arXiv subjects
Publications and source records attributed to Florian Eisele.
We determine the Morita equivalence classes of $2$-blocks with quaternion defect groups of arbitrary $2$-power order, thereby completing the proof of Donovan's conjecture for blocks of tame representation type.
We introduce block pro-fusion systems for blocks of profinite groups, prove a profinite version of Puig's structure theorem for nilpotent blocks, and use it to show that there is only one Morita equivalence class of blocks having the infinite dihedral pro-$2$ group as their defect group.
We obtain restrictions on units of even order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ by studying their actions on the reductions modulo $4$ of lattices over the $2$-adic group ring $\mathbb{Z}_2G$. This improves the "lattice method" which considers reductions modulo primes $p$, but is of limited use for $p=2$ essentially due to the fact that $1\equiv -1 \ (\textrm{mod }2)$. Our methods yield results in cases where $\mathbb Z_2 G$ has blocks whose defect groups are Klein four groups or dihedral groups of order $8$. This allows us to disprove the existence of units of order $2p$ for almost simple groups with socle $\operatorname{PSL}(2,p^f)$ where $p^f\equiv \pm 3 \ (\textrm{mod } 8)$ and to answer the Prime Graph Question affirmatively for many such groups.
We show that there are finite monoids $M$ such that the Cartan matrix of the monoid algebra $\mathbb C M$ is non-singular, whilst the Cartan matrix of $kM$ is singular for some field $k$ of positive characteristic, disproving a recent conjecture of Steinberg.
We introduce a method that produces a bijection between the posets ${\rm silt-}{A}$ and ${\rm silt-}{B}$ formed by the isomorphism classes of basic silting complexes over finite-dimensional $k$-algebras $A$ and $B$, by lifting $A$ and $B$ to two $k[[X]]$-orders which are isomorphic as rings. We apply this to a class of algebras generalising Brauer graph and weighted surface algebras, showing that their silting posets are multiplicity-independent in most cases. Under stronger hypotheses we also prove the existence of large multiplicity-independent subgroups in their derived Picard groups as well as multiplicity-invariance of $\rm TrPicent$. As an application to the modular representation theory of finite groups we show that if $B$ and $C$ are blocks with $|{\rm IBr}(B)|=|{\rm IBr}(C)|$ whose defect groups are either both cyclic, both dihedral or both quaternion, then the posets ${\rm tilt-}{B}$ and ${\rm tilt-}{C}$ are isomorphic (except, possibly, in the quaternion case with $|{\rm IBr}(B)|=2$) and ${\rm TrPicent}(B)\cong{\rm TrPicent}(C)$ (except, possibly, in the quaternion and dihedral cases with $|{\rm IBr}(B)|=2$).
We construct blocks of finite groups with arbitrarily large Morita Frobenius numbers, an invariant which determines the size of the minimal field of definition of the associated basic algebra. This answers a question of Benson and Kessar. This also improves upon a result of the second author where arbitrarily large $\mathcal{O}$-Morita Frobenius numbers are constructed.
Let $(K,\mathcal O, k)$ be a $p$-modular system with $k$ algebraically closed and $\mathcal O$ unramified, and let $Λ$ be an $\mathcal O$-order in a separable $K$-algebra. We call a $Λ$-lattice $L$ rigid if ${\rm Ext}^1_Λ(L,L)=0$, in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the $Λ$-lattices of a given dimension into "varieties of lattices", we show that there are only finitely many rigid $Λ$-lattices $L$ of any given dimension. As a consequence we show that if the first Hochschild cohomology of $Λ$ vanishes, then the Picard group and the outer automorphism group of $Λ$ are finite. In particular the Picard groups of blocks of finite groups defined over $\mathcal O$ are always finite.
We give a reduction to quasisimple groups for Donovan's conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring $\mathcal{O}$. Consequences are that Donovan's conjecture holds for $\mathcal{O}$-blocks with abelian defect groups for the prime two, and that, using recent work of Farrell and Kessar, for arbitrary primes Donovan's conjecture for $\mathcal{O}$-blocks with abelian defect groups reduces to bounding the Cartan invariants of blocks of quasisimple groups in terms of the defect. A result of independent interest is that in general (i.e. for arbitrary defect groups) Donovan's conjecture for $\mathcal{O}$-blocks is a consequence of conjectures predicting bounds on the $\mathcal{O}$-Frobenius number and on the Cartan invariants, as was proved by Kessar for blocks defined over an algebraically closed field.
For an arbitrary finite-dimensional algebra $A$, we introduce a general approach to determining when its first Hochschild cohomology ${\rm HH}^1(A)$, considered as a Lie algebra, is solvable. If $A$ is moreover of tame or finite representation type, we are able to describe ${\rm HH}^1(A)$ as the direct sum of a solvable Lie algebra and a sum of copies of $\mathfrak{sl}_2$. We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of $A$. As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll and Solotar.
Let $(K,\mathcal O,k)$ be a $p$-modular system and assume $k$ is algebraically closed. We show that if $Λ$ is an $\mathcal O$-order in a separable $K$-algebra, then $\textrm{Pic}_{\mathcal O}(Λ)$ carries the structure of an algebraic group over $k$. As an application to the modular representation theory of finite groups, we show that a reduction theorem by Külshammer concerned with Donovan's conjecture remains valid over $\mathcal O$.
