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Sumanta Sarkar - One of the best experts on this subject based on the ideXlab platform.
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Involutions over the galois field mathbb f _ 2 n
IEEE Transactions on Information Theory, 2016Co-Authors: Pascale Charpin, Sihem Mesnager, Sumanta SarkarAbstract:An Involution is a permutation, such that its inverse is itself (i.e., cycle length ≤ 2). Due to this property, Involutions have been used in many applications, including cryptography and coding theory. In this paper, we provide a systematic study of Involutions that are defined over a finite field of characteristic 2. We characterize the Involution property of several classes of polynomials and propose several constructions. Furthermore, we study the number of fixed points of Involutions, which is a pertinent question related to permutations with short cycle. In this paper, we mostly have used combinatorial techniques.
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Involutions over the Galois field ${\mathbb F}_{2^{n}}$
IEEE Transactions on Information Theory, 2016Co-Authors: Pascale Charpin, Sihem Mesnager, Sumanta SarkarAbstract:An Involution is a permutation such that its inverse is itself (i.e., cycle length less than 2). Due to this property Involutions have been used in many applications including cryptography and coding theory. In this paper we provide a systematic study of Involutions that are defined over finite field of characteristic 2. We characterize the Involution property of several classes of polynomials and propose several constructions. Further we study the number of fixed points of Involutions which is a pertinent question related to permutations with short cycle. In this paper we mostly have used combinatorial techniques.
Pascale Charpin - One of the best experts on this subject based on the ideXlab platform.
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Involutions over the galois field mathbb f _ 2 n
IEEE Transactions on Information Theory, 2016Co-Authors: Pascale Charpin, Sihem Mesnager, Sumanta SarkarAbstract:An Involution is a permutation, such that its inverse is itself (i.e., cycle length ≤ 2). Due to this property, Involutions have been used in many applications, including cryptography and coding theory. In this paper, we provide a systematic study of Involutions that are defined over a finite field of characteristic 2. We characterize the Involution property of several classes of polynomials and propose several constructions. Furthermore, we study the number of fixed points of Involutions, which is a pertinent question related to permutations with short cycle. In this paper, we mostly have used combinatorial techniques.
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Involutions over the Galois field ${\mathbb F}_{2^{n}}$
IEEE Transactions on Information Theory, 2016Co-Authors: Pascale Charpin, Sihem Mesnager, Sumanta SarkarAbstract:An Involution is a permutation such that its inverse is itself (i.e., cycle length less than 2). Due to this property Involutions have been used in many applications including cryptography and coding theory. In this paper we provide a systematic study of Involutions that are defined over finite field of characteristic 2. We characterize the Involution property of several classes of polynomials and propose several constructions. Further we study the number of fixed points of Involutions which is a pertinent question related to permutations with short cycle. In this paper we mostly have used combinatorial techniques.
Brendan Pawlowski - One of the best experts on this subject based on the ideXlab platform.
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schur p positivity and Involution stanley symmetric functions
International Mathematics Research Notices, 2019Co-Authors: Zachary Hamaker, Eric Marberg, Brendan PawlowskiAbstract:The Involution Stanley symmetric functions $\hat{F}_y$ are the stable limits of the analogues of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for Involution words, and are indexed by the Involutions in the symmetric group. By construction each $\hat{F}_y$ is a sum of Stanley symmetric functions and therefore Schur positive. We prove the stronger fact that these power series are Schur $P$-positive. We give an algorithm to efficiently compute the decomposition of $\hat{F}_y$ into Schur $P$-summands, and prove that this decomposition is triangular with respect to the dominance order on partitions. As an application, we derive pattern avoidance conditions which characterize the Involution Stanley symmetric functions which are equal to Schur $P$-functions. We deduce as a corollary that the Involution Stanley symmetric function of the reverse permutation is a Schur $P$-function indexed by a shifted staircase shape. These results lead to alternate proofs of theorems of Ardila-Serrano and DeWitt on skew Schur functions which are Schur $P$-functions. We also prove new Pfaffian formulas for certain related Involution Schubert polynomials.
