Related papers: Alvis-Curtis duality, central characters, and real…

We generalize the Alvis-Curtis duality to the abstract representations of reductive groups with Frobenius maps. Similar to the case of representations of finite reductive groups, we show that the Alvis-Curtis duality of infinite type which…

Let $p$ be an odd prime and let $B$ be a $p$-block of a finite group which has cyclic defect groups. We show that all exceptional characters in $B$ have the same Frobenius-Schur indicators. Moreover the common indicator can be computed,…

We formulate a strong positivity conjecture on characters afforded by the Alvis-Curtis dual of the intersection cohomology of Deligne-Lusztig varieties. This conjecture provides a powerful tool to determine decomposition numbers of…

We prove that the number of irreducible real characters in a nilpotent block of a finite group is locally determined. We further conjecture that the Frobenius-Schur indicators of those characters can be computed for p=2 in terms of the…

Let B be a real 2-block of a finite group G. Then B has a real defect class. Let g be an element of such a class. A defect couple of B is (D,E), where E is a Sylow 2-subgroup of the extended centralizer C^*(g) of g, and D is the…

The Frobenius--Schur indicators of characters in a real 2-block with dihedral defect groups have been determined by Murray. We show that two infinite families described in his work do not exist and we construct examples for the remaining…

We prove that the finite exceptional groups $F_4(q)$, $E_7(q)_{\mathrm{ad}}$, and $E_8(q)$ have no irreducible complex characters with Frobenius-Schur indicator $-1$, and we list exactly which irreducible characters of these groups are not…

Let $\bf{G}$ be a connected reductive group with connected center defined over $\mathbb{F}_q$, with Frobenius morphism $F$. We parameterize all of the real-valued irreducible complex characters of ${\bf G}^F$ using the Jordan decomposition…

We determine the rationality properties of unipotent characters of finite reductive groups arising as fixed points of disconnected reductive groups under a Frobenius map. In the proof we use realisations of characters in $\ell$-adic…

We introduce the Frobenius-Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius-Schur theorem, including that for semisimple quasi-Hopf algebras, weak Hopf…

Let $W$ be a finite Weyl group and $\sg$ be a non-trivial graph automorphism of $W$. We show a remarkable relation between the $\sg$-twisted involution module for $W$ and the Frobenius--Schur indicators of the unipotent characters of a…

We show that for general linear groups ${\rm GL}_n(q)$ as well as for $q$-Schur algebras the knowledge of the modular Alvis-Curtis duality over fields of characteristic $\ell$, $\ell \nmid q$, is equivalent to the knowledge of the…

Let $G$ be a quasi-split tamely ramified connected reductive group defined over a $p$-adic field $F$. We show that if $-1$ is in the $F$-points of the absolute Weyl group of $G$, then self-dual supercuspidal representations of $G(F)$ exist.…

We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne-Lusztig (or Alvis-Curtis) duality for $p$-adic groups and the homological duality. This provides a new way to introduce an involution on…

In this work, we generalize the notion of character for 2-representations of finite 2-groups. The properties of 2-characters bear strong similarities to those classical characters of finite groups, including conjugation invariance,…

It is well-known that the value of the Frobenius-Schur indicator $|G|^{-1} \sum_{g\in G} \chi(g^2)=\pm1$ of a real irreducible representation of a finite group $G$ determines which of the two types of real representations it belongs to,…

We obtain the structure of weight 2 blocks and [2:1]-pairs of q-Schur algebras, and compute explicitly the modular Alvis-Curtis duality for weight 2 blocks of finite general linear groups in non-defining characteristic.

It is well known that the number of real irreducible characters of a finite group G coincides with the number of real conjugacy classes of G. Richard Brauer has asked if the number of irreducible characters with Frobenius-Schur indicator 1…

The classical Frobenius-Schur indicators for finite groups are character sums defined for any representation and any integer m greater or equal to 2. In the familiar case m=2, the Frobenius-Schur indicator partitions the irreducible…

This article is concerned with the relative McKay conjecture for finite reductive groups. Let G be a connected reductive group defined over the finite field F_q of characteristic p>0 with corresponding Frobenius map F. We prove that if the…