Hans J. Zassenhaus conjectured that for any unit $u$ of finite order in the integral group ring of a finite group $G$ there exists a unit $a$ in the rational group algebra of $G$ such that $a^{-1}\cdot u \cdot a=\pm g$ for some $g\in G$. We disprove this conjecture by first proving general results that help identify counterexamples and then providing an infinite number of examples where these results apply. Our smallest example is a metabelian group of order $2^7 \cdot 3^2 \cdot 5 \cdot 7^2 \cdot 19^2$ whose integral group ring contains a unit of order $7 \cdot 19$ which, in the rational group algebra, is not conjugate to any element of the form $\pm g$.
We identify a class of symmetric algebras over a complete discrete valuation ring $\mathcal O$ of characteristic zero to which the characterisation of Knörr lattices in terms of stable endomorphism rings in the case of finite group algebras, can be extended. This class includes finite group algebras, their blocks and source algebras and Hopf orders. We also show that certain arithmetic properties of finite group representations extend to this class of algebras. Our results are based on an explicit description of Tate duality for lattices over symmetric $\mathcal O$-algebras whose extension to the quotient field of $\mathcal O$ is separable.
We prove a theorem which gives a bijection between the support $τ$-tilting modules over a given finite-dimensional algebra $A$ and the support $τ$-tilting modules over $A/I$, where $I$ is the ideal generated by the intersection of the center of $A$ and the radical of $A$. This bijection is both explicit and well-behaved. We give various corollaries of this, with a particular focus on blocks of group rings of finite groups. In particular we show that there are $τ$-tilting finite wild blocks with more than one simple module. We then go on to classify all support $τ$-tilting modules for all algebras of dihedral, semidihedral and quaternion type, as defined by Erdmann, which include all tame blocks of group rings. Note that since these algebras are symmetric, this is the same as classifying all basic two-term tilting complexes, and it turns out that a tame block has at most $32$ different basic two-term tilting complexes. We do this by using the aforementioned reduction theorem, which reduces the problem to ten different algebras only depending on the ground field $k$, all of which happen to be string algebras. To deal with these ten algebras we give a combinatorial classification of all $τ$-rigid modules over (not necessarily symmetric) string algebras.
We show that two blocks of generalized quaternion defect with three simple modules over a sufficiently large $2$-adic ring $\mathcal O$ are Morita-equivalent if and only if the corresponding blocks over the residue field of $\mathcal O$ are Morita-equivalent. As a corollary we show that any two blocks defined over $\mathcal O$ with three simple modules and the same generalized quaternion defect group are derived equivalent.
We restrict the type of $2 \times 2$-matrices which can occur as simple components in the Wedderburn decomposition of the rational group algebra of a finite group. This results in a description up to commensurability of the group of units of the integral group ring $\mathbb Z G$ for all finite groups $G$ that do not have a non-commutative Frobenius complement as a quotient.
A finite group $G$ is called Involutive Yang-Baxter (IYB) if there exists a bijective 1-cocycle $χ: G \longrightarrow M$ for some $\mathbb Z G$-module $M$. It is known that every IYB-group is solvable, but it is still an open question whether the converse holds. A characterization of the IYB property by the existence of an ideal $I$ in the augmentation ideal $ω\mathbb Z G$ complementing the set $1-G$ lead to some speculation that there might be a connection with the isomorphism problem for $\mathbb Z G$. In this paper we show that if $N$ is a nilpotent group of class two and $H$ is an IYB-group of order coprime to that of $N$, then $N\rtimes H$ is IYB. The class of groups that can be obtained in that way (and hence are IYB) contains in particular Hertweck's famous counterexample to the isomorphism conjecture as well as all of its subgroups. We then investigate what an IYB structure on Hertweck's counterexample looks like concretely.
In this article we show that the $\Z_p[ζ_{p^f-1}]$-order $\Z_p[ζ_{p^f-1}]\SL_2(p^f)$ can be recognized among those orders whose reduction modulo $p$ is isomorphic to $\F_{p^f}\SL_2(p^f)$ using only ring-theoretic properties (in other words we show that $\F_{p^f}\SL_2(p^f)$ lifts uniquely to a $\Z_p[ζ_{p^f-1}]$-order, provided certain reasonable conditions are imposed on the lift). This proves a conjecture made by Nebe concerning the basic order of $\Z_p[ζ_{p^f-1}]\SL_2(p^f)$.
For two derived equivalent $k$-algebras $\barΛ$ and $\barΓ$, we introduce a correspondence between $\OO$-orders reducing to $\barΛ$ and $\OO$-orders reducing to $\barΓ$. We outline how this may be used to transfer properties like uniqueness (or non-existence) of a lift between $\barΛ$ and $\barΓ$. As an application, we look at tame algebras of dihedral type with two simple modules, where, most notably, we are able to show that among those algebras only the algebras $\mathcal D^{κ,0}(2A)$ and $\mathcal D^{κ,0}(2B)$ can actually occur as basic algebras of blocks of group rings of finite groups.