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Involution words counting problems and connections to schubert calculus for symmetric orbit closures
Journal of Combinatorial Theory Series A, 2018Co-Authors: Zachary Hamaker, Eric Marberg, Brendan PawlowskiAbstract:Abstract Involution words are variations of reduced words for Involutions in Coxeter groups, first studied under the name of “admissible sequences” by Richardson and Springer. They are maximal chains in Richardson and Springer's weak order on Involutions. This article is the first in a series of papers on Involution words, and focuses on their enumerative properties. We define Involution analogues of several objects associated to permutations, including Rothe diagrams, the essential set, Schubert polynomials, and Stanley symmetric functions. These definitions have geometric interpretations for certain intervals in the weak order on Involutions. In particular, our definition of “Involution Schubert polynomials” can be viewed as a Billey–Jockusch–Stanley type formula for cohomology class representatives of O n - and Sp 2 n -orbit closures in the flag variety, defined inductively in recent work of Wyser and Yong. As a special case of a more general theorem, we show that the Involution Stanley symmetric function for the longest element of a finite symmetric group is a product of staircase-shaped Schur functions. This implies that the number of Involution words for the longest element of a finite symmetric group is equal to the dimension of a certain irreducible representation of a Weyl group of type B.
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Transition formulas for Involution Schubert polynomials
Selecta Mathematica, 2018Co-Authors: Zachary Hamaker, Eric Marberg, Brendan PawlowskiAbstract:The orbits of the orthogonal and symplectic groups on the flag variety are in bijection, respectively, with the Involutions and fixed-point-free Involutions in the symmetric group \(S_n\). Wyser and Yong have described polynomial representatives for the cohomology classes of the closures of these orbits, which we denote as \({\hat{\mathfrak S}}_y\) (to be called Involution Schubert polynomials) and \(\hat{\mathfrak S}^\mathtt{{FPF}}_y\) (to be called fixed-point-free Involution Schubert polynomials). Our main results are explicit formulas decomposing the product of \({\hat{\mathfrak S}}_y\) (respectively, \(\hat{\mathfrak S}^\mathtt{{FPF}}_y\)) with any y-invariant linear polynomial as a linear combination of other Involution Schubert polynomials. These identities serve as analogues of Lascoux and Schutzenberger’s transition formula for Schubert polynomials, and lead to a self-contained algebraic proof of the nontrivial equivalence of several definitions of \({\hat{\mathfrak S}}_y\) and \( \hat{\mathfrak S}^\mathtt{{FPF}}_y\) appearing in the literature. Our formulas also imply combinatorial identities about Involution words, certain variations of reduced words for Involutions in \(S_n\). We construct operators on Involution words based on the Little map to prove these identities bijectively. The proofs of our main theorems depend on some new technical results, extending work of Incitti, about covering relations in the Bruhat order of \(S_n\) restricted to Involutions.
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Involution words II: braid relations and atomic structures
Journal of Algebraic Combinatorics, 2017Co-Authors: Zachary Hamaker, Eric Marberg, Brendan PawlowskiAbstract:Involution words are variations of reduced words for twisted Involutions in Coxeter groups. They arise naturally in the study of the Bruhat order, of certain Iwahori–Hecke algebra modules, and of orbit closures in flag varieties. Specifically, to any twisted Involutions x , y in a Coxeter group W with automorphism $$*$$ ∗ , we associate a set of Involution words $$\hat{\mathcal {R}}_*(x,y)$$ R ^ ∗ ( x , y ) . This set is the disjoint union of the reduced words of a set of group elements $$\mathcal {A}_*(x,y)$$ A ∗ ( x , y ) , which we call the atoms of y relative to x . The atoms, in turn, are contained in a larger set $$\mathcal {B}_*(x,y) \subset W$$ B ∗ ( x , y ) ⊂ W with a similar definition, whose elements are referred to as Hecke atoms. Our main results concern some interesting properties of the sets $$\hat{\mathcal {R}}_*(x,y)$$ R ^ ∗ ( x , y ) and $$\mathcal {A}_*(x,y) \subset \mathcal {B}_*(x,y)$$ A ∗ ( x , y ) ⊂ B ∗ ( x , y ) . For finite Coxeter groups, we prove that $$\mathcal {A}_*(1,y)$$ A ∗ ( 1 , y ) consists of exactly the minimal-length elements $$w \in W$$ w ∈ W such that $$w^* y \le w$$ w ∗ y ≤ w in Bruhat order, and we conjecture a more general property for arbitrary Coxeter groups. In type A , we describe a simple set of conditions characterizing the sets $$\mathcal {A}_*(x,y)$$ A ∗ ( x , y ) for all Involutions $$x,y \in S_n$$ x , y ∈ S n , giving a common generalization of three recent theorems of Can et al. We show that the atoms of a fixed Involution in the symmetric group (relative to $$x=1$$ x = 1 ) naturally form a graded poset, while the Hecke atoms surprisingly form an equivalence class under the “Chinese relation” studied by Cassaigne et al. These facts allow us to recover a recent theorem of Hu and Zhang describing a set of “braid relations” spanning the Involution words of any self-inverse permutation. We prove a generalization of this result giving an analogue of Matsumoto’s theorem for Involution words in arbitrary Coxeter groups.
