arXiv subjects

David J. Benson

Publications and source records attributed to David J. Benson.

14 recordsLinked to original sources

arXiv2026

Simple modules for affine nilCoxeter algebras

We study the representation theory of the affine nilCoxeter algebra $A$ of type $\tilde A_{n-1}$, over a field $k$ of any characteristic. Our main theorem states that this is a Noetherian prime affine PI algebra of PI degree $n!$. As a consequence, the simple $A$-modules are all finite dimensional, and the maximum dimension of a simple module is $n!$ over a suitable finite extension of $k$. To achieve this, we investigate a large commutative subalgebra $C$ which is finitely generated as an algebra and over which $A$ is finitely generated as a module. We show that the associated primes of $C$ are minimal primes, and there are $n!$ of them, regularly permuted by $\mathfrak{S}_n$. The algebra $R=C^{\mathfrak{S}_n}$ is equal to the centre of $A$, and isomorphic to $C/\mathfrak{p}$ for each of the minimal primes $\mathfrak{p}$. We prove that the ring $R$ is isomorphic to $k[X_1,\dots,X_{n-1}]^{\mu_n}$, where $\mu_n$ is the finite group scheme of $n$th roots of unity, acting so that $X_i$ has degree $i$ modulo $n$. The ring $R$ is Cohen--Macaulay, and is Gorenstein if and only if $n$ is odd or $n=2$. It is a toric ring, with divisor class group $\mathsf{Cl}(R)\cong\mathbb{Z}/n$, and every projective $R$-module is free.

math.RT↗

arXiv2025

Classifying spaces of finite groups of tame representation type

Thanks to the work of Karin Erdmann, we know a great deal about the representation theory of blocks of finite groups with tame representation type. Our purpose here is to examine the $p$-completed classifying spaces of these blocks and their loop spaces. We pay special attention to the $A_\infty$ algebra structures, and singularity and cosingularity categories.

math.RT↗

arXiv2025

Projective Modules and Cohomology for Integral Basic Algebras

Algebras defined over fields of characteristic zero and positive characteristic usually do not behave the same way. However, for certain algebras, for example the group algebras, they behave the same way as the characteristic zero case at "good enough" prime. In this paper, we initiate the study of this topic by imposing increasingly strong hypotheses on basic algebras. When the algebras satisfy the right hypotheses, we have equalities of the dimensions of their cohomology groups between simple modules and equalities of graded Cartan numbers. The examples include the Solomon descent algebras of finite Coxeter groups at large enough primes, nilCoxeter algebra, and certain finite semigroup algebras at an arbitrary prime.

math.RT↗

arXiv2024

The cohomology of the nilCoxeter algebra

The nilCoxeter algebra $\mathcal{N}S_n$ of the symmetric group $S_n$ is the algebra over $\mathbb{Z}$ with generators $Y_i$ ($1\leqslant i\leqslant n-1$), satisfying the braid relations $Y_iY_{i+1}Y_i=Y_{i+1}Y_iY_{i+1}$, $Y_iY_j=Y_jY_i$ ($|j-i|\geqslant 2$), together with the relations $Y_i^2=0$. We describe an explicit presentation for the cohomology ring $Z\cong\mathsf{Ext}^*_{\mathcal{N}S_n}(\mathbb{Z},\mathbb{Z})$, with $n-i$ new generators in degree $i$ for $0< i<n$, and all relations are quadratic. We show that this $\mathsf{Ext}$ ring is $\mathbb{Z}$-free, and that it is a semiprime Noetherian affine polynomial identity (PI) ring with Poincaré series $1/(1-t)^{n-1}$ and PI degree $2^{n-2}$. For any field of coefficients $\mathbf{k}$, we show that $\mathsf{Ext}^*_{\mathbf{k}\mathcal{N}S_n}(\mathbf{k},\mathbf{k})$ is $\mathbf{k}\otimes_{\mathbb{Z}} Z$. Similar results hold for other finite Coxeter types. In the final section we show that $Z$ is a Koszul algebra whose Koszul dual is a signed version of the nilcactus algebra, an algebra closely related to the cactus group.

