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[^1]: Our choice is in this sense analogous to choosing polar coordinates for a sphere, which allows easy explicit calculation but obscures the rotational symmetry.

[^2]: Jakob Gath informs us (private communication) that for sufficiently large values of the dilaton coupling these bounds are satisfied/violated in the same manner as we have found: is always satisfied, and is always violated, for all values of the charge density.


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abstract: 'Let $G$ be a finite group and $R$ be a commutative ring. The Mackey algebra $\mu_{R}(G)$ shares a lot of properties with the group algebra $RG$ however, there are some differences. For example, the group algebra is a symmetric algebra and this is not always the case for the Mackey algebra. In this paper we present a systematic approach to the question of the symmetry of the Mackey algebra, by producing symmetric associative bilinear forms for the Mackey algebra. Using the fact that the category of Mackey functors is a closed symmetric monoidal category, we prove that the Mackey algebra $\mu_{R}(G)$ is a symmetric algebra if and only if the family of Burnside algebras $(RB(H))_{H\leqslant G}$ is a family of symmetric algebras with a compatibility condition. As a corollary, we recover the well known fact that over a field of characteristic zero, the Mackey algebra is always symmetric. Over the ring of integers the Mackey algebra of $G$ is symmetric if and only if the order of $G$ is square free. Finally, if $(K,\mathcal{O},k)$ is a $p$-module system for $G$, we show that the Mackey algebras $\mu_{\mathcal{O}}(G)$ and $\mu_{k}(G)$ are symmetric if and only if the Sylow $p$-subgroups of $G$ are of order $1$ or $p$.'
author:
- Baptiste Rognerud
title: 'Trace maps for Mackey algebra