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Involution words ii braid relations and atomic structures
arXiv: Combinatorics, 2016Co-Authors: Zachary Hamaker, Eric Marberg, Brendan PawlowskiAbstract:Involution words are variations of reduced words for twisted Involutions in Coxeter groups. They arise naturally in the study of the Bruhat order, of certain Iwahori-Hecke algebra modules, and of orbit closures in flag varieties. Specifically, to any twisted Involutions $x$, $y$ in a Coxeter group $W$ with automorphism $*$, we associate a set of Involution words $\hat{\mathcal{R}}_*(x,y)$. This set is the disjoint union of the reduced words of a set of group elements $\mathcal{A}_*(x,y)$, which we call the atoms of $y$ relative to $x$. The atoms, in turn, are contained in a larger set $\mathcal{B}_*(x,y) \subset W$ with a similar definition, whose elements we refer to as Hecke atoms. Our main results concern some interesting properties of the sets $\hat{\mathcal{R}}_*(x,y)$ and $\mathcal{A}_*(x,y) \subset \mathcal{B}_*(x,y)$. For finite Coxeter groups we prove that $\mathcal{A}_*(1,y)$ consists of exactly the minimal-length elements $w \in W$ such that $w^* y \leq w$ in Bruhat order, and conjecture a more general property for arbitrary Coxeter groups. In type $A$, we describe a simple set of conditions characterizing the sets $\mathcal{A}_*(x,y)$ for all Involutions $x,y \in S_n$, giving a common generalization of three recent theorems of Can, Joyce, and Wyser. We show that the atoms of a fixed Involution in the symmetric group (relative to $x=1$) naturally form a graded poset, while the Hecke atoms surprisingly form an equivalence class under the "Chinese relation" studied by Cassaigne, Espie, et al. These facts allow us to recover a recent theorem of Hu and Zhang describing a set of "braid relations" spanning the Involution words of any self-inverse permutation. We prove a generalization of this result giving an analogue of Matsumoto's theorem for Involution words in arbitrary Coxeter groups.
Vinicius Casteluber Laass - One of the best experts on this subject based on the ideXlab platform.
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the borsuk ulam property for homotopy classes of selfmaps of surfaces of euler characteristic zero
Journal of Fixed Point Theory and Applications, 2019Co-Authors: Daciberg Lima Goncalves, John Guaschi, Vinicius Casteluber LaassAbstract:Let M and N be topological spaces such that M admits a free Involution τ. A homotopy class β ∈ [M, N ] is said to have the Borsuk-Ulam property with respect to τ if for every representative map f : M → N of β, there exists a point x ∈ M such that f (τ (x)) = f (x). In the case where M is a compact, connected manifold without boundary and N is a compact, connected surface without boundary different from the 2-sphere and the real projective plane, we formulate this property in terms of the pure and full 2-string braid groups of N , and of the fundamental groups of M and the orbit space of M with respect to the action of τ. If M = N is either the 2-torus T^2 or the Klein bottle K^2 , we then solve the problem of deciding which homotopy classes of [M, M ] have the Borsuk-Ulam property. First, if τ : T^2 → T^2 is a free Involution that preserves orientation, we show that no homotopy class of [T^2 , T^2 ] has the Borsuk-Ulam property with respect to τ. Secondly, we prove that up to a certain equivalence relation, there is only one class of free Involutions τ : T^2 → T^2 that reverse orientation, and for such Involutions, we classify the homotopy classes in [T^2 , T^2 ] that have the Borsuk-Ulam property with respect to τ in terms of the induced homomorphism on the fundamental group. Finally, we show that if τ : K^2 → K^2 is a free Involution, then a homotopy class of [K^2 , K^2 ] has the Borsuk-Ulam property with respect to τ if and only if the given homotopy class lifts to the torus.
