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Sergey Shpectorov

Publications and source records attributed to Sergey Shpectorov.

25 records · Page 2Linked to original sources
arXiv2014

Recovering the Lie algebra from its extremal geometry

An element $x$ of a Lie algebra $L$ over the field $F$ is extremal if $[x,[x,L]]=Fx$. Under minor assumptions, it is known that, for a simple Lie algebra $L$, the extremal geometry ${\cal{E}}(L)$ is a subspace of the projective geometry of $L$ and either has no lines or is the root shadow space of an irreducible spherical building $Δ$. We prove that if $Δ$ is of simply-laced type, then $L$ is a quotient of a Chevalley algebra of the same type.

math.RA↗
arXiv2014

On Sidki's presentation for orthogonal groups

We study presentations, defined by Sidki, resulting in groups $y(m,n)$ that are conjectured to be finite orthogonal groups of dimension $m+1$ in characteristic two. This conjecture, if true, shows an interesting pattern, possibly connected with Bott periodicity. It would also give new presentations for a large family of finite orthogonal groups in characteristic two, with no generator having the same order as the cyclic group of the field. We generalise the presentation to an infinite version $y(m)$ and explicitly relate this to previous work done by Sidki. The original groups $y(m,n)$ can be found as quotients over congruence subgroups of $y(m)$. We give two representations of our group $y(m)$. One into an orthogonal group of dimension $m+1$ and the other, using Clifford algebras, into the corresponding pin group, both defined over a ring in characteristic two. Hence, this gives two different actions of the group. Sidki's homomorphism into $SL_{2^{m-2}}(R)$ is recovered and extended as an action on a submodule of the Clifford algebra.

math.GR↗
arXiv2013

A class of 2-groups admitting an action of the symmetric group of degree 3

A biextraspecial group of rank $m$ is an extension of a special 2-group $Q$ of the form $2^{2 + 2m}$ by $L\cong L_2(2)$, such that the 3-element from $L$ acts on $Q$ fixed-point-freely. Subgroups of this type appear in at least the sporadic groups $J_2$, $J_3$, $McL$, $Suz$, and $Co_1$. In this paper we completely classify biextraspecial groups, namely, we show that the rank $m$ must be even and for each such $m$ there exist exactly two biextraspecial groups $B^\varepsilon(m)$ up to isomorphism where $\varepsilon\in{+,-}$. We also prove that $\Out(B^\varepsilon(m))$ is an extension of the $m$-dimensional orthogonal GF(2)-space of type $\varepsilon$ by the corresponding orthogonal group. The extension is non-split except in a few small cases.

math.GR↗
arXiv2011

Generating sets of Affine groups of low genus

We describe a new algorithm for computing braid orbits on Nielsen classes. As an application we classify all families of affine genus zero systems; that is all families of coverings of the Riemann sphere by itself such that the monodromy group is a primitive affine permutation group.

math.GR↗
arXiv2011

AG-groups and other classes of right Bol quasigroups

By a result of Sharma, right Bol quasigroups are obtainable from right Bol loops via an involutive automorphism. We prove that the class of AG-groups, introduced by Kamran, is obtained via the same construction from abelian groups. We further introduce a new class of Bol* quasigroups, which turns out to correspond, as above, to the class of groups. Sharma's correspondence allows an efficient implementation and we present some enumeration results for the above three classes.

math.GR↗
arXiv2008

Hypercube embedding of Wythoffians

The Wythoff construction takes a $d$-dimensional polytope $P$, a subset $S$ of $\{0,..., d\}$ and returns another $d$-dimensional polytope $P(S)$. If $P$ is a regular polytope, then $P(S)$ is vertex-transitive. This construction builds a large part of the Archimedean polytopes and tilings in dimension 3 and 4. We want to determine, which of those Wythoffians $P(S)$ with regular $P$ have their skeleton or dual skeleton isometrically embeddable into the hypercubes $H_m$ and half-cubes ${1/2}H_m$. We find six infinite series, which, we conjecture, cover all cases for dimension $d>5$ and some sporadic cases in dimension 3 and 4 (see Tables \ref{WythoffEmbeddable3} and \ref{WythoffEmbeddable4}). Three out of those six infinite series are explained by a general result about the embedding of Wythoff construction for Coxeter groups. In the last section, we consider the Euclidean case; also, zonotopality of embeddable $P(S)$ are addressed throughout the text.

math.CO↗