Finite groups with the same character tables, Drinfel'd algebras and Galois algebras

On the Galois symmetries for the character table of an integral fusion category

In this paper we show that integral fusion categories with rational structure constants admit a natural group of symmetries given by the Galois group of their character tables. We also generalize a well known result of Burnside from representation theory of finite groups. More precisely, we show that any row corresponding to a non invertible object in the character table of a weakly integral fusion category contains a zero entry.

The Galois action on character tables

A geometric interpretation and generalisation for the Galois action on finite group character tables is sketched. The generalisation is a Galois action on the space Map_G(G^n,\bar{Q})/S_n for each finite G, where G acts by simultaneous conjugation on the n-tuples G^n and the symmetric group S_n permutes the components.

Higher frobenius-schur indicators for drinfeld doubles of finite groups through characters of centralizers

We present a new approach to calculating the higher Frobenius-Schur indicators for the simple modules over the Drinfeld double of a finite group. In contrast to the formula by Kashina-Sommerh{ä}user-Zhu that involves a sum over all group elements satisfying a certain condition, our formula operates on the level of conjugacy classes and character tables. It can be implemented in the computer algebra system GAP, efficiently enough to deal, on a laptop, with symmetric groups up to $S_{18}$ (providing further evidence that indicators are non-negative in this case) or simple groups of order up to $…

Galois作用于主块和Sylow 3-子群的生成

Galois action on the principal block and generation of Sylow 3-subgroups

本文证明了关于有限群Sylow 3-子群生成的猜想，并验证了块级Galois-McKay猜想的另一个版本。

The Semisimplicity Conjecture for A-Motives

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V_p(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Akio Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G_p(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G_p(M) may be…

Caracteres de rigidite du groupe de Grothendieck-Teichmuller

Let $\k$ be a (topological) field of characteristic 0. Using a Drinfeld associator $Φ$, a representation $Φ(ρ)$ of the braid group over the field $\k((h))$ of Laurent series can be associated to any representation $ρ$ of a certain Hopf algebra $\mathfrak{B}_n(\k)$. We investigate the dependance in $Φ$ of $Φ(ρ)$ for a certain class of representations -- so-called GT-rigid representations -- and deduce from it (continuous) projective representations of the Grothendieck-Teichmuller group $GT_1(\k)$, hence for $\k = \Q_l$ representations of the absolute Galois group of $\Q(μ_{l^{\infty}})$. In mos…