A. Vershik, M. Vsemirnov
We give the canonical normal form for the elements of the finite or infinite alternating groups using local stationary presentation of these groups.
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For a Coxeter system (G,S) the multi-parametric alternating subalgebra H^+(G) of the Hecke algebra and the alternating subgroup B^+(G) of the braid group are defined. Two presentations for H^+(G) and B^+(G) are given; one generalizes the Bourbaki presentation for the alternating subgroups of Coxeter groups, another one uses generators related to edges of the Coxeter graph.
Every tame, prime and alternating knot is equivalent to a tame, prime and alternating knot in regular position, with a common projection. In this work, we show that the Dehn presentation of the knot group of a tame, prime, alternating knot, with a regular and common projection has a finite and complete rewriting system. Although there are rules in the rewriting system with left-hand side a generator and which increase the length of the words we show that the system is terminating.
There were defined by R. Shwartz OGS for non-abelian groups, as an interesting generalization of the basis of finite abelian groups. The definition of OGS states that that every element of a group has a unique presentation as a product of some powers of the OGS, in a specific given order. In case of the symmetric groups S_{n} there is a paper of R. Shwartz, which demonstrates a strong connection between the OGS and the standard Coxeter presentation of the symmetric group. The OGS presentation helps us to find the Coxeter length and the descent set of an arbitrary element of the symmetric group…
Let $G$ be a discrete Coxeter group, $G^+$ its alternating subgroup and $\tilde{G}^+$ the spinor cover of $G^+$. A presentation of the groups $G^+$ and $\tilde{G}^+$ is proved for an arbitrary Coxeter system $(G,S)$; the generators are related to edges of the Coxeter graph. Results of the Coxeter--Todd algorithm - with this presentation - for the chains of alternating groups of types A, B and D are given.
We study combinatorial properties of the alternating subgroup of a Coxeter group, using a presentation of it due to Bourbaki.
The group of alternating colored permutations is the natural analogue of the classical alternating group, inside the wreath product $\mathbb{Z}_r \wr S_n$. We present a 'Coxeter-like' presentation for this group and compute the length function with respect to that presentation. Then, we present this group as a covering of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ and use this point of view to give another expression for the length function. We also use this covering to lift several known parameters of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ to the group of alternating colored permutations.
If a finite group G has a presentation with d generators and r relations, it is well-known that r - d is at least the rank of the Schur multiplier of G; a presentation is called efficient if equality holds. There is an analogous definition for proficient profinite presentations. We show that many perfect groups have proficient presentations. Moreover, we prove that infinitely many alternating groups, symmetric groups and their double covers have proficient presentations
The main result of this paper shows that, over large enough fields of characteristic different from $2$, the alternating Hecke algebras are $\mathbb{Z}$-graded algebras that are isomorphic to fixed-point subalgebras of the quiver Hecke algebra of the symmetric group $\mathfrak{S}_n$. As a special case, this shows that the group algebra of the alternating group, over large enough fields of characteristic different from $2$, is a $\mathbb{Z}$-graded algebra. We give a homogeneous presentation for these algebras, compute their graded dimension and show that the blocks of the quiver Hecke algebras…
The mod 2 cohomology rings of the alternating subgroups of the Coxeter groups of Type B
本研究证明B型Coxeter群的交错子群上同调群直和具殆Hopf环结构，运用Giusti和Sinha的技术算出模2系数下该结构的表示。
We show that one can naturally describe elements of R. Thompson's finitely presented infinite simple group $V$, known by Thompson to have a presentation with four generators and fourteen relations, as products of permutations analogous to transpositions. This perspective provides an intuitive explanation towards the simplicity of $V$ and also perhaps indicates a reason as to why it was one of the first discovered infinite finitely presented simple groups: it is (in some basic sense) a relative of the finite alternating groups. We find a natural infinite presentation for $V$ as a group generate…
The size of minimal generating set for commutator of Sylow 2-subgroup of alternating group was found. Given a permutational wreath product of finite cyclic groups sequence we prove that the commutator width of such groups is 1 and we research some properties of its commutator subgroup. It was shown that $(Syl_2 A_{2^k})^2 = Syl'_2 (A_{2^k}), \, k>2$. A new approach to presentation of Sylow 2-subgroups of alternating group ${A_{2^{k}}}$ was applied. As a result the short proof that the commutator width of Sylow 2-subgroups of alternating group ${A_{2^{k}}}$, permutation group ${S_{2^{k}}}$ and …
An ideal triangulation $\mathcal{T}$ of a hyperbolic 3-manifold $M$ with one cusp is non-peripheral if no edge of $\mathcal{T}$ is homotopic to a curve in the boundary torus of $M$. For such a triangulation, the gluing and completeness equations can be solved to recover the hyperbolic structure of $M$. A planar projection of a knot gives four ideal cell decompositions of its complement (minus 2 balls), two of which are ideal triangulations that use 4 (resp., 5) ideal tetrahedra per crossing. Our main result is that these ideal triangulations are non-peripheral for all planar, reduced, alternat…
Given a permutational wreath product sequence of cyclic groups of prime order we research a commutator width of such groups and some properties of its commutator subgroup. Commutator width of Sylow 2-subgroups of alternating group $A_{2^{k}}$, permutation group $S_{2^k}$ and $C_p \wr B$ were founded. The result of research was extended on subgroups $(Syl_2 {A_{2^k}})'$, $p>2$. The paper presents a construction of commutator subgroup of Sylow 2-subgroups of symmetric and alternating groups. Also minimal generic sets of Sylow 2-subgroups of $A_{2^k}$ were founded. Elements presentation of $(Syl_…
Tame automorphism groups of polynomial rings with property (T) and infinitely many alternating group quotients
本文构造了具有性质（T）且有无穷多个交错群商的新群族，包括由驯顺自同构生成的多项式环自同构子群，并由此得到显式的 Gromov 双曲群表示及交错群的度为4的扩张器 Cayley 图无限族。
A triangle group is denoted by $Δ(p,q,r)$ and has finite presentation $$ Δ(p,q,r)=\langle x,y | x^p=y^q=(xy)^r=1 \rangle .$$ We examine a method for composition of permutation representations of a triangle group $Δ(p,q,r)$ that was used in Everitt's proof of Higman's 1968 conjecture that every Fuchsian group has amongst its homomorphic images all but finitely many alternating groups. We see that some of these compositions must give imprimitive representations, and in particular situations, where the representations being composed are all equivalent copies of an alternating group in the same de…
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