Permutation Modules Research Articles

Realization of Permutation Modules via Alexandroff Spaces

We raise the question of the realizability of permutation modules in the context of Kahn’s realizability problem for abstract groups and the G-Moore space problem. Specifically, given a finite group G, we consider a collection {Mi}i=1n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{M_i\\}_{i=1}^n$$\\end{document} of finitely generated ZG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}G$$\\end{document}-modules that admit a submodule decomposition on which G acts by permuting the summands. Then we prove the existence of connected finite spaces X that realize each Mi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_i$$\\end{document} as its i-th homology, G as its group of self-homotopy equivalences E(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {E}(X)$$\\end{document}, and the action of G on each Mi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_i$$\\end{document} as the action of E(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {E}(X)$$\\end{document} on Hi(X;Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_i(X; \\mathbb {Z})$$\\end{document}.

Software and hardware realizations for different designs of chaos-based secret image sharing systems

Secret image sharing (SIS) conveys a secret image to mutually suspicious receivers by sending meaningless shares to the participants, and all shares must be present to recover the secret. This paper proposes and compares three systems for secret sharing, where a visual cryptography system is designed with a fast recovery scheme as the backbone for all systems. Then, an SIS system is introduced for sharing any type of image, where it improves security using the Lorenz chaotic system as the source of randomness and the generalized Arnold transform as a permutation module. The second SIS system further enhances security and robustness by utilizing SHA-256 and RSA cryptosystem. The presented architectures are implemented on a field programmable gate array (FPGA) to enhance computational efficiency and facilitate real-time processing. Detailed experimental results and comparisons between the software and hardware realizations are presented. Security analysis and comparisons with related literature are also introduced with good results, including statistical tests, differential attack measures, robustness tests against noise and crop attacks, key sensitivity tests, and performance analysis.

TOP-ReID: Multi-Spectral Object Re-identification with Token Permutation

Multi-spectral object Re-identification (ReID) aims to retrieve specific objects by leveraging complementary information from different image spectra. It delivers great advantages over traditional single-spectral ReID in complex visual environment. However, the significant distribution gap among different image spectra poses great challenges for effective multi-spectral feature representations. In addition, most of current Transformer-based ReID methods only utilize the global feature of class tokens to achieve the holistic retrieval, ignoring the local discriminative ones. To address the above issues, we step further to utilize all the tokens of Transformers and propose a cyclic token permutation framework for multi-spectral object ReID, dubbled TOP-ReID. More specifically, we first deploy a multi-stream deep network based on vision Transformers to preserve distinct information from different image spectra. Then, we propose a Token Permutation Module (TPM) for cyclic multi-spectral feature aggregation. It not only facilitates the spatial feature alignment across different image spectra, but also allows the class token of each spectrum to perceive the local details of other spectra. Meanwhile, we propose a Complementary Reconstruction Module (CRM), which introduces dense token-level reconstruction constraints to reduce the distribution gap across different image spectra. With the above modules, our proposed framework can generate more discriminative multi-spectral features for robust object ReID. Extensive experiments on three ReID benchmarks (i.e., RGBNT201, RGBNT100 and MSVR310) verify the effectiveness of our methods. The code is available at https://github.com/924973292/TOP-ReID.

Permutation Modules with Nakayama Endomorphism Rings

Permutation Modules with Nakayama Endomorphism Rings

High-quality color image compression-encryption using chaos and block permutation

This paper proposes a high-quality color image compression-encryption scheme based on chaos and block permutation. In this scheme, the color digital image is first converted and sampled in the YCbCr gamut, and then the coefficients in the sub-blocks are extracted for compression coding after 8×8 post-blocking discrete cosine transformation (DCT) to the frequency domain. Then, an image encryption algorithm using chaos-based block permutation and two-round row-column diffusion is designed for the compressed frequency domain information. In terms of security, a chaos-based block permutation module is designed, which includes the encryption operations of block scrambling, block rotation and inversion, pixel value complement and color component interchange to enhance security. In addition, a mechanism for associating plaintext and intermediate ciphertext is introduced to generate dynamic chaotic sequences to effectively resist various cryptographic attacks. Moreover, the joint compression-encryption reduces the computational complexity of the encrypted object significantly and improves the efficiency. The simulation results show that the joint compression-encryption scheme has the advantages of high compression ratio and high image recovery quality, and has a fairly high security level to resist common cryptographic attacks. Therefore, the compression-encryption scheme proposed in this paper is a preferred and promising method for protecting the privacy of digital images.

