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Jean-yves Thibon - One of the best experts on this subject based on the ideXlab platform.
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The Hopf algebras of signed permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer
Advances in Mathematics, 2020Co-Authors: Li Guo, Jean-yves ThibonAbstract:Abstract This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra S Sym of permutations, mapped onto QSym by taking descents of permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions. We extend these two surjective Hopf algebra homomorphisms into a Commutative Diagram by introducing a Hopf algebra H Sym , linearly spanned by signed permutations from the hyperoctahedral groups, equipped with the shifted quasi-shuffle product and deconcatenation coproduct. Extracting a permutation from a signed permutation defines a Hopf algebra surjection form H Sym to S Sym and taking a suitable descent from a signed permutation defines a linear surjection from H Sym to RQSym. The notion of weak P-partitions from signed permutations is introduced which, by taking generating functions, gives fundamental weak quasi-symmetric functions and sends the shifted quasi-shuffle product to the product of the corresponding generating functions. Together with the existing Hopf algebra surjections from S Sym and RQSym to QSym, we obtain a Commutative Diagram of Hopf algebras revealing the close relationship among compositions, weak compositions, permutations and signed permutations.
Li Guo - One of the best experts on this subject based on the ideXlab platform.
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The Hopf algebras of signed permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer
Advances in Mathematics, 2020Co-Authors: Li Guo, Jean-yves ThibonAbstract:Abstract This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra S Sym of permutations, mapped onto QSym by taking descents of permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions. We extend these two surjective Hopf algebra homomorphisms into a Commutative Diagram by introducing a Hopf algebra H Sym , linearly spanned by signed permutations from the hyperoctahedral groups, equipped with the shifted quasi-shuffle product and deconcatenation coproduct. Extracting a permutation from a signed permutation defines a Hopf algebra surjection form H Sym to S Sym and taking a suitable descent from a signed permutation defines a linear surjection from H Sym to RQSym. The notion of weak P-partitions from signed permutations is introduced which, by taking generating functions, gives fundamental weak quasi-symmetric functions and sends the shifted quasi-shuffle product to the product of the corresponding generating functions. Together with the existing Hopf algebra surjections from S Sym and RQSym to QSym, we obtain a Commutative Diagram of Hopf algebras revealing the close relationship among compositions, weak compositions, permutations and signed permutations.
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The Hopf algebras of signed permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer
2019Co-Authors: Li Guo, Thibon Jean-yves, Yu HouyiAbstract:This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra SSym of permutations, mapped onto QSym by taking descents of permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions. We extend these two surjective Hopf algebra homomorphisms into a Commutative Diagram by introducing a Hopf algebra HSym, linearly spanned by signed permutations from the hyperoctahedral groups, equipped with the shifted quasi-shuffle product and deconcatenation coproduct. Extracting a permutation from a signed permutation defines a Hopf algebra surjection form HSym to SSym and taking a suitable descent from a signed permutation defines a linear surjection from HSym to RQSym. The notion of signed $P$-partitions from signed permutations is introduced which, by taking generating functions, gives fundamental weak quasi-symmetric functions and sends the shifted quasi-shuffle product to the product of the corresponding generating functions. Together with the existing Hopf algebra surjections from SSym and RQSym to QSym, we obtain a Commutative Diagram of Hopf algebras revealing the close relationship among compositions, weak compositions, permutations and signed permutations.Comment: 25 pages, 2 figure
David Wagner - One of the best experts on this subject based on the ideXlab platform.
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Towards a unifying view of block cipher cryptanalysis
Fast Software Encryption, 2004Co-Authors: David WagnerAbstract:We introduce Commutative Diagram cryptanalysis, a frame-work for expressing certain kinds of attacks on product ciphers. We show that many familiar attacks, including linear cryptanalysis, differential cryptanalysis, differential-linear cryptanalysis, mod n attacks, truncated differential cryptanalysis, impossible differential cryptanalysis, higher-order differential cryptanalysis, and interpolation attacks can be ex-pressed within this framework. Thus, we show that Commutative Diagram attacks provide a unifying view into the field of block cipher cryptanal-ysis. Then, we use the language of Commutative Diagram cryptanalysis to compare the power of many previously known attacks. Finally, we introduce two new attacks, generalized truncated differential cryptanaly-sis and bivariate interpolation, and we show how these new techniques generalize and unify many previous attack methods.
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FSE - Towards a Unifying View of Block Cipher Cryptanalysis
Fast Software Encryption, 2004Co-Authors: David WagnerAbstract:We introduce Commutative Diagram cryptanalysis, a framework for expressing certain kinds of attacks on product ciphers. We show that many familiar attacks, including linear cryptanalysis, differential cryptanalysis, differential-linear cryptanalysis, mod n attacks, truncated differential cryptanalysis, impossible differential cryptanalysis, higher-order differential cryptanalysis, and interpolation attacks can be expressed within this framework. Thus, we show that Commutative Diagram attacks provide a unifying view into the field of block cipher cryptanalysis. Then, we use the language of Commutative Diagram cryptanalysis to compare the power of many previously known attacks. Finally, we introduce two new attacks, generalized truncated differential cryptanalysis and bivariate interpolation, and we show how these new techniques generalize and unify many previous attack methods.
Bengt E. W. Nilsson - One of the best experts on this subject based on the ideXlab platform.
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Off-shell structure of twisted (2,0) theory
Journal of High Energy Physics, 2014Co-Authors: Ulf Gran, Hampus Linander, Bengt E. W. NilssonAbstract:A $Q$-exact off-shell action is constructed for twisted abelian (2,0) theory on a Lorentzian six-manifold of the form $M_{1,5} = C\times M_4$, where $C$ is a flat two-manifold and $M_4$ is a general Euclidean four-manifold. The properties of this formulation, which is obtained by introducing two auxiliary fields, can be summarised by a Commutative Diagram where the Lagrangian and its stress-tensor arise from the $Q$-variation of two fermionic quantities $V$ and $\lambda^{\mu\nu}$. This completes and extends the analysis in [arXiv:1311.3300].
Ulf Gran - One of the best experts on this subject based on the ideXlab platform.
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Off-shell structure of twisted (2,0) theory
Journal of High Energy Physics, 2014Co-Authors: Ulf Gran, Hampus Linander, Bengt E. W. NilssonAbstract:A $Q$-exact off-shell action is constructed for twisted abelian (2,0) theory on a Lorentzian six-manifold of the form $M_{1,5} = C\times M_4$, where $C$ is a flat two-manifold and $M_4$ is a general Euclidean four-manifold. The properties of this formulation, which is obtained by introducing two auxiliary fields, can be summarised by a Commutative Diagram where the Lagrangian and its stress-tensor arise from the $Q$-variation of two fermionic quantities $V$ and $\lambda^{\mu\nu}$. This completes and extends the analysis in [arXiv:1311.3300].