arXiv subjects

David Benson

Publications and source records attributed to David Benson.

14 recordsLinked to original sources

arXiv2023

Hochschild cohomology of symmetric groups and generating functions,II

We relate the generating functions of the dimensions of the Hochschild cohomology in any fixed degree of the symmetric groups with those of blocks of the symmetric groups. We show that the first Hochschild cohomology of a positive defect block of a symmetric group is non-zero, answering in the affirmative a question of the third author. To do this, we prove a formula expressing the dimension of degree one Hochschild cohomology as a sum of dimensions of centres of blocks of smaller symmetric groups. This in turn is a consequence of a general formula that makes more precise a theorem of our previous paper describing the generating functions for the dimensions of Hochschild cohomology of symmetric groups.

math.GR↗

arXiv2022

Formality of cochains on BG

Let $G$ be a compact Lie group with maximal torus $T$. If $|N_G(T)/T|$ is invertible in the field $k$ then the algebra of cochains $C^*(BG;k)$ is formal as an $A_\infty$ algebra, or equivalently as a DG algebra.

math.AT↗

arXiv2022

Hochschild cohomology of symmetric groups in low degrees

We compute the dimensions of the Hochschild cohomology of symmetric groups over prime fields in low degrees. This involves us in studying some partition identities and generating functions of the dimensions in any fixed degree of the Hochschild cohomology of symmetric groups.

math.GR↗

arXiv2022

Structure of blocks with normal defect and abelian $p'$ inertial quotient

Let $k$ be an algebraically closed field of prime characteristic $p$. Let $kGe$ be a block of a group algebra of a finite group $G$, with normal defect group $P$ and abelian $p'$ inertial quotient $L$. Then we show that $kGe$ is a matrix algebra over a quantised version of the group algebra of a semidirect product of $P$ with a certain subgroup of $L$. To do this, we first examine the associated graded algebra, using a Jennings--Quillen style theorem. As an example, we calculate the associated graded of the basic algebra of the non-principal block in the case of a semidirect product of an extraspecial $p$-group $P$ of exponent $p$ and order $p^3$ with a quaternion group of order eight with the centre acting trivially. In the case $p=3$ we give explicit generators and relations for the basic algebra as a quantised version of $kP$. As a second example, we give explicit generators and relations in the case of a group of shape $2^{1+4}:3^{1+2}$ in characteristic two.

math.RT↗

arXiv2020

On cohomology in symmetric tensor categories in prime characteristic

We describe graded commutative Gorenstein algebras ${\mathcal E}_n(p)$ over a field of characteristic $p$, and we conjecture that $\mathrm{Ext}^\bullet_{\mathsf{Ver}_{p^{n+1}}}(1,1)\cong{\mathcal E}_{n}(p)$, where $\mathsf{Ver}_{p^{n+1}}$ are the new symmetric tensor categories recently constructed in \cite{Benson/Etingof:2019a,Benson/Etingof/Ostrik,Coulembier}. We investigate the combinatorics of these algebras, and the relationship with Minc's partition function, as well as possible actions of the Steenrod operations on them. Evidence for the conjecture includes a large number of computations for small values of $n$. We also provide some theoretical evidence. Namely, we use a Koszul construction to identify a homogeneous system of parameters in ${\mathcal E}_n(p)$ with a homogeneous system of parameters in $\mathrm{Ext}^\bullet_{\mathsf{Ver}_{p^{n+1}}}(1,1)$. These parameters have degrees $2^i-1$ if $p=2$ and $2(p^i-1)$ if $p$ is odd, for $1\le i \le n$. This at least shows that $\mathrm{Ext}^\bullet_{\mathsf{Ver}_{p^{n+1}}}(1,1)$ is a finitely generated graded commutative algebra with the same Krull dimension as ${\mathcal E}_n(p)$. For $p=2$ we also show that $\mathrm{Ext}^\bullet_{\mathsf{Ver}_{2^{n+1}}}(1,1)$ has the expected rank $2^{n(n-1)/2}$ as a module over the subalgebra of parameters.

math.RT↗

arXiv2020

Modular representation theory and commutative Banach algebras

In a recent paper of Benson and Symonds, a new invariant was introduced for modular representations of a finite group. An interpretation was given as a spectral radius with respect to a Banach algebra completion of the representation ring. Our purpose here is to take these notions further, and investigate the structure of the resulting Banach algebras. Some of the material in that paper is repeated here in greater generality, and for clarity of exposition. We give an axiomatic definition of an abstract representation ring, and representation ideal. The completion is then a commutative Banach algebra, and the techniques of Gelfand from the 1940s are applied in order to study the space of algebra homomorphisms to $\mathbb C$. One surprising consequence of this investigation is that the Jacobson radical and the nil radical of a (complexified) representation ring always coincide. These notes are intended for representation theorists. So background material on commutative Banach algebras is given in detail, whereas representation theoretic background is more condensed.

math.RT↗

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 $P\rtimes L$.

math.RT↗

arXiv2016

A Kernel-based Lagrangian Method for Imperfectly-mixed Chemical Reactions

Current Lagrangian (particle-tracking) algorithms used to simulate diffusion-reaction equations must employ a certain number of particles to properly emulate the system dynamics---particularly for imperfectly-mixed systems. The number of particles is tied to the statistics of the initial concentration fields of the system at hand. Systems with shorter-range correlation and/or smaller concentration variance require more particles, potentially limiting the computational feasibility of the method. For the well-known problem of bimolecular reaction, we show that using kernel-based, rather than Dirac-delta, particles can significantly reduce the required number of particles. We derive the fixed width of a Gaussian kernel for a given reduced number of particles that analytically eliminates the error between kernel and Dirac solutions at any specified time. We also show how to solve for the fixed kernel size by minimizing the squared differences between solutions over any given time interval. Numerical results show that the width of the kernel should be kept below about 12% of the domain size, and that the analytic equations used to derive kernel width suffer significantly from the neglect of higher-order moments. The simulations with a kernel width given by least squares minimization perform better than those made to match at one specific time. A heuristic time-variable kernel size, based on the previous results, performs on a par with the least squares fixed kernel size.