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the borsuk ulam property for homotopy classes of selfmaps of surfaces of euler characteristic zero
arXiv: Geometric Topology, 2016Co-Authors: Daciberg Lima Goncalves, John Guaschi, Vinicius Casteluber LaassAbstract:Let M and N be topological spaces such that M admits a free Involution $\\tau$. A homotopy class $\beta$ $\in$ [M, N ] is said to have the Borsuk-Ulam property with respect to $\\tau$ if for every representative map f : M $\rightarrow$ N of $\beta$, there exists a point x $\in$ M such that f ($\\tau$ (x)) = f (x). In the case where M is a compact, connected manifold without boundary and N is a compact, connected surface without boundary different from the 2-sphere and the real projective plane, we formulate this property in terms of the pure and full 2-string braid groups of N , and of the fundamental groups of M and the orbit space of M with respect to the action of $\\tau$. If M = N is either the 2-torus T^2 or the Klein bottle K^2 , we then solve the problem of deciding which homotopy classes of [M, M ] have the Borsuk-Ulam property. First, if $\\tau$ : T^2 $\rightarrow$ T^2 is a free Involution that preserves orientation, we show that no homotopy class of [T^2 , T^2 ] has the Borsuk-Ulam property with respect to $\\tau$. Secondly, we prove that up to a certain equivalence relation, there is only one class of free Involutions $\\tau$ : T^2 $\rightarrow$ T^2 that reverse orientation, and for such Involutions, we classify the homotopy classes in [T^2 , T^2 ] that have the Borsuk-Ulam property with respect to $\\tau$ in terms of the induced homomorphism on the fundamental group. Finally, we show that if $\\tau$ : K^2 $\rightarrow$ K^2 is a free Involution, then a homotopy class of [K^2 , K^2 ] has the Borsuk-Ulam property with respect to $\\tau$ if and only if the given homotopy class lifts to the torus.
Sihem Mesnager - One of the best experts on this subject based on the ideXlab platform.
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Involutions over the galois field mathbb f _ 2 n
IEEE Transactions on Information Theory, 2016Co-Authors: Pascale Charpin, Sihem Mesnager, Sumanta SarkarAbstract:An Involution is a permutation, such that its inverse is itself (i.e., cycle length ≤ 2). Due to this property, Involutions have been used in many applications, including cryptography and coding theory. In this paper, we provide a systematic study of Involutions that are defined over a finite field of characteristic 2. We characterize the Involution property of several classes of polynomials and propose several constructions. Furthermore, we study the number of fixed points of Involutions, which is a pertinent question related to permutations with short cycle. In this paper, we mostly have used combinatorial techniques.
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Involutions over the Galois field ${\mathbb F}_{2^{n}}$
IEEE Transactions on Information Theory, 2016Co-Authors: Pascale Charpin, Sihem Mesnager, Sumanta SarkarAbstract:An Involution is a permutation such that its inverse is itself (i.e., cycle length less than 2). Due to this property Involutions have been used in many applications including cryptography and coding theory. In this paper we provide a systematic study of Involutions that are defined over finite field of characteristic 2. We characterize the Involution property of several classes of polynomials and propose several constructions. Further we study the number of fixed points of Involutions which is a pertinent question related to permutations with short cycle. In this paper we mostly have used combinatorial techniques.
