Permutation Modules Research Articles

3-setwise intersecting families of the symmetric group

Given two positive integers n≥3 and t≤n, the permutations σ,π∈Sym(n) are t-setwise intersecting if they agree (setwise) on a t-subset of {1,2,…,n}. A family F⊂Sym(n) is t-setwise intersecting if any two permutations of F are t-setwise intersecting. Ellis (2012) [6] conjectured that if t≤n and F⊂Sym(n) is a t-setwise intersecting family, then |F|≤t!(n−t)! and equality holds only if F is a coset of a setwise stabilizer of a t-subset of {1,2,…,n}.In this paper, we prove that if n≥11 and F⊂Sym(n) is 3-setwise intersecting, then |F|≤6(n−3)!. Moreover, we prove that the characteristic vector of a 3-setwise intersecting family of maximum size lies in the sum of the eigenspaces induced by the permutation module of Sym(n) acting on the 3-subsets of {1,2,…,n}.

APM: Adaptive permutation module for point cloud classification

Despite deep convolutional networks have been highly successful in 2D vision tasks, extending them to point cloud classification task is still challenging due to irregularities of point cloud data relative to image grids. Recent methods usually take advantage of symmetric operators like max-pooling, to deal with point cloud order ambiguity. However, this kind of treatment does not consider the latent geometric information contained in the spatial order, which may limit the performance of feature learning. To address this issue, we present an adaptive permutation module (APM) that calculates a particular permutation from the input point clouds to achieve permutation invariance, as demonstrated by the visualization of the APM output feature maps. Thorough experiments are conducted to demonstrate the superiority of APM as well. In addition, the APM can be plugged into other state-of-the-art approaches flexibly to further improve performance in classification task. We build an end-to-end deep convolutional neural network applying PointCNN as our backbone combined with the adaptive permutation module and achieve state-of-the-art performance in point cloud classification task. Our work demonstrates that the latent spatially-local correlations play a critical role in feature learning on point clouds.

Irreducible representations of the symmetric groups from slash homologies of p-complexes

In the 40s, Mayer introduced a construction of (simplicial) $p$-complex by using the unsigned boundary map and taking coefficients of chains modulo $p$. We look at such a $p$-complex associated to an $(n-1)$-simplex; in which case, this is also a $p$-complex of representations of the symmetric group of rank $n$ - specifically, of permutation modules associated to two-row compositions. In this article, we calculate the so-called slash homology - a homology theory introduced by Khovanov and Qi - of such a $p$-complex. We show that every non-trivial slash homology group appears as an irreducible representation associated to two-row partitions, and how this calculation leads to a basis of these irreducible representations given by the so-called $p$-standard tableaux.

The Benson - Symonds Invariant for Permutation Modules

In a recent paper, Dave Benson and Peter Symonds defined a new invariant γG(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 permutation modules of the symmetric group corresponding to two-part partitions using tools from representation theory and combinatorics.

A note on finite group 1-cohomology via semi-direct products with applications to permutation modules

A note on finite group 1-cohomology via semi-direct products with applications to permutation modules

Cellularity of endomorphism algebras of Young permutation modules

Let E be an n-dimensional vector space. Then the symmetric group Sym(n) acts on E by permuting the elements of a basis and hence on the r-fold tensor product E⊗r. Bowman, Doty and Martin ask, in [1], whether the endomorphism algebra EndSym(n)(E⊗r) is cellular. The module E⊗r is the permutation module for a certain Young Sym(n)-set. We shall show that the endomorphism algebra of the permutation module on an arbitrary Young Sym(n)-set is a cellular algebra. We determine, in terms of the point stabilisers which appear, when the endomorphism algebra is quasi-hereditary.

On signed p-Kostka matrices

We show that the signed p -Kostka numbers depend just on p -Kostka numbers and the multiplicities of projective indecomposable modules in certain signed Young permutation modules. We then examine the signed p -Kostka number k ( α | β ) , ( λ | p μ ) in the case when | β | = p | μ | . This allows us to explicitly describe the multiplicities of direct summands of a signed Young permutation module lying in the principal block of F S m p in terms of the p -Kostka numbers.