Bases of the equivariant cohomologies of regular semisimple Hessenberg varieties

We consider bases for the cohomology space of regular semisimple Hessenberg varieties, consisting of the classes that naturally arise from the Białynicki-Birula decomposition of the Hessenberg varieties. We give an explicit combinatorial description of the support of each class, which enables us to compute the symmetric group actions on the classes in our bases. We then successfully apply the results to the permutohedral varieties to explicitly write down each class and to construct permutation submodules that constitute summands of a decomposition of cohomology space of each degree. This resolves the problem posed by Stembridge on the geometric construction of permutation module decomposition and also the conjecture posed by Chow on the construction of bases for the equivariant cohomology spaces of permutohedral varieties.

MMDA: Multi‐person marginal distribution awareness for monocular 3D pose estimation

AbstractMost existing 3D pose representations cannot completely decouple the overlapping two or more human joints of the same type. In this paper, the authors propose a novel 2.5 D representation of the human pose by projecting human joints in 3D space onto the three orthogonal planes. The authors apply for the first time the permutation module to a multi‐person 3D human pose estimation task and use Geometric Constraints Loss (GCL) to guide the learning of the model. The authors overcome the negative effects of the inductive bias of convolutional neural networks (CNNs) by aligning the intermediate feature space with the output feature space. The effectiveness of the authors’ approach is validated on the carnegie mellon university (CMU) panoptic dataset and MuPoTS‐3D dataset. The authors’ proposed representations can effectively decouple the human joints in their selected data from overlapping human joints.

Multiplication formulas and isomorphism theorem of ıSchur superalgebras

We introduce the notion of ı Schur superalgebra, which can be regarded as a type B/C counterpart of the q -Schur superalgebra (of type A) formulated as centralizer algebras of certain signed q -permutation modules over Hecke algebras. Some multiplication formulas for ı Schur superalgebra are obtained to construct their canonical bases. Furthermore, we established an isomorphism theorem between the ı Scuhr superalgebras and the q -Schur superalgebras of type A, which helps us derive semisimplicity criteria of the ı Schur superalgebras.

Projective Indecomposable Permutation Modules

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.

Permutation Module Decomposition of the Second Cohomology of a Regular Semisimple Hessenberg Variety

Abstract Regular semisimple Hessenberg varieties admit actions of associated Weyl groups on their cohomology spaces of each degree. In this paper, we consider the module structure of the cohomology spaces of regular semisimple Hessenberg varieties of type $A$. We define a subset of the Białynicki–Birula basis of the cohomology space, which becomes a module generator set of the cohomology module of each degree. We use these generators to construct permutation submodules of the degree two cohomology module to form a permutation module decomposition. Our construction is consistent with a known combinatorial result by Chow on chromatic quasisymmetric functions.

A Chaotic Image Encryption Scheme Based on Genetic Central Dogma and KMP Method

In this paper, an image cryptosystem based on genetic central dogma (GCD), Knuth–Morria–Pratt (KMP) algorithm and a chaotic system is developed. The KMP algorithm is firstly used to bind DNA strings to obtain the next array, which participates in the design of the chaotic initial condition, and then the secure chaotic sequences are produced by employing the sliding idea in pattern string matching. In the present procedure, a DNA-level two-way pixel’s shuffle is achieved by a shared stack push operation and it is adopted in the permutation module for the purpose of accelerating the overall pixel’s shuffle. Subsequently, the pixel values are substituted by simulating the process of protein synthesis in GCD, in which the DNA replication and RNA replication form the basis of the DNA-triploid mutation and new RNA mutation rules, respectively. Experimental simulations and extensive cryptanalysis fully vindicate that superior security effects in addition to satisfactory low time complexity can be simultaneously obtained by the proposed confusion-substitution scheme.

