arXiv subjects

Robert Boltje

Publications and source records attributed to Robert Boltje.

At least 19 recordsLinked to original sources

arXiv2026

The $D^{\Delta}$-pair biset category

Let $p$ be a prime number. A $D^{\Delta}$-pair is a pair consisting of a finite $p$-group and a $p'$-automorphism of the group. In this paper, we introduce diagonal $D^\Delta$-pair bisets and a category whose objects are $D^\Delta$-pairs and whose morphism groups are Grothendieck groups of diagonal $D^\Delta$-pair bisets. Our main result shows that, over suitable coefficient rings, the category of diagonal $p$-permutation functors is equivalent to the category of linear functors on a natural quotient of this new category. In this way, diagonal $p$-permutation functors can be studied through a category built only from finite $p$-groups and their automorphisms of $p'$-order.

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arXiv2025

Fixed points of extended tensor products

For a $p$-permutation equivalence between two block algebras of finite groups, we introduce new square diagrams that link the $p$-permutation equivalence via the Brauer construction to local equivalences between stabilizers of corresponding Brauer pairs. These diagrams can be viewed as lifts of the square diagrams in the definition of isotypies. The proof of the commutativity requires new technical tools, namely a formula for how taking fixed points commutes with extended tensor products of finite sets with group actions and how the Brauer construction commutes with taking extended tensor products of $p$-permutation modules. These fundamental formulas, generalizing earlier results by Boltje-Danz and by Boltje-Perepelitsky, should be of independent interest.

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arXiv2025

Feit's conjecture, the canonical Brauer induction formula, and Adams operations

This paper is motivated by a strong version of Feit's conjecture, first formulated by the authors in joint work with A. Kleshchev and P. H. Tiep in 2025, concerning the conductor $c(χ)$ of an irreducible character $χ$ of a finite group $G$. We connect the conjecture with the following construction: For any positive integer $n$ dividing the exponent of $G$ and for any character $χ$ of $G$, we introduce an integer-valued invariant $S(G,χ,n)$ which can be defined as the sum of certain coefficients of the canonical Brauer induction formula of $χ$, or alternatively as the multiplicity of the trivial character in a specified integral linear combination of Adams operations of $χ$. We show two facts about this invariant. The first seems of independent interest (apart from Feit's conjecture): $S(G,χ,n)$ is always non-negative, and it is positive if and only if a representation affording $χ$ involves an eigenvalue of order $n$. Secondly, the strong version of Feit's conjecture holds for an irreducible character $χ$ if and only if $S(G,χ, c(χ))>0$.

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arXiv2025

Functorial equivalence classes of $2$-blocks of tame representation type

For any block of a finite group over an algebraically closed field of characteristic $2$ which has dihedral, semidihedral, or generalized quaternion defect groups, we determine explicitly the decomposition of the associated diagonal $p$-permutation functor over an algebraically closed field $\mathbb{F}$ of characteristic $0$ into a direct sum of simple functors. As a consequence we see that two blocks with dihedral, semidihedral, or generalized quaternion defect groups are functorially equivalent over $\mathbb{F}$ if and only if their fusion systems are isomorphic. It is an open question if two blocks (with arbitrary defect groups) that are functorially equivalent over $\mathbb{F}$ must have isomorphic fusion systems. The converse is wrong in general.

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arXiv2025

On Alperin's conjecture and functorial equivalence of blocks

Let $k$ be an algebraically closed field of positive characteristic $p$ and let $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider Alperin's weight conjecture (over $k$) from the point of view of (stable) functorial equivalence of blocks over $\mathbb{F}$. We formulate a functorial version of Alperin's blockwise weight conjecture, and show that it is equivalent to the original one. We also show that this conjecture holds stably, i.e., in the category of stable diagonal $p$-permutation functors over $\mathbb{F}$.

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arXiv2025

A reduction theorem for the Feit conjecture

We prove that if all the simple groups involved in a finite group $G$ satisfy the `inductive Feit condition', then Walter Feit's conjecture from 1980 holds for $G$. In particular, this would solve Brauer's Problem 41 from 1963 in the affirmative. This inductive Feit condition implies that some features of all the irreducible characters of finite groups can be found locally.

