Conjugacy classes in maximal parabolic subgroups of general linear groups

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a ``matrix problem''. Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a perfect field when one of the two blocks has dimension less than 6. In particular, this includes every maximal parabolic subgroup of GL_n(k) for n < 12 and k a perfect field. If our field is finite of size q, we also show that the number of conjugacy classes, and so the number of characters, of these groups is a polynomial in $q$ with integral coefficients.Comment: 23 pages, 6 figures. See also http://zaphod.uchicago.edu/~murray/research/index.html . Submitted to Journal of Algebr

Constructive homomorphisms for classical groups

Let Omega be a quasisimple classical group in its natural representation over a finite vector space V, and let Delta be its normaliser in the general linear group. We construct the projection from Delta to Delta/Omega and provide fast, polynomial-time algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent Delta/Omega as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of Omega. A key ingredient of our algorithms is a new, asymptotically fast method for constructing isometries between spaces with forms. Our results are useful for the matrix group recognition project, can be used to solve element conjugacy problems, and can improve algorithms to construct maximal subgroups

Fundamental domains for congruence subgroups of SL2 in positive characteristic

In this work, we construct fundamental domains for congruence subgroups of $SL_2(F_q[t])$ and $PGL_2(F_q[t])$. Our method uses Gekeler's description of the fundamental domains on the Bruhat- Tits tree $X = X_{q+1}$ in terms of cosets of subgroups. We compute the fundamental domains for a number of congruence subgroups explicitly as graphs of groups using the computer algebra system Magma

Computing in unipotent and reductive algebraic groups

The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on the maximal unipotent subgroup of a split reductive group and show how this improves computation in the reductive group itself.Comment: 22 page

Spontaneous γH2AX Foci in Human Solid Tumor-Derived Cell Lines in Relation to p21WAF1 and WIP1 Expression.

Phosphorylation of H2AX on Ser139 (γH2AX) after exposure to ionizing radiation produces nuclear foci that are detectable by immunofluorescence microscopy. These so-called γH2AX foci have been adopted as quantitative markers for DNA double-strand breaks. High numbers of spontaneous γH2AX foci have also been reported for some human solid tumor-derived cell lines, but the molecular mechanism(s) for this response remains elusive. Here we show that cancer cells (e.g., HCT116; MCF7) that constitutively express detectable levels of p21WAF1 (p21) exhibit low numbers of γH2AX foci (&lt;3/nucleus), whereas p21 knockout cells (HCT116p21-/-) and constitutively low p21-expressing cells (e.g., MDA-MB-231) exhibit high numbers of foci (e.g., &gt;50/nucleus), and that these foci are not associated with apoptosis. The majority (&gt;95%) of cells within HCT116p21-/- and MDA-MB-231 cultures contain high levels of phosphorylated p53, which is localized in the nucleus. We further show an inverse relationship between γH2AX foci and nuclear accumulation of WIP1, an oncogenic phosphatase. Our studies suggest that: (i) p21 deficiency might provide a selective pressure for the emergence of apoptosis-resistant progeny exhibiting genomic instability, manifested as spontaneous γH2AX foci coupled with phosphorylation and nuclear accumulation of p53; and (ii) p21 might contribute to positive regulation of WIP1, resulting in dephosphorylation of γH2AX

Infinite dimensional Chevalley groups and Kac-Moody groups over Z\mathbb{Z}

Let $A$ be a symmetrizable generalized Cartan matrix, which is not of finite or affine type. Let $\mathfrak{g}$ be the corresponding Kac-Moody algebra over a commutative ring $R$ with $1$. We construct an infinite-dimensional group $G_V(R)$ analogous to a finite-dimensional Chevalley group over $R$. We use a $\mathbb{Z}$-form of the universal enveloping algebra of $\mathfrak{g}$ and a $\mathbb{Z}$-form of an integrable highest-weight module $V$. We construct groups $G_V(\mathbb{Z})$ analogous to arithmetic subgroups in the finite-dimensional case. We also consider a universal representation-theoretic Kac-Moody group $G$ and its completion $\widetilde{G}$. For the completion we prove a Bruhat decomposition $\widetilde{G}({\mathbb{Q}})=\widetilde{G}({\mathbb{Z}})\widetilde{B}({\mathbb{Q}})$ over $\mathbb{Q}$, and that the arithmetic subgroup $\widetilde{\Gamma}(\mathbb{Z})$ coincides with the subgroup of integral points $\widetilde{G}(\mathbb{Z})$Comment: Submitte

A Lie group analog for the Monster Lie algebra

The Monster Lie algebra $\mathfrak{m}$, which admits an action of the Monster finite simple group $\mathbb{M}$, was introduced by Borcherds as part of his work on the Conway-Norton Monstrous Moonshine conjecture. Here we construct an analog $G(\frak m)$ of a Lie group, or Kac-Moody group, associated to $\frak m$. The group $G(\frak m)$ is given by generators and relations, analogous to the Tits construction of a Kac-Moody group. In the absence of local nilpotence of the adjoint representation of $\frak m$, we introduce the notion of pro-summability of an infinite sum of operators. We use this to construct a complete pro-unipotent group $\widehat{U}^+$ of automorphisms of a completion $\widehat{\mathfrak{m}}=\frak n^-\ \oplus\ \frak h\ \oplus\ \widehat{\frak n}^+$ of $\mathfrak{m}$, where $\widehat{\frak n}^+$ is the formal product of the positive root spaces of $\frak m$. The elements of $\widehat{U}^+$ are pro-summable infinite series with constant term 1. The group $\widehat{U}^+$ has a subgroup $\widehat{U}^+_{\text{im}}$, which is an analog of a complete unipotent group corresponding to the positive imaginary roots of $\frak m$. We construct analogs Exp:$\widehat{\mathfrak{n}}^+\to\widehat{U}^+$ and Ad:$\widehat{U}^+ \to Aut(\widehat{\frak{n}}^+)$ of the classical exponential map and adjoint representation. Although the group $G(\mathfrak m)$ is not a group of automorphisms, it contains the analog of a unipotent subgroup $U^+$, which conjecturally acts as automorphisms of $\widehat{\mathfrak{m}}$. We also construct groups of automorphisms of $\mathfrak{m}$, of certain $\mathfrak{gl}_2$ subalgebras of $\mathfrak{m}$, of the completion $\widehat{\mathfrak{m}}$ and of similar completions of $\frak m$ that are conjecturally identified with subgroups of~$G(\mathfrak m)$