math.RA↗

arXiv2023

Modules with finitely generated cohomology

Let $G$ be a finite group and $\mathsf{k}$ a field of characteristic $p$. It is conjectured in a paper of the first author and John Greenlees that the thick subcategory of the stable module category StMod$(\mathsf{k}G)$ consisting of modules whose cohomology is finitely generated over $\mathsf{H}^*(G,\mathsf{k})$ is generated by finite dimensional modules and modules with no cohomology. If the centraliser of every element of order $p$ in $G$ is $p$-nilpotent, this statement follows from previous work. Our purpose here is to prove this conjecture in two cases with non $p$-nilpotent centralisers. The groups involved are ${\mathbb Z}/3^r\timesΣ_3$ ($r> 0$) in characteristic three and ${\mathbb Z}/2\times A_4$ in characteristic two. As a consequence, in these cases the bounded derived category of $C^*BG$ (cochains on $BG$ with coefficients in $\mathsf{k}$) is generated by $C^*BS$, where $S$ is a Sylow $p$-subgroup of $G$.

math.RT↗

arXiv2023

Matrices for finite group representations that respect Galois automorphisms

We are given a finite group $H$, an automorphism $τ$ of $H$ of order $r$, a Galois extension $L/K$ of fields of characteristic zero with cyclic Galois group $\langleσ\rangle$ of order $r$, and an absolutely irreducible representation $ρ\colon H\to\operatorname{\sf GL}(n,L)$ such that the action of $τ$ on the character of $ρ$ is the same as the action of $σ$. Then the following are equivalent. $\bullet$ $ρ$ is equivalent to a representation $ρ'\colon H\to\operatorname{\sf GL}(n,L)$ such that the action of $σ$ on the entries of the matrices corresponds to the action of $τ$ on $H$, and $\bullet$ the induced representation $\operatorname{\sf ind}_{H,H\rtimes\langleτ\rangle}(ρ)$ has Schur index one; that is, it is similar to a representation over $K$. As examples, we discuss a three dimensional irreducible representation of $A_5$ over $\mathbb{Q}[\sqrt5]$ and a four dimensional irreducible representation of the double cover of $A_7$ over $\mathbb{Q}[\sqrt{-7}]$.

math.RT↗

arXiv2023

Modules with finitely generated cohomology, and singularities of $C^*BG$

Let $G$ be a finite group and $k$ a field of characteristic $p$. We conjecture that if $M$ is a $kG$-module with $H^*(G,M)$ finitely generated as a module over $H^*(G,k)$ then as an element of the stable module category $\mathsf{StMod}(kG)$, $M$ is contained in the thick subcategory generated by the finitely generated $kG$-modules and the modules $M'$ with $H^*(G,M')=0$. We show that this is equivalent to a conjecture of the second author about generation of the bounded derived category of cochains $C^*(BG;k)$, and we prove the conjecture in the case where the centraliser of every element of $G$ of order $p$ is $p$-nilpotent. In this case some stronger statements are true, that probably fail for more general finite groups.

math.RT↗

arXiv2023

The socle of the group algebra of a finite $p$-group

Let $G$ be a finite $p$-group, and $α$ an automorphism of the group algebra ${\mathbb F}_pG$. Then $α$ fixes the socle of ${\mathbb F}_pG$ pointwise. More generally, if $k$ is a field of characteristic $p$, and $α$ is a $k$-algebra automorphism of $kG$, then $α$ induces a linear action on the dimension subquotients of the group, and the action on the socle is scalar multiplication by the $(p-1)$st power of the product of the determinants of this action. The scalar is thus an element of $(k^\times)^{p-1}$.

math.RT↗

arXiv2022

Centralisers of finite groups in locally finite simple groups

We answer in the negative a question of Hartley about representations of finite groups, by constructing examples of finite simple groups with arbitrarily large representations whose endomorphism ring consists of just the scalars. We show as a consequence that there are finite simple groups of automorphisms of the locally finite simple group $SL(\infty,\mathbb{F}_q)$ with trivial centraliser. The smallest of our examples is $A_6$ with $q=9$.

