The Schur-Clifford subgroup of the Brauer-Clifford group

We define a Schur-Clifford subgroup of Turull's Brauer-Clifford group, similar to the Schur subgroup of the Brauer group. The Schur-Clifford subgroup contains exactly the equivalence classes coming from the intended application to Clifford theory of finite groups. We show that the Schur-Clifford subgroup is indeed a subgroup of the Brauer-Clifford group, as are certain naturally defined subsets. We also show that this Schur-Clifford subgroup behaves well with respect to restriction and corestriction maps between Brauer-Clifford groups.Comment: Corrected a few typos. Final versio

Lattices of finite abelian groups

We study certain lattices constructed from finite abelian groups. We show that such a lattice is eutactic, thereby confirming a conjecture by B\"ottcher, Eisenbarth, Fukshansky, Garcia, Maharaj. Our methods also yield simpler proofs of two known results: First, such a lattice is strongly eutactic if and only if the abelian group has odd order or is elementary abelian. Second, such a lattice has a basis of minimal vectors, except for the cyclic group of order 4.Comment: v2: Minor corrections due to referee reports, 14 page

On Clifford theory with Galois action

Let $\widehat{G}$ be a finite group, $N$ a normal subgroup of $\widehat{G}$ and $\theta\in \operatorname{Irr}N$. Let $\mathbb{F}$ be a subfield of the complex numbers and assume that the Galois orbit of $\theta$ over $\mathbb{F}$ is invariant in $\widehat{G}$. We show that there is another triple $(\widehat{G}_1,N_1,\theta_1)$ of the same form, such that the character theories of $\widehat{G}$ over $\theta$ and of $\widehat{G}_1$ over $\theta_1$ are essentially "the same" over the field $\mathbb{F}$ and such that the following holds: $\widehat{G}_1$ has a cyclic normal subgroup $C$ contained in $N_1$, such that $\theta_1=\lambda^{N_1}$ for some linear character $\lambda$ of $C$, and such that $N_1/C$ is isomorphic to the (abelian) Galois group of the field extension $\mathbb{F}(\lambda)/\mathbb{F}(\theta_1)$. More precisely, "the same" means that both triples yield the same element of the Brauer-Clifford group $\operatorname{BrCliff}(G,\mathbb{F}(\theta))$ defined by A. Turull.Comment: v3: Referee's comments included, and a few other small correction

The Problem of Post-Truth. Rethinking the Relationship between Truth and Politics

‘Post-truth’ is a failed concept, both epistemically and politically because its simplification of the relationship between truth and politics cripples our understanding and encourages authoritarianism. This makes the diagnosis of our ‘post-truth era’ as dangerous to democratic politics as relativism with its premature disregard for truth. In order to take the step beyond relativism and ‘post-truth’, we must conceptualise the relationship between truth and politics differently by starting from a ‘non-sovereign’ understanding of truth

Corestriction for algebras with group action

We define a corestriction map for equivariant Brauer groups in the sense of Fr\"ohlich and Wall, which contain as a special case the Brauer-Clifford groups introduced by Turull. We show that this corestriction map has similar properties as the corestriction map in group cohomology (especially Galois cohomology). In particular, composing corestriction and restriction associated to a subgroup $H\leq G$ amounts to powering with the index $\lvert G:H \rvert$.Comment: Typos corrected, final versio

Affine Symmetries of Orbit Polytopes

An orbit polytope is the convex hull of an orbit under a finite group $G \leq \operatorname{GL}(d,\mathbb{R})$. We develop a general theory of possible affine symmetry groups of orbit polytopes. For every group, we define an open and dense set of generic points such that the orbit polytopes of generic points have conjugated affine symmetry groups. We prove that the symmetry group of a generic orbit polytope is again $G$ if $G$ is itself the affine symmetry group of some orbit polytope, or if $G$ is absolutely irreducible. On the other hand, we describe some general cases where the affine symmetry group grows. We apply our theory to representation polytopes (the convex hull of a finite matrix group) and show that their affine symmetries can be computed effectively from a certain character. We use this to construct counterexamples to a conjecture of Baumeister et~al.\ on permutation polytopes [Advances in Math. 222 (2009), 431--452, Conjecture~5.4].Comment: v2: Referee comments implemented, last section updated. Numbering of results changed only in Sections 9 and 10. v3: Some typos corrected. Final version as published. 36 pages, 5 figures (TikZ