On the symmetric and exterior powers of Young permutation modules

Let n be a positive integer and λ be a partition of n. Let Mλ be the Young permutation module labelled by λ. In this paper, we study symmetric and exterior powers of Mλ in positive characteristic case. We determine the symmetric and exterior powers of Mλ that are projective. All the indecomposable exterior powers of Mλ are also classified. We then prove some results for indecomposable direct summands that have the largest complexity in direct sum decompositions of some symmetric and exterior powers of Mλ. We end by parameterizing all the Scott modules that are isomorphic to direct summands of the symmetric or exterior square of Mλ and determining their corresponding multiplicities explicitly.

Resolutions by permutation modules

We prove that, up to adding a complement, every modular representation of a finite group admits a finite resolution by permutation modules.

Permutation modules for cellularly stratified algebras

Permutation modules play an important role in the representation theory of the symmetric group. Hartmann and Paget defined permutation modules for Brauer algebras. We generalise their construction to a wider class of algebras, namely cellularly stratified algebras, satisfying certain conditions. We give a decomposition into indecomposable summands, the Young modules, and show that permutation modules and Young modules admit cell filtrations (with well-defined filtration multiplicities). Partition algebras are shown to satisfy the given conditions, provided the characteristic of the underlying field is large enough. Thus we obtain a definition of permutation modules for partition algebras as an application.

A Characterization of Permutation Modules Extending a Theorem of Weiss

Let G be a finite p -group with normal subgroup N . A celebrated theorem of A. Weiss gives a sufficient condition for a \mathbb{Z}_pG -lattice to be a permutation module, looking only at its restriction to N and its N -fixed points. In case N has order p , we extend the condition of Weiss to a characterization.

Minimal binary 2-neighbour-transitive codes

The main result here is a characterisation of binary 2-neighbour-transitive codes with minimum distance at least 5 via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of codes comes primarily from their relationship to the class of completely regular codes. The results contained here yield many more examples of 2-neighbour-transitive codes than previous classification results of families of 2-neighbour-transitive codes. In the process, new lower bounds on the minimum distance of particular sub-families are produced. Several results on the structure of 2-neighbour-transitive codes with arbitrary alphabet size are also proved. The proofs of the main results apply the classification of minimal and pre-minimal submodules of the permutation modules over F2 for finite 2-transitive permutation groups.

Endomorphism algebras of 2-row permutation modules in characteristic 3

Given r∈N, let λ be a partition of r with at most two parts. Let F be a field of characteristic 3. Write Mλ for the FSr-permutation module corresponding to the action of the symmetric group Sr on the cosets of the maximal Young subgroup Sλ. We construct a full set of central primitive idempotents in EndFSr(Mλ) in this case. We also determine the Young module corresponding to each primitive idempotent that we construct.

The multistep homology of the simplex and representations of symmetric groups

AbstractThe symmetric group on a set acts transitively on the set of its subsets of a fixed size. We define homomorphisms between the corresponding permutation modules, defined over a field of characteristic two, which generalize the boundary maps from simplicial homology. The main results determine when these chain complexes are exact and when they are split exact. As a corollary we obtain a new explicit construction of the basic spin modules for the symmetric group.

Affine quiver Schur algebras and p-adic $${\textit{GL}}_n$$ GL n

In this paper we consider the (affine) Schur algebra which arises as the endomorphism algebra of certain permutation modules for the Iwahori–Matsumoto Hecke algebra. This algebra describes, for a general linear group over a p-adic field, a large part of the unipotent block over fields of characteristic different from p. We show that this Schur algebra is, after a suitable completion, isomorphic to the quiver Schur algebra attached to the cyclic quiver. The isomorphism is explicit, but nontrivial. As a consequence, the completed (affine) Schur algebra inherits a grading. As a byproduct we obtain a detailed description of the algebra with a basis adapted to the geometric basis of quiver Schur algebras. We illustrate the grading in the explicit example of $${\text {GL}}_2({\mathbb {Q}}_5)$$ in characteristic 3.

