arXiv subjects

Matthew C. Clarke

Publications and source records attributed to Matthew C. Clarke.

5 recordsLinked to original sources

arXiv2011

The Hesselink stratification of nullcones and base change

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $p \ge 0$. We give a case-free proof of Lusztig's conjectures [Unipotent elements in small characteristic, {\em Transform. Groups} 10 (2005), 449--487] on so-called unipotent pieces. This presents a uniform picture of the unipotent elements of $G$ which can be viewed as an extension of the Dynkin--Kostant theory, but is valid without restriction on $p$. We also obtain analogous results for the adjoint action of $G$ on its Lie algebra $\gl$ and the coadjoint action of $G$ on $\gl^*$.

math.RT↗

arXiv2011

Computing nilpotent and unipotent canonical forms: a symmetric approach

Let $k$ be an algebraically closed field of any characteristic except 2, and let $G = \GL_n(k)$ be the general linear group, regarded as an algebraic group over $k$. Using an algebro-geometric argument and Dynkin-Kostant theory for $G$ we begin by obtaining a canonical form for nilpotent $\Ad(G)$-orbits in $\glł_n(k)$ which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map $f : (x_{i,j})\mapsto (x_{n+1-j,n+1-i})$), with entries in $\{0,1\}$. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing $G$ by any simple classical algebraic group we thus obtain a unified approach to computing representatives for nilpotent orbits of all classical Lie algebras. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in $G$. As a corollary we obtain a complete set of generic canonical representatives for the unipotent classes in finite general unitary groups $\GU_n(\F_q)$ for all prime powers $q$.

math.GR↗

arXiv2011

On the endomorphism algebra of generalised Gelfand-Graev representations

Let $G$ be a connected reductive algebraic group defined over the finite field $\F_q$, where $q$ is a power of a good prime for $G$, and let $F$ denote the corresponding Frobenius endomorphism, so that $G^F$ is a finite reductive group. Let $u \in G^F$ be a unipotent element and let $Γ_u$ be the associated generalised Gelfand-Graev representation of $G^F$. Under the assumption that $G$ has a connected centre, we show that the dimension of the endomorphism algebra of $Γ_u$ is a polynomial in $q$, with degree given by $\dim C_G(u)$. When the centre of $G$ is disconnected, it is impossible, in general, to parametrise the (isomorphism classes of) generalised Gelfand-Graev representations independently of $q$, unless one adopts a convention of considering separately various congruence classes of $q$. Subject to such a convention we extend our result.

math.RT↗

arXiv2011

On the algebra structure of some bismash products

We study several families of semisimple Hopf algebras, arising as bismash products, which are constructed from finite groups with a certain specified factorization. First we associate a bismash product $H_q$ of dimension $q(q-1)(q+1)$ to each of the finite groups $PGL_2(q)$ and show that these $H_q$ do not have the structure (as algebras) of group algebras (except when $q =2,3$). As a corollary, all Hopf algebras constructed from them by a comultiplication twist also have this property and are thus non-trivial. We also show that bismash products constructed from Frobenius groups do have the structure (as algebras) of group algebras.

math.RT↗

arXiv2009

On the Chances of Completing the Game of "Perpetual Motion"

This brief paper describes the single-player card game called "Perpetual Motion" and reports on a computational analysis of the game's outcome. The analysis follows a Monte Carlo methodology based on a sample of 10,000 randomly generated games. The key result is that 54.55% +/- 0.89% of games can be completed (by a patient player!) but that the remaining 45.45% result in non-terminating cycles. The lengths of these non-terminating cycles leave some outstanding questions.

cs.GT↗