If f is a nonzero complex-valued function defined on a finite abelian group A and \hat f is its Fourier transform, then |Supp (f)||Supp {\hat f)| \ge |A|, where Supp (f) and Supp (\hat f) are the supports of f and \hat f. In this paper we generalize this known result in several directions. In particular, we prove an analogous inequality where the abelian group A is replaced by a transitive right G-set, where G is an arbitrary finite group. We obtain stronger inequalities when the G-set is primitive and we determine the primitive groups for which equality holds. We also explore connections between inequalities of this type and a result of Chebotarëv on complex roots of unity, and we thereby obtain a new proof of Chebotarëv's theorem.
We introduce a notion of permutation presentations of modules over finite groups, and completely determine finite groups over which every module has a permutation presentation. To get this result, we prove that every coflasque module over a cyclic p-group is permutation projective.
We show that the permutation module over $\mathbb{C}$ afforded by the action of $Sp_{2m}(2^f)$ on its natural module is isomorphic to the permutation module over $\mathbb{C}$ afforded by the action of $Sp_{2m}(2^f)$ on the union of the right cosets of $O_{2m}^+(2^f)$ and $O_{2m}^-(2^f)$.
We investigate finite non-Abelian simple groups $G$ for which the projective cover of the trivial module coincides with the permutation module on a subgroup and classify all cases unless $G$ is of Lie type in defining characteristic.
We study the odd-characteristic structure of permutation modules for the rank 3 natural actions of $O_{2n}^{\pm}(2)$ ($n\geq3$) and $U_{m}(2)$ ($m\geq4$) on nonsingular points of their standard modules.
We describe the structure, including composition factors and submodule lattices, of cross-characteristic permutation modules for the natural actions of the orthogonal groups $O_{m}^{\pm}(3)$ with $m\geq6$ on nonsingular points of their standard modules. These actions together with those studied in \cite{HN} are all examples of primitive rank 3 actions of finite classical groups on nonsingular points.
Permutation modules are fundamental in the representation theory of symmetric groups $\Sym_n$ and their corresponding Iwahori--Hecke algebras $\He = \He(\Sym_n)$. We find an explicit combinatorial basis for the annihilator of a permutation module in the "integral" case -- showing that it is a cell ideal in G.E. Murphy's cell structure of $\He$. The same result holds whenever $\He$ is semisimple, but may fail in the non-semisimple case.
The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant $γ_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics.
The minimal faithful permutation degree of a finite group $G$, denote by $μ(G)$ is the least non-negative integer $n$ such that $G$ embeds inside the symmetric group $\Sym(n)$. In this paper, we outline a Magma proof that 10 is the smallest degree for which there are groups $G$ and $H$ such that $μ(G \times H) < μ(G)+ μ(H)$.
Using cohomological methods, we prove the existence of a subgroup isomorphic to SL(2,q), q = -1 (mod 4), in the permutation module for PSL(2,q) in characteristic 2 that arises from the action on the projective line. A similar problem for q = 1 (mod 4) is reduced to the determination of certain first cohomology groups.
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.
Let $K$ be a field of characteristic two, and let $λ$ be a two-part partition of some natural number $r$. Denote the permutation module corresponding to the (maximal) Young subgroup $Σ_λ$ in $Σ_r$ by $M^λ$. We construct a full set of orthogonal primitive idempotents of the centraliser subalgebra $S_K(λ) = 1_λS_K(2,r) 1_λ= End_{KΣ_r}(M^λ)$ of the Schur algebra $S_K(2,r)$. These idempotents are naturally in one-to-one correspondence with the 2-Kostka numbers.
By means of a certain module V and its tensor powers in a finite tensor category, we study a question of whether the depth of a Hopf subalgebra R of a finite-dimensional Hopf algebra H is finite. The module V is the counit representation induced from R to H, which is then a generalized permutation module, as well as a module coalgebra. We show that if in the subalgebra pair either Hopf algebra has finite representation type, or V is either semisimple with R* pointed, projective, or its tensor powers satisfy a Burnside ring formula over a finite set of Hopf subalgebras including R, then the dep…
In [Boltje,Hartmann: Permutation resolutions for Specht modules, J. Algebraic Combin. 34 (2011), 141-162], a chain complex was constructed in a combinatorial way which conjecturally is a resolution of the (dual of the) integral Specht module for the symmetric group in terms of permutation modules. In this paper we extend the definition of the chain complex to the integral Iwahori Hecke algebra and prove the same partial exactness results that were proved in the symmetric group case. A complete proof of the exactness conjecture in the symmetric group case was recently given by Santana and Yudin…
The present survey aims at being a list of Conjectures and Problems in an area of model-theoretic algebra wide open for research, not a list of known results. To keep the text compact, it focuses on structures of finite Morley rank, although the same questions can be asked about other classes of objects, for example, groups definable in $ω$-stable and $o$-minimal theories. In many cases, answers are not known even in the classical category of algebraic groups over algebraically closed fields.
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.
In this paper, I show that if $p$ is an odd prime, and if $P$ is a finite $p$-group, then there exists an exact sequence of abelian groups $$0\to T(P)\to D(P)\to\lproj{P}\to H^1\big(\apdeux(P),\Z\big)^{(P)},$$ where $D(P)$ is the Dade group of $P$ and $T(P)$ is the subgroup of endo-trivial modules. Here $\lproj{P}$ is the group of sequences of compatible elements in the Dade groups $D\big(N_P(Q)/Q\big)$ for non trivial subgroups $Q$ of $P$. The poset $\apdeux(P)$ is the set of elementary abelian subgroups of rank at least 2 of $P$, ordered by inclusion. The group $H^1\big(\apdeux(P),\Z\big)^{(…
Let G be a finite group, let p be a prime number, and let K be a field of characteristic 0 and k be a field of characteristic p, both large enough. In this note we state explicit formulae for the primitive idempotents of K\otimes pp_k(G), where pp_k(G) is the ring of p-permutation kG-modules.
The group algebra of the permutation group is spanned by a set of elements called projectors. The coordinates of permutations expanded in projectors are matrix elements of irreducible representations. The projectors of the permutation group are a product of a Young symmetriser and a Young antisymmetriser. They form non-orthogonal bases of right and left modules. The non-orthogonality is compensated by a constant matrix. It turns out that this reduces the matrix entries to -1, 0,+1. An algorithm to compute the projectors is given.
It is proved that each Hoeffding space associated with a random permutation (or, equivalently, with extractions without replacement from a finite population) carries an irreducible representation of the symmetric group, equivalent to a two-block Specht module.
In this paper we classify all maximal subgroups M of a nearly simple primitive rank 3 group G of type L=Omega_{2m+1}(3), m > 3; acting on an L-orbit E of non-singular points of the natural module for L such that 1_P^G <=1_M^G where P is a stabilizer of a point in E. This result has an application to the study of minimal genera of algebraic curves which admit group actions.
We extend the notion of a {$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 fini…
Let $k$ be an algebraically closed field of positive characteristic $p$, and $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider the $\mathbb{F}$-linear category $\mathbb{F} pp_k^Δ$ of finite groups, in which the set of morphisms from $G$ to $H$ is the $\mathbb{F}$-linear extension $\mathbb{F} T^Δ(H,G)$ of the Grothendieck group $T^Δ(H,G)$ of $p$-permutation $(kH,kG)$-bimodules with (twisted) diagonal vertices. The $\mathbb{F}$-linear functors from $\mathbb{F} pp_k^Δ$ to $\mathbb{F}\hbox{-Mod}$ are called {\em diagonal $p$-permutation functors}. They form an abelian …