An efficient high dimensional quantum Schur transform

The Schur transform is a unitary operator that block diagonalizes the action of the symmetric and unitary groups on an n fold tensor product V⊗n of a vector space V of dimension d. Bacon, Chuang and Harrow [5] gave a quantum algorithm for this transform that is polynomial in n, d and log⁡ϵ−1, where ϵ is the precision. In a footnote in Harrow's thesis [18], a brief description of how to make the algorithm of [5] polynomial in log⁡d is given using the unitary group representation theory (however, this has not been explained in detail anywhere). In this article, we present a quantum algorithm for the Schur transform that is polynomial in n, log⁡d and log⁡ϵ−1 using a different approach. Specifically, we build this transform using the representation theory of the symmetric group and in this sense our technique can be considered a ''dual" algorithm to [5]. A novel feature of our algorithm is that we construct the quantum Fourier transform over the so called permutation modules, which could have other applications.

Presentations of categories of modules using the Cautis–Kamnitzer–Morrison principle

We study commuting actions of a complex reductive Lie algebra, \mathfrak g , and an algebra A on a finite dimensional vector space V . We outline how such commuting actions can be used to give a presentation of the category of A -modules isomorphic to \mathfrak g -weight spaces of V . This generalizes a result of Cautis–Kamnitzer–Morrison (2014) in the non-quantum setting. As examples, we apply this to classical Schur–Weyl duality and the Brauer Schur–Weyl dualities. In particular we obtain a diagrammatic presentation, in terms of generators and relations, of the category of permutation modules of the symmetric group, S_d . This is used to give a new description of the Kronecker product of the S_d -equivariant morphisms between permutation modules.

Relations between permutation representations in positive characteristic

Given a finite group G and a field F, a G-set X gives rise to an F[G]-permutation module F[X]. This defines a map from the Burnside ring of G to its representation ring over F. It is an old problem in representation theory, with wide-ranging applications in algebra, number theory, and geometry, to give explicit generators of the kernel K_F(G) of this map, i.e. to classify pairs of G-sets X, Y such that F[X] is isomorphic to F[Y]. When F has characteristic 0, a complete description of K_F(G) is now known. In this paper, we give a similar description of K_F(G) when F is a field of characteristic p>0 in all but the most complicated case, which is when G has a subquotient that is a non-p-hypo-elementary (p,p)-Dress group.

Straightening rule for an m′-truncated polynomial ring

We consider a certain quotient of a polynomial ring categorified by both the isomorphic Green rings of the symmetric groups and Schur algebras generated by the signed Young permutation modules and mixed powers respectively. They have bases parametrised by pairs of partitions whose second partitions are multiples of the odd prime p the characteristic of the underlying field. We provide an explicit formula rewriting a signed Young permutation module (respectively, mixed power) in terms of signed Young permutation modules (respectively, mixed powers) labelled by those pairs of partitions. As a result, for each partition λ, we discovered the number of compositions δ such that δ can be rearranged to λ and whose partial sums of δ are not divisible by p.

THE VERTICES OF THE COMPONENTS OF THE PERMUTATION MODULE INDUCED FROM PARABOLIC GROUPS

We consider the permutation module $k_P{\uparrow^{{\rm GL}_n(p^f)}}$, where $P$ is a parabolic group in the general linear group ${\rm GL}_n(p^f)$ and $k$ is an algebraically closed field of prime characteristic $p$. The vertices of the components of these modules have been calculated in [9] by Tinberg, who studied these modules for all groups with split BN-pairs in characteristic $p$. In this paper we show that the idea of suitability is strong enough to find all $p$-groups that are vertex of some component of $k_P{\uparrow^{{\rm GL}_n(p^f)}}$. Furthermore using a result of Burry and Carlson we show that all components have a different vertex.