On the annihilator ideal in the bt-algebra of tensor space

We study the representation theory of the braids and ties algebra, or the bt -algebra, E n ( q ) . Using the cellular basis { m s t } for E n ( q ) obtained in previous joint work with J. Espinoza we introduce two kinds of permutation modules M ( λ ) and M ( Λ ) for E n ( q ) . We show that the tensor product module V ⊗ n for E n ( q ) is a direct sum of M ( λ ) 's. We introduce the dual cellular basis { n s t } for E n ( q ) and study its action on M ( λ ) and M ( Λ ) . We show that the annihilator ideal I in E n ( q ) of V ⊗ n enjoys a nice compatibility property with respect to { n s t } . We finally study the quotient algebra E n ( q ) / I , showing in particular that it is a simultaneous generalization of Härterich's ‘generalized Temperley-Lieb algebra’ and Juyumaya's ‘partition Temperley-Lieb algebra’.

The Green polynomials via vertex operators

An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length $\leq 3$ and the diagonal lengths $\leq 3$; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group $G$ and the Iwahori-Hecke algebra of type $A$ on the permutation module of $G$ by its Borel subgroup.

Permutation modules and cohomological singularity

We define a new invariant of finitely generated representations of a finite group, with coefficients in a commutative noetherian ring. This invariant uses group cohomology and takes values in the singularity category of the coefficient ring. It detects which representations are controlled by permutation modules.

MODUL WEYL SEBAGAI REPRESENTASI GRUP SIMETRI DARI PEWARNAAN SUATU GRAF

This research identifies the properties of Weyl module which had some similar properties with Specht module. Specht module as an irreducible representation of permutation module for some partition

Cohomology of PSL2(q)

In 2011, Guralnick and Tiep proved that if G was a Chevalley group with Borel subgroup B and V an irreducible G-module in cross characteristic with VB=0, then the dimension of H1(G,V) is determined by the structure of the permutation module on the cosets of B. We generalise this theorem to higher cohomology and an arbitrary finite group, so that if H≤G such that Or′(H)=Or(H) and VH=0 for V a G-module in characteristic r then dim⁡H1(G,V) is determined by the structure of the permutation module on cosets of H, and Hn(G,V) by ExtGn−1(V⁎,M) for some kG-module M dependent on H. We also determine ExtGn(V,W) for all irreducible kG-modules V, W for G∈{PSL2(q),PGL2(q),SL2(q)} in cross characteristic.

Bounded complexes of permutation modules

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

Double centralisers and annihilator ideals of Young permutation modules

We study the double centraliser algebras and annihilator ideals of certain permutation modules for symmetric groups and their quantum analogues. Of particular interest is the behaviour of these algebras and ideals under base change. By exploiting connections with polynomial representation theory over a field we establish the double centraliser property and dimension invariance of annihilator ideals over fields. By elementary localisation techniques we obtain the corresponding results over arbitrary rings. The modules considered include the permutation modules relevant to partition algebras, as studied in the recent paper by Bowman, Doty and Martin, [3], and we obtain in particular their main results on the double centraliser property in this context. Much of our development is equally valid for the corresponding modules over Hecke algebras of type A and, for the most part, we give it at this level of generality.

Computational Bounds for Doing Harmonic Analysis on Permutation Modules of Finite Groups

We develop an approach to finding upper bounds for the number of arithmetic operations necessary for doing harmonic analysis on permutation modules of finite groups. The approach takes advantage of the intrinsic orbital structure of permutation modules, and it uses the multiplicities of irreducible submodules within individual orbital spaces to express the resulting computational bounds. We conclude by showing that these bounds are surprisingly small when dealing with certain permutation modules arising from the action of the symmetric group on tabloids.

The Benson - Symonds invariant for ordinary and signed permutation modules

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 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 all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics.