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A M Vershik - One of the best experts on this subject based on the ideXlab platform.
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the serpentine representation of the infinite Symmetric Group and the basic representation of the affine lie algebra widehat mathfrak sl _2
Letters in Mathematical Physics, 2015Co-Authors: N V Tsilevich, A M VershikAbstract:We introduce and study the so-called serpentine representations of the infinite Symmetric Group \({\mathfrak{S}_\mathbb{N}}\), which turn out to be closely related to the basic representation of the affine Lie algebra \({\widehat{\mathfrak{sl}_2}}\) and representations of the Virasoro algebra.
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the serpentine representation of the infinite Symmetric Group and the basic representation of the affine lie algebra widehat mathfrak sl _2
arXiv: Representation Theory, 2014Co-Authors: N V Tsilevich, A M VershikAbstract:We introduce and study the so-called serpentine representations of the infinite Symmetric Group $\sinf$, which turn out to be closely related to the basic representation of the affine Lie algebra $\widehat{\mathfrak{sl}_2}$ and representations of the Virasoro algebra. This is a new version of the manuscript of the same authors "On a relation between the basic representation of the affine Lie algebra $\widehat\sl$ and a Schur--Weyl representation of the infinite Symmetric Group" (arXiv:1403.1558)
N V Tsilevich - One of the best experts on this subject based on the ideXlab platform.
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the serpentine representation of the infinite Symmetric Group and the basic representation of the affine lie algebra widehat mathfrak sl _2
Letters in Mathematical Physics, 2015Co-Authors: N V Tsilevich, A M VershikAbstract:We introduce and study the so-called serpentine representations of the infinite Symmetric Group \({\mathfrak{S}_\mathbb{N}}\), which turn out to be closely related to the basic representation of the affine Lie algebra \({\widehat{\mathfrak{sl}_2}}\) and representations of the Virasoro algebra.
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the serpentine representation of the infinite Symmetric Group and the basic representation of the affine lie algebra widehat mathfrak sl _2
arXiv: Representation Theory, 2014Co-Authors: N V Tsilevich, A M VershikAbstract:We introduce and study the so-called serpentine representations of the infinite Symmetric Group $\sinf$, which turn out to be closely related to the basic representation of the affine Lie algebra $\widehat{\mathfrak{sl}_2}$ and representations of the Virasoro algebra. This is a new version of the manuscript of the same authors "On a relation between the basic representation of the affine Lie algebra $\widehat\sl$ and a Schur--Weyl representation of the infinite Symmetric Group" (arXiv:1403.1558)
Dirk Zeindler - One of the best experts on this subject based on the ideXlab platform.
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central limit theorem for multiplicative class functions on the Symmetric Group
Journal of Theoretical Probability, 2013Co-Authors: Dirk ZeindlerAbstract:Hambly, Keevash, O’Connell, and Stark have proven a central limit theorem for the characteristic polynomial of a permutation matrix with respect to the uniform measure on the Symmetric Group. We generalize this result in several ways. We prove here a central limit theorem for multiplicative class functions on the Symmetric Group with respect to the Ewens measure and compute the covariance of the real and the imaginary part in the limit. We also estimate the rate of convergence with the Wasserstein distance.
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central limit theorem for multiplicative class functions on the Symmetric Group
arXiv: Probability, 2010Co-Authors: Dirk ZeindlerAbstract:Hambly, Keevash, O'Connell and Stark have proven a central limit theorem for the characteristic polynomial of a permutation matrix with respect to the uniform measure on the Symmetric Group. We generalize this result in several ways. We prove here a central limit theorem for multiplicative class functions on Symmetric Group with respect to the Ewens measure and compute the covariance of the real and the imaginary part in the limit. We also estimate the rate of convergence with the Wasserstein distance.
David J Hemmer - One of the best experts on this subject based on the ideXlab platform.
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frobenius twists in the representation theory of the Symmetric Group
arXiv: Representation Theory, 2012Co-Authors: David J HemmerAbstract:For the general linear Group $GL_n(k)$ over an algebraically closed field $k$ of characteristic $p$, there are two types of "twisting" operations that arise naturally on partitions. These are of the form $\lambda \rightarrow p\lambda$ and $\lambda \rightarrow \lambda + p^r\tau$ The first comes from the Frobenius twist, and the second arises in various tensor product situations, often from tensoring with the Steinberg module. This paper surveys and adds to an intriguing series of seemingly unrelated Symmetric Group results where this partition combinatorics arises, but with no structural explanation for it. This includes cohomology of simple, Specht and Young modules, support varieties for Specht modules, homomorphisms between Specht modules, the Mullineux map, $p$-Kostka numbers and tensor products of Young modules.
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stable decompositions for some Symmetric Group characters arising in braid Group cohomology
Journal of Combinatorial Theory Series A, 2011Co-Authors: David J HemmerAbstract:We prove that certain permutation characters for the Symmetric Group @S"n decompose in a manner that is independent of n for large n. This result is a key ingredient in the recent work of T. Church and B. Farb, who obtain a ''representation stability'' theorem for the character of @S"n acting on the cohomology H^p(P"n,C) of the pure braid Group P"n.
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on the cohomology of young modules for the Symmetric Group
Advances in Mathematics, 2010Co-Authors: F R Cohen, David J Hemmer, Daniel K NakanoAbstract:Abstract The main result of this paper is an application of the topology of the space Q ( X ) to obtain results for the cohomology of the Symmetric Group on d letters, Σ d , with ‘twisted’ coefficients in various choices of Young modules and to show that these computations reduce to certain natural questions in representation theory. The authors extend classical methods for analyzing the homology of certain spaces Q ( X ) with mod-p coefficients to describe the homology H • ( Σ d , V ⊗ d ) as a module for the general linear Group GL ( V ) over an algebraically closed field k of characteristic p. As a direct application, these results provide a method of reducing the computation of Ext Σ d • ( Y λ , Y μ ) (where Y λ , Y μ are Young modules) to a representation theoretic problem involving the determination of tensor products and decomposition numbers. In particular, in characteristic two, for many d, a complete determination of H • ( Σ d Y λ ) can be found. This is the first nontrivial class of Symmetric Group modules where a complete description of the cohomology in all degrees can be given. For arbitrary d the authors determine H i ( Σ d , Y λ ) for i = 0 , 1 , 2 . An interesting phenomenon is uncovered-namely a stability result reminiscent of generic cohomology for algebraic Groups. For each i the cohomology H i ( Σ p a d , Y p a λ ) stabilizes as a increases. The methods in this paper are also powerful enough to determine, for any p and λ, precisely when H • ( Σ d , Y λ ) = 0 . Such modules with vanishing cohomology are of great interest in representation theory because their support varieties constitute the representation theoretic nucleus.
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on the cohomology of young modules for the Symmetric Group
arXiv: Representation Theory, 2008Co-Authors: F R Cohen, David J Hemmer, Daniel K NakanoAbstract:The main result of this paper is an application of the topology of the space $Q(X)$ to obtain results for the cohomology of the Symmetric Group on $d$ letters, $\Sigma_d$, with `twisted' coefficients in various choices of Young modules and to show that these computations reduce to certain natural questions in representation theory. The authors extend classical methods for analyzing the homology of certain spaces $Q(X)$ with mod-$p$ coefficients to describe the homology $\HH_{\bullet}(\Sigma_d, V^{\otimes d})$ as a module for the general linear Group $GL(V)$ over an algebraically closed field $k$ of characteristic $p$. As a direct application, these results provide a method of reducing the computation of $\text{Ext}^{\bullet}_{\Sigma_{d}}(Y^{\lambda},Y^{\mu})$ (where $Y^{\lambda}$, $Y^{\mu}$ are Young modules) to a representation theoretic problem involving the determination of tensor products and decomposition numbers. In particular, in characteristic two, for many $d$, a complete determination of $\Hs Y^\lambda)$ can be found. This is the first nontrivial class of Symmetric Group modules where a complete description of the cohomology in all degrees can be given. For arbitrary $d$ the authors determine $\HH^i(\Sigma_d,Y^\lambda)$ for $i=0,1,2$. An interesting phenomenon is uncovered--namely a stability result reminiscent of generic cohomology for algebraic Groups. For each $i$ the cohomology $\HH^i(\Sigma_{p^ad}, Y^{p^a\lambda})$ stabilizes as $a$ increases. The methods in this paper are also powerful enough to determine, for any $p$ and $\lambda$, precisely when $\HH^{\bullet}(\sd,Y^\lambda)=0$. Such modules with vanishing cohomology are of great interest in representation theory because their support varieties constitute the representation theoretic nucleus.
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the ext1 quiver for completely splittable representations of the Symmetric Group
Journal of Group Theory, 2001Co-Authors: David J HemmerAbstract:Kleshchev has recently [7] classi®ed those modules for the Symmetric Group which have semisimple restriction to any Young subGroup. We determine Ext K Sd D ;D m where D l and D m are K Sd -modules of this type, called completely splittable. As a corollary of this and recent work of Kleshchev and Nakano, we can determine ExtGL n K L l;L m for certain simple GLn K-modules L l and L m. 1 Introduction Let Sd denote the Symmetric Group on d letters. The complex irreducible Sd -modules correspond bijectively with partitions l of d, and we denote by S l the irreducible module corresponding to l. We work over an algebraically closed ®eld K of positive characteristic p > 2. The simple K Sd -modules are indexed by p-regular partitions, and we denote the corresponding simple module by D. These modules can also be indexed by column p-regular partitions, and for l column p-regular we denote the corresponding simple module by Dl. For a comprehensive treatment of the theory, see [4]. For any composition m m1; m2; . . . mk of d there is a standard Young subGroup de®ned by Sm Sm1 Sm2 Smk : where h l is the height of l and hij l1; l2; . . . ; lk li l 0 j 1y i y j is the i; j hook length. This theorem together with the work of [8] gives a corresponding result for GLn K when nd d. Let L l denote the simple, polynomial GLn K-module with highest weight l. Let m denote the Mullineaux map on p-regular partitions de®ned by
Xianwei Zhou - One of the best experts on this subject based on the ideXlab platform.
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s boxes construction based on the cayley graph of the Symmetric Group for uasns
IEEE Access, 2019Co-Authors: Li Shuai, Lina Wang, Li Miao, Xianwei ZhouAbstract:Modern block ciphers can be used for designing security schemes to meet the security requirements in underwater acoustic sensor networks. The substitution boxes (S-boxes) used in block ciphers must have good cryptographic properties to resist various attacks. The difficulty in the design of S-boxes is that the cryptographic properties of S-boxes have no obvious distribution. In this paper, we investigate the certain subgraph of the Cayley graph of the Symmetric Group to get some information about the distribution of the cryptographic properties of S-boxes. Based on the information obtained, an algorithm for designing S-boxes with good cryptographic properties is proposed. The security analysis shows that the preferred S-box constructed by the proposed algorithm has good cryptographic properties.
