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Radha Kessar

Publications and source records attributed to Radha Kessar.

At least 37 records · Page 2Linked to original sources
arXiv2018

Blocks with normal abelian defect and abelian p' inertial quotient

Let $k$ be an algebraically closed field of characteristic $p$, and let $\mathcal{O}$ be either $k$ or its ring of Witt vectors $W(k)$. Let $G$ a finite group and $B$ a block of $\mathcal{O}G$ with normal abelian defect group and abelian $p'$ inertial quotient. We show that $B$ is isomorphic to its second Frobenius twist. This is motivated by the fact that bounding Frobenius numbers is one of the key steps towards Donovan's conjecture. For $\mathcal{O}=k$, we give an explicit description of the basic algebra of $B$ as a quiver with relations. It is a quantised version of the group algebra of the semidirect product $P\rtimes L$.

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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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arXiv2018

Descent of equivalences and character bijections

Categorical equivalences between block algebras of finite groups - such as Morita and derived equivalences - are well-known to induce character bijections which commute with the Galois groups of field extensions. This is the motivation for attempting to realise known Morita and derived equivalences over non splitting fields. This article presents various result on the theme of descent. We start with the observation that perfect isometries induced by a virtual Morita equivalence induce isomorphisms of centers in non-split situations, and explain connections with Navarro's generalisation of the Alperin-McKay conjecture. We show that Rouquier's splendid Rickard complex for blocks with cyclic defect groups descends to the non-split case. We also prove a descent theorem for Morita equivalences with endopermutation source.

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arXiv2016

On Tate duality and a projective scalar property for symmetric algebras

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.

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arXiv2016

On blocks of defect two and one simple module, and Lie algebra structure of $HH^1$

Let $k$ be a field of odd prime characteristic $p$. We calculate the Lie algebra structure of the first Hochschild cohomology of a class of quantum complete intersections over $k$. As a consequence, we prove that if $B$ is a defect $2$-block of a finite group algebra $kG$ whose Brauer correspondent $C$ has a unique isomorphism class of simple modules, then a basic algebra of $B$ is a local algebra which can be generated by at most $2\sqrt I$ elements, where $I$ is the inertial index of $B$, and where we assume that $k$ is a splitting field for $B$ and $C$.

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arXiv2015

On the Brauer indecomposability of Scott modules

Let $k$ be an algebraically closed field of prime characteristic $p$, and let $P$ be a $p$-subgroup of a finite group $G$. We give sufficient conditions for the $kG$-Scott module $\mathrm{Sc}(G,P)$ with vertex $P$ to remain indcomposable under the Brauer construction with respect to any subgroup of $P$. This generalizes similar results for the case where $P$ is abelian. The background motivation for this note is the fact that the Brauer indecomposability of a $p$-permutation bimodule is a key step towards showing that the module under consideration induces a stable equivalence of Morita type, which then may possibly be lifted to a derived equivalence.

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arXiv2015

Anchors of irreducible characters

Given a prime number $p$, every irreducible character $χ$ of a finite group $G$ determines a unique conjugacy class of $p$-subgroups of $G$ which we will call the anchors of $χ$. This invariant has been considered by L. Barker in the context of finite $p$-solvable groups. Besides proving some basic properties of these anchors, we investigate the relation to other $p$-groups which can be attached to irreducible characters, such as defect groups, vertices in the sense of J. A. Green and vertices in the sense of G. Navarro.

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arXiv2015

Brauer's height zero conjecture for quasi-simple goups

We show that Brauer's height zero conjecture holds for blocks of finite quasi-simple groups. This result is used in Navarro-Späth's reduction of this conjecture for general groups to the inductive Alperin-McKay condition for simple groups.

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arXiv2014

Blocks with transitive fusion systems

Suppose that all nontrivial subsections of a $p$-block $B$ are conjugate (where $p$ is a prime). By using the classification of the finite simple groups, we prove that the defect groups of $B$ are either extraspecial of order $p^3$ with $p \in \{3,5\}$ or elementary abelian.

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arXiv2014

A characterisation of nilpotent blocks

Let $B$ be a $p$-block of a finite group, and set $m=$ $\sum χ(1)^2$, the sum taken over all height zero characters of $B$. Motivated by a result of M. Isaacs characterising $p$-nilpotent finite groups in terms of character degrees, we show that $B$ is nilpotent if and only if the exact power of $p$ dividing $m$ is equal to the $p$-part of $|G:P|^2|P:R|$, where $P$ is a defect group of $B$ and where $R$ is the focal subgroup of $P$ with respect to a fusion system $\CF$ of $B$ on $P$. The proof involves the hyperfocal subalgebra $D$ of a source algebra of $B$. We conjecture that all ordinary irreducible characters of $D$ have degree prime to $p$ if and only if the $\CF$-hyperfocal subgroup of $P$ is abelian.

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arXiv2013

On symmetric quotients of symmetric algebras

We investigate symmetric quotient algebras of symmetric algebras, with an emphasis on finite group algebras over a complete discrete valuation ring ${\mathcal O}$. Using elementary methods, we show that if an ordinary irreducible character $χ$ of a finite group $G$ gives rise to a symmetric quotient over ${\mathcal O}$ which is not a matrix algebra, then the decomposition numbers of the row labelled by $χ$ are all divisible by the characteristic $p$ of the residue field of ${\mathcal O}$.

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arXiv2013

2-blocks with abelian defect groups

We give a classification, up to Morita equivalence, of 2-blocks of quasi-simple groups with abelian defect groups. As a consequence, we show that Donovan's conjecture holds for elementary abelian 2-groups, and that the entries of the Cartan matrices are bounded in terms of the defect for arbitrary abelian 2-groups. We also show that a block with defect groups of the form $C_{2^m} \times C_{2^m}$ for $m \geq 2$ has one of two Morita equivalence types and hence is Morita equivalent to the Brauer correspondent block of the normaliser of a defect group. This completes the analysis of the Morita equivalence types of 2-blocks with abelian defect groups of rank 2, from which we conclude that Donovan's conjecture holds for such 2-groups. A further application is the completion of the determination of the number of irreducible characters in a block with abelian defect groups of order 16. The proof uses the classification of finite simple groups.

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arXiv2010

Bounds for Hochschild cohomology of block algebras

We show that for any block algebra B of a finite group over an algebraically closed field of prime characteristic the dimension of HH^n(B) is bounded by a function depending only on the nonnegative integer n and the defect of B. The proof uses in particular a theorem of Brauer and Feit which implies the result for n=0.

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