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arXiv2024

The ring of perfect $p$-permutation bimodules for blocks with cyclic defect groups

Let $B$ be a block algebra of a group algebra $FG$ of a finite group $G$ over a field $F$ of characteristic $p>0$. This paper studies ring theoretic properties of the representation ring $T^Δ(B,B)$ of perfect $p$-permutation $(B,B)$-bimodules and properties of the $k$-algebra $k\otimes_\mathbb{Z} T^Δ(B,B)$, for a field $k$. We show that if the Cartan matrix of $B$ has $1$ as an elementary divisor then $[B]$ is not primitive in $T^Δ(B,B)$. If $B$ has cyclic defect groups we determine a primitive decomposition of $[B]$ in $T^Δ(B,B)$. Moreover, if $k$ is a field of characteristic different from $p$ and $B$ has cyclic defect groups of order $p^n$ we describe $k\otimes_\mathbb{Z} T^Δ(B,B)$ explicitly as a direct product of a matrix algebra and $n$ group algebras.

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arXiv2023

Monomial structures, I

The goal of a series of papers is to define $G$-actions on various $A$-fibered structures, where $G$ is a finite group and $A$ is an abelian group. One prominent such example is the $A$-fibered Burnside ring. If $A=\mathbb{C}^\times$, it is also called the ring of monomial representations (introduced by Dress in \cite{Dress1971}) and is the natural home for the canonical induction formula (see \cite{Boltje1990}). In this first part of the series, motivated by constructions in \cite{BoucMutlu}, we introduce $A$-fibered structures on posets, on abstract simplicial complexes, and on $A$-bundles over topological spaces, together with natural notions of homotopy, and functors between these structures respecting homotopy. In a sequel we will continue with $G$-representations in these $A$-fibered structures and associate to them elements in the $A$-fibered Burnside ring.

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arXiv2023

Groups with isomorphic fibered Burnside rings

Let $G$ and $H$ be finite groups. We give a condition on $G$ and $H$ that implies that the $A$-fibered Burnside rings $B^A(G)$ and $B^A(H)$ are isomorphic. As a consequence, we show the existence of non-isomorphic groups $G$ and $H$ such that $B^A(G)$ and $B^A(H)$ are isomorphic rings. Here, the abelian fiber group $A$ can be chosen in a non-trivial way, that is, such that $B^A(G)$ and $B^A(H)$ are strictly bigger than the Burnside rings of $G$ and $H$, for which such counterexamples are already known.

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arXiv2022

The orthogonal unit group of the trivial source ring

Let $G$ be a finite group, $p$ a prime, and $(K,\mathcal{O},F)$ a $p$-modular system. We prove that the trivial source ring of $\mathcal{O} G$ is isomorphic to the ring of {\em coherent} $G$-stable tuples $(χ_P)$, where $χ_P$ is a virtual character of $K[N_G(P)/P]$, $P$ runs through all $p$-subgroups of $G$, and the coherence condition is the equality of certain character values. We use this result to describe the group of orthogonal units of the trivial source ring as the product of the unit group of the Burnside ring of the fusion system of $G$ with the group of coherent $G$-stable tuples $(φ_P)$ of homomorphisms $N_G(P)/P\to F^\times$. The orthogonal unit group of the trivial source ring of $\mathcal{O} G$ is of interest, since it embeds into the group of $p$-permutation autoequivalences of $\mathcal{O} G$.

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arXiv2022

Anti-involutions on Green biset functors

In this article, we propose a concept of anti-involution for Green biset functors and we provide equivalent definitions. We present $\star$-Green biset functors and we study their orthogonal units and the orthogonal automorphisms in their associated categories.

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arXiv2022

The Broué invariant of a $p$-permutation equivalence

A perfect isometry $I$ (introduced by Broué) between two blocks $B$ and $C$ is a frequent phenomenon in the block theory of finite groups. It maps an irreducible character $ψ$ of $C$ to $\pm$ an irreducible character of $B$. Broué proved that the ratio of the codegrees of $ψ$ and $I(ψ)$ is a rational number with $p$-value zero and that its class in $\mathbb{F}_p$ is independent of $ψ$. We call this element the Broué invariant of $I$. The goal of this paper is to show that if $I$ comes from a $p$-permutation equivalence or a splendid Rickard equivalence between $B$ and $C$ then, up to a sign, the Broué invariant of $I$ is determined by local data of $B$ and $C$ and therefore, up to a sign, is independent of the $p$-permutation equivalence or splendid Rickard equivalence. Apart from results on $p$-permutation equivalences, our proof requires new results on extended tensor products and bisets that are also proved in this paper. As application of the theorem on the Broué invariant we show that various refinements of the Alperin-McKay Conjecture, introduced by Isaacs-Navarro, Navarro, and Turull are consequences of $p$-permutation equivalences or Rickard equivalences over a sufficiently large complete discrete valuation ring or over $\mathbb{Z}_p$, depending on the refinement.

