### An application of Burnside rings in elementary finite group theory.

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Let ${H}_{8}$ be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra $\tilde{{H}_{8}}$ based on ${H}_{8}$, then we investigate the structure of the representation ring of $\tilde{{H}_{8}}$. Finally, we prove that the automorphism group of $r\left(\tilde{{H}_{8}}\right)$ is just isomorphic to ${D}_{6}$, where ${D}_{6}$ is the dihedral group with order 12.

If $G$ is a non-cyclic finite group, non-isomorphic $G$-sets $X,Y$ may give rise to isomorphic permutation representations $\u2102\left[X\right]\cong \u2102\left[Y\right]$. Equivalently, the map from the Burnside ring to the rational representation ring of $G$ has a kernel. Its elements are called Brauer relations, and the purpose of this paper is to classify them in all finite groups, extending the Tornehave–Bouc classification in the case of $p$-groups.

Let $G$ be a finite group with a Sylow 2-subgroup $P$ which is either quaternion or semi-dihedral. Let $k$ be an algebraically closed field of characteristic 2. We prove the existence of exotic endotrivial $kG$-modules, whose restrictions to $P$ are isomorphic to the direct sum of the known exotic endotrivial $kP$-modules and some projective modules. This provides a description of the group $T\left(G\right)$ of endotrivial $kG$-modules.

Let $kG$ be a group algebra, and $D\left(kG\right)$ its quantum double. We first prove that the structure of the Grothendieck ring of $D\left(kG\right)$ can be induced from the Grothendieck ring of centralizers of representatives of conjugate classes of $G$. As a special case, we then give an application to the group algebra $k{D}_{n}$, where $k$ is a field of characteristic $2$ and ${D}_{n}$ is a dihedral group of order $2n$.

A new calculus of planar diagrams involving diagrammatics for biadjoint functors and degenerate affine Hecke algebras is introduced. The calculus leads to an additive monoidal category whose Grothendieck ring contains an integral form of the Heisenberg algebra in infinitely many variables. We construct bases of the vector spaces of morphisms between products of generating objects in this category.

Let $k$ be a field of characteristic $p$. Let $G$ be a finite group of order divisible by $p$ and $P$ a $p$-Sylow subgroup of $G$. We describe the kernel of the restriction homomorphism $T\left(G\right)\to T\left(P\right)$, for $T\left(-\right)$ the group of endotrivial representations. Our description involves functions $G\to {k}^{\times}$ that we call weak $P$-homomorphisms. These are generalizations to possibly non-normal $P\le G$ of the classical homomorphisms $G/P\to {k}^{\times}$ appearing in the normal case.

The automorphisms of a quasigroup or Latin square are permutations of the set of entries of the square, and thus belong to conjugacy classes in symmetric groups. These conjugacy classes may be recognized as being annihilated by symmetric group class functions that belong to a $\lambda $-ideal of the special $\lambda $-ring of symmetric group class functions.

Given a tuple $({\mathcal{X}}_{1},...,{\mathcal{X}}_{k})$ of irreducible characters of $G{L}_{n}\left({F}_{q}\right)$ we define a star-shaped quiver $\Gamma $ together with a dimension vector $v$. Assume that $({\mathcal{X}}_{1},...,{\mathcal{X}}_{k})$ is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product ${\mathcal{X}}_{1}\otimes \cdots \otimes {\mathcal{X}}_{k}$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(\Gamma ,v)$. The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied, we...