T}_{\bm\mu}(1,q)$ and $U'_{\bm\mu}(q)=(-1)^{\frac{1}{2}d_{\bm\mu}+n}\mathcal{T}_{\bm\mu}(-1,-q)$. The paper further connects these multiplicities to the cohomology of quiver varieties, Fourier transforms, and the theory of regular semisimple orbits, culminating in a result where the top-degree coefficient in $u$ yields a Kronecker coefficient of the symmetric group, $\langle\chi^{\mu^1}\otimes\cdots\otimes\chi^{\mu^k},1\rangle_{S_n}$. Overall, it provides a unified, geometry-grounded framework for Ennola duality across generic and unipotent regimes, along with explicit polynomial interpolations and illustrative examples.
Abstract
The aim of this paper is to investigate Ennola duality for decomposition of tensor products of irreducible characters of finite general linear groups and finite unitary groups. We prove that Ennola duality holds generically and give a geometric interpretation using the cohomology of quiver varieties. For non-generic characters (like unipotent characters), Ennola duality does not work just by replacing q by -q. We construct two-variable polynomials that interpolate multiplicities for finite general linear groups and finite unitary groups in the unipotent case (which can be considered as Ennola duality).