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arXiv2021

Galois descent of equivalences between blocks of $p$-nilpotent groups

We give sufficient conditions on $p$-blocks of $p$-nilpotent groups over $\mathbb{F}_p$ to be splendidly Rickard equivalent and $p$-permutation equivalent to their Brauer correspondents. The paper also contains Galois descent results on $p$-permutation modules and $p$-permutation equivalences that hold for arbitrary groups.

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arXiv2020

The $A$-fibered Burnside ring as $A$-fibered biset functor in characteristic zero

Let $A$ be an abelian group such that $\mathrm{Hom}(G,A)$ is finite for all finite groups $G$, and let $\mathbb{K}$ be a field of characteristic zero containing roots of unity of all orders equal to finite element orders in $A$. In this paper we prove foundational properties of the $A$-fibered Burnside ring functor $B_{\mathbb{K}}^A$ as an $A$-fibered biset functor over $\mathbb{K}$. This includes the determination of the lattice of subfunctors of $B_{\mathbb{K}}^A$ and the determination of the composition factors of $B_{\mathbb{K}}^A$. The results of the paper extend results of Coşkun and Yılmaz for the $A$-fibered Burnside ring functor restricted to $p$-groups and results of Bouc in the case that $A$ is trivial, i.e., the case of the Burnside ring functor over fields of characteristic zero.

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arXiv2020

$p$-permutation equivalences between blocks of group algebras

We extend the notion of a {$p$-permutation equivalence} between two $p$-blocks $A$ and $B$ of finite groups $G$ and $H$, from the definition in [Boltje-Xu 2008] to a virtual $p$-permutation bimodule whose components have twisted diagonal vertices. It is shown that various invariants of $A$ and $B$ are preserved, including defect groups, fusion systems, and Külshammer-Puig classes. Moreover it is shown that $p$-permutation equivalences have additional surprising properties. They have only one constituent with maximal vertex and the set of $p$-permutation equivalences between $A$ and $B$ is finite (possibly empty). The paper uses new methods: a consequent use of module structures on subgroups of $G\times H$ arising from Brauer constructions which in general are not direct product subgroups, the necessary adaptation of the notion of tensor products between bimodules, and a general formula (stated in these new terms) for the Brauer construction of a tensor product of $p$-permutation bimodules.

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arXiv2019

Fusion systems of blocks of finite groups over arbitrary fields

To any block idempotent $b$ of a group algebra $kG$ of a finite group $G$ over a field $k$ of characteristic $p>0$, Puig associated a fusion system and proved that it is saturated if the $k$-algebra $kC_G(P)e$ is split, where $(P,e)$ is a maximal $kGb$-Brauer pair. We investigate in the non-split case how far the fusion system is from being saturated by describing it in an explicit way as being generated by the fusion system of a related block idempotent over a larger field together with a single automorphism of the defect group.

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arXiv2018

The $-_+$ and $-^+$ constructions for biset functors

In this article we define the $-_+$-construction and the $-^+$-construction, that was crucial in the theory of canonical induction formulas (see \cite{Boltje1998b}), in the setting of biset functors, thus providing the necessary framework to define and construct canonical induction formulas for representation rings that are most naturally viewed as biset functors. Additionally, this provides a unified approach to the study of a class of functors including the Burnside ring, the monomial Burnside ring and global representation ring.

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arXiv2018

On Picard groups of blocks of finite groups

We show that the subgroup of the Picard group of a $p$-block of a finite group given by bimodules with endopermutation sources modulo the automorphism group of a source algebra is determined locally in terms of the fusion system on a defect group. We show that the Picard group of a block over the a complete discrete valuation ring ${\mathcal O}$ of characteristic zero with an algebraic closure $k$ of ${\mathbb F}_p$ as residue field is a colimit of finite Picard groups of blocks over $p$-adic subrings of ${\mathcal O}$. We apply the results to blocks with an abelian defect group and Frobenius inertial quotient, and specialise this further to blocks with cyclic or Klein four defect groups.

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