math.NA↗

arXiv2015

A Counterexample for Lightning Flash Modules over E(e1,e2)

We give a counterexample to Theorem 5 in Section 18.2 of Margolis' book, "Spectra and the Steenrod Algebra", and make remarks about the proofs of some later theorems in the book that depend on it. The counterexample is a module which does not split as a sum of lightning flash modules and free modules.

math.RT↗

arXiv2013

Group cohomology and control of p-fusion

We show that if an inclusion of finite groups H < G of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F-isomorphism in mod p cohomology, then H controls p-fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p-subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p=2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories. The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p-fusion systems over the same finite p-group are detected on elementary abelian p-groups if p odd and abelian 2-groups of exponent at most 4 if p=2.

math.AT↗

arXiv2009

Schur-Weyl duality over finite fields

We prove a version of Schur--Weyl duality over finite fields. We prove that for any field $k$, if $k$ has at least $r+1$ elements, then Schur--Weyl duality holds for the $r$th tensor power of a finite dimensional vector space $V$. Moreover, if the dimension of $V$ is at least $r+1$, the natural map $k\Sym_r \to End\_{GL(V)}(V^{\otimes r})$ is an isomorphism. This isomorphism may fail if $\dim_k V$ is not strictly larger than $r$.

math.GR↗

arXiv2007

Local helioseismology and correlation tracking analysis of surface structures in realistic simulations of solar convection

We apply time-distance helioseismology, local correlation tracking and Fourier spatial-temporal filtering methods to realistic supergranule scale simulations of solar convection and compare the results with high-resolution observations from the SOHO Michelson Doppler Imager (MDI). Our objective is to investigate the surface and sub-surface convective structures and test helioseismic measurements. The size and grid of the computational domain are sufficient to resolve various convective scales from granulation to supergranulation. The spatial velocity spectrum is approximately a power law for scales larger than granules, with a continuous decrease in velocity amplitude with increasing size. Aside from granulation no special scales exist, although a small enhancement in power at supergranulation scales can be seen. We calculate the time-distance diagram for f- and p-modes and show that it is consistent with the SOHO/MDI observations. From the simulation data we calculate travel time maps for surface gravity waves (f-mode). We also apply correlation tracking to the simulated vertical velocity in the photosphere to calculate the corresponding horizontal flows. We compare both of these to the actual large-scale (filtered) simulation velocities. All three methods reveal similar large scale convective patterns and provide an initial test of time-distance methods.

astro-ph↗

arXiv2006

Validation of Time-Distance Helioseismology by Use of Realistic Simulations of Solar Convection

Recent progress in realistic simulations of solar convection have given us an unprecedented opportunity to evaluate the robustness of solar interior structures and dynamics obtained by methods of local helioseismology. We present results of testing the time-distance method using realistic simulations. By computing acoustic wave propagation time and distance relations for different depths of the simulated data, we confirm that acoustic waves propagate into the interior and then turn back to the photosphere. This demonstrates that in the numerical simulations properties of acoustic waves (p-modes) are similar to the solar conditions, and that these properties can be analyzed by the time-distance technique. For the surface gravity waves (f-mode), we calculate perturbations of their travel times, caused by localized downdrafts, and demonstrate that the spatial pattern of these perturbations (representing so-called sensitivity kernels) is similar to the patterns obtained from the real Sun, displaying characteristic hyperbolic structures. We then test the time-distance measurements and inversions by calculating acoustic travel times from a sequence of vertical velocities at the photosphere of the simulated data, and inferring a mean 3D flow fields by performing inversion based on the ray approximation. The inverted horizontal flow fields agree very well with the simulated data in subsurface areas up to 3 Mm deep, but differ in deeper areas. Due to the cross-talk effects between the horizontal divergence and downward flows, the inverted vertical velocities are significantly different from the mean convection velocities of the simulation dataset.

astro-ph↗

arXiv2003

Dickson invariants, regularity and computation in group cohomology

In this paper, we investigate the commutative algebra of the cohomology ring $H^*(G,k)$ of a finite group $G$ over a field $k$. We relate the concept of quasi-regular sequence, introduced by Benson and Carlson, to the local cohomology of the cohomology ring. We give some slightly strengthened versions of quasi-regularity, and relate one of them to Castelnuovo--Mumford regularity. We prove that the existence of a quasi-regular sequence in either the original sense or the strengthened ones is true if and only if the Dickson invariants form a quasi-regular sequence in the same sense. The proof involves the notion of virtual projectivity, introduced by Carlson, Peng and Wheeler. As a by-product of this investigation, we give a new proof of the Bourguiba--Zarati theorem on depth and Dickson invariants, in the context of finite group cohomology, without using the machinery of unstable modules over the Steenrod algebra. Finally, we describe an improvement of Carlson's algorithm for computing the cohomology of a finite group using a finite initial segment of a projective resolution of the trivial module. In contrast to Carlson's algorithm, ours does not depend on verifying any conjectures during the course of the calculation, and is always guaranteed to work.

math.GR↗