math.GR↗

arXiv2020

Bounded complexes of permutation modules

Let $k$ be a field of characteristic $p > 0$. For $G$ an elementary abelian $p$-group, there exist collections of permutation module such that if $C^*$ is any exact bounded complex whose terms are sums of copies of modules from the collection, then $C^*$ is contractible. A consequence is that if $G$ is any finite group whose Sylow $p$-subgroups are not cyclic or quaternion, and if $C^*$ is a bounded exact complex such that each $C^i$ is direct sum of one dimensional modules and projective modules, then $C^*$ is contractible.

math.GR↗

arXiv2015

Nature of short high amplitude pulses in a periodic dissipative laminate metamaterial

We study the evolution of high amplitude stress pulses in periodic dissipative laminates taking into account the nonlinear constitutive equations of the components and their dissipative behavior. Aluminum and Tungsten laminate was taken as an example due to the large difference in acoustic impedances of aluminum and tungsten, the significant nonlinearity of aluminum constitutive equation at the investigated range of stresses, and its possible practical applications. Laminates with different cell size, which controls internal time scale, impacted by the plates with different thicknesses, determining the incoming pulse duration, were investigated. It has been observed that the ratio of the duration of the incoming pulse to the internal characteristic time determines the nature of the high amplitude dissipative propagating waves, the oscillatory shock like profile, the train of localized pulses or a single localized pulse. These localized quasistationary waves resemble solitary waves even in the presence of dissipation: the similar pulses emerged from different initial conditions, indicating that they are inherent properties of the corresponding laminates, their characteristic length scale is determined by the mesostructural scale and the stress amplitude, and they mostly recover their shapes after collision with phase shift and a linear relationship exists between their speed and amplitude. A theoretical description approximating the shape, length scale and speed of these high amplitude dissipative pulses was proposed based on the Korteweg de Vries type equation with a dispersive term dictated by the mesostructure. The nonlinear term is derived from nonlinear constitutive equations of Aluminum and Tungsten, which were extracted from their Hugoniot curves.

cond-mat.mtrl-sci↗

arXiv2010

A realization theorem for modules of constant Jordan type and vector bundles

Let E be an elementary abelian p-group of rank r and let k be a field of characteristic p. We introduce functors F_i from finitely generated kE-modules of constant Jordan type to vector bundles over projective space of dimension r-1. The fibers of these functors encode complete information about the Jordan type of the module. We prove that given any vector bundle of rank s on P^{r-1}, there is a kE-module M of stable constant Jordan type [1]^s such that the functor F_1 applied to M yields the original vector bundle for p=2 and the Frobenius twist of the original vector bundle for p>2. We also prove that the theorem cannot be improved if p is odd, because if M is any module of stable constant Jordan type [1]^s then the Chern numbers c_1, ... ,c_{p-2} of F_1(M) are divisible by p.

math.RT↗

arXiv2007

The generating hypothesis for the stable module category of a $p$-group

Freyd's generating hypothesis, interpreted in the stable module category of a finite p-group G, is the statement that a map between finite-dimensional kG-modules factors through a projective if the induced map on Tate cohomology is trivial. We show that Freyd's generating hypothesis holds for a non-trivial finite p-group G if and only if G is either C_2 or C_3. We also give various conditions which are equivalent to the generating hypothesis.

math.RT↗

arXiv2003

Symmetries of Kirchberg algebras

Let A be a separable unital nuclear purely infinite simple C*-algebra satisfying the Universal Coefficient Theorem, and such that the K_0-class of the identity is zero. We prove that every automorphism of order two of the K-theory of A is implemented by an automorphism of A of order two. As a consequence, we prove that every countable Z/2Z-graded module over the representation ring of Z/2Z is isomorphic to the equivariant K-theory for some action of Z/2Z on a separable unital nuclear purely infinite simple C*-algebra. Along the way, we prove that every not necessarily finitely generated module over the group ring of Z/2Z which is free as an abelian group has a direct sum decomposition with only three kinds of summands, namely the group ring itself and Z on which the nontrivial element of Z/2Z acts either trivially or by multiplication by -1.

math.OA↗