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Let $G(\mathbb{Q})$ be a simply connected Chevalley group over $\mathbb{Q}$ corresponding to a simple Lie algebra $\mathfrak g$ over $\mathbb{C}$. Let $V$ be a finite dimensional faithful highest weight $\mathfrak g$-module and let $V_\mathbb{Z}$ be a Chevalley $\mathbb{Z}$-form of $V$. Let $\Gamma(\mathbb{Z})$ be the subgroup of $G(\mathbb{Q})$ that preserves $V_{\mathbb{Z}}$ and let $G(\mathbb{Z})$ be the group of $\mathbb{Z}$-points of $G(\mathbb{Q})$. Then $G(\mathbb{Q})$ is \emph{integral} if $G(\mathbb{Z})=\Gamma(\mathbb{Z})$. Chevalley's original work constructs a scheme-theoretic integral form of $G(\mathbb{Q})$ which equals $\Gamma(\mathbb{Z})$. Here we give a representation-theoretic proof of integrality of $G(\mathbb{Q})$ using only the action of $G(\mathbb{Q})$ on $V$, rather than the language of group schemes. We discuss the challenges and open problems that arise in trying to extend this to a proof of integrality for Kac-Moody groups over $\mathbb{Q}$.

Let $\mathfrak{g}$ be a symmetrizable Kac--Moody algebra. Given a root $\alpha$ and a real root $\beta$ of $\mathfrak{g}$, it is known that the $\beta$-string through $\alpha$, denoted $R_\alpha(\beta)$, is finite. Given an imaginary root $\beta$, we show that $R_\alpha(\beta)=\{\beta\}$ or $R_\alpha(\beta)$ is infinite. If $(\beta,\beta)<0$, we also show that the multiplicity of the root ${\alpha+n\beta}$ grows at least exponentially as $n\to\infty$. If $(\beta,\beta)=(\alpha, \beta) = 0$, we show that $R_\alpha(\beta)$ is bi-infinite and the multiplicities of $\alpha+n\beta$ are bounded. If $(\beta,\beta)=0$ and $(\alpha, \beta) \neq 0$, we show that $R_\alpha(\beta)$ is semi-infinite and the muliplicity of $\alpha+n\beta$ or $\alpha-n\beta$ grows faster than every polynomial as $n\to\infty$. We also prove that $\dim \mathfrak{g}_{\alpha+\beta} \geq \dim \mathfrak{g}_\alpha + \dim \mathfrak{g}_\beta -1$ whenever $\alpha \neq \beta$ with $(\alpha, \beta)<0$.

The Monster Lie algebra $\frak 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 a construction of a Kac--Moody group given by Tits. 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 $\Uhp$ 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 $\text{Exp}: \widehat{\mathfrak{n}}^+\to\widehat{U}^+$ and $\text{Ad} :\widehat{U}^+ \to \Aut(\widehat{\frak{n}}^+)$ of the classical exponential map and adjoint representation. We show that the action of $\mathbb{M}$ on $\mathfrak m$ induces an action of~$\mathbb{M}$ on~$\widehat{\frak m}$, and that this in turn induces an action of $\mathbb{M}$ on~$\widehat{U}^+$. We also show that the action of $\mathbb{M}$ on $\widehat{\mathfrak n}^+$ is compatible with the action of $\widehat{U}^+$ on $\widehat{\mathfrak n}^+$.

The Monster Lie algebra $\mathfrak m$ is a quotient of the physical space of the vertex algebra $V=V^\natural\otimes V_{1,1}$, where $V^\natural$ is the Moonshine module vertex operator algebra of Frenkel, Lepowsky, and Meurman, and $V_{1,1}$ is the vertex algebra corresponding to the rank 2 even unimodular lattice $\textrm{II}_{1,1}$. We construct vertex algebra elements that project to bases for subalgebras of $\mathfrak m$ isomorphic to $\mathfrak{gl}_{2}$, corresponding to each imaginary simple root, denoted $(1,j)$ for $j>0$. Our method requires the existence of pairs of primary vectors in $V^{\natural}$ satisfying some natural conditions, which we prove. We show that the action of the Monster finite simple group $\mathbb{M}$ on the subspace of primary vectors in $V^\natural$ induces an $\mathbb{M}$-action on the set of $\mathfrak{gl}_2$ subalgebras corresponding to a fixed imaginary simple root. We use the generating function for dimensions of subspaces of primary vectors of $V^\natural$ to prove that this action is non-trivial for small values of $j$., Comment: Final version, to appear in the Journal of Pure and Applied Algebra

Let $A$ be a symmetrizable generalized Cartan matrix with corresponding Kac--Moody algebra $\frak{g}$ over ${\mathbb Q}$. Let $V=V^{\lambda}$ be an integrable highest weight $\frak{g}$-module and let $V_{\mathbb Z}=V^{\lambda}_{\mathbb Z}$ be a ${\mathbb Z}Z$-form of $V$. Let $G$ be an associated minimal representation-theoretic Kac--Moody group and let $G({\mathbb Z})$ be its integral subgroup. Let $\Gamma({\mathbb Z})$ be the Chevalley subgroup of $G$, that is, the subgroup that stabilizes the lattice $V_{{\mathbb Z}}$ in $V$. For a subgroup $M$ of $G$, we say that $M$ is integral if $M\cap G({\mathbb Z})=M\cap \Gamma({\mathbb Z})$ and that $M$ is strongly integral if there exists $v\in V^{\lambda}_{\mathbb Z}$ such that, for all $g\in M$, $g\cdot v\in V_{\mathbb{Z}}$ implies $g\in G({\mathbb{Z}})$. We prove strong integrality of inversion subgroups $U_{(w)}$ of $G$ where, for $w\in W$, $U_{(w)}$ is the the group generated by positive real root groups that are flipped to negative roots by $w^{-1}$. We use this to prove strong integrality of subgroups of the unipotent subgroup $U$ of $G$ generated by commuting real root groups. When $A$ has rank 2, this gives strong integrality of subgroups $U_1$ and $U_2$ where $U=U_{1}{\Large{*}}\ U_{2}$ and each $U_{i}$ is generated by `half' the positive real roots.

We describe various approaches to constructing groups which may serve as Lie group analogs for the monster Lie algebra of Borcherds., Comment: To appear in Letters in Math. Phys

We completely determine the structure constants between real root vectors in a rank 2 Kac--Moody algebra $\mathfrak{g}$. Our description is computationally efficient, even in the rank 2 hyperbolic case where the coefficients of roots on the root lattice grow exponentially with height. Our approach is to extend Carter's method of finding structure constants from those on extraspecial pairs to the rank 2 Kac--Moody case. We also determine all commutator relations involving only real root vectors in all rank 2 Kac-Moody algebras. The generalized Cartan matrix of $\mathfrak{g}$ is of the form $H(a,b)= \left(\begin{smallmatrix} ~2 & -b\\ -a & ~2 \end{smallmatrix}\right)$ where $a,b\in\mathbb{Z}$ and $ab\geq 4$. If $ab=4$, then $\mathfrak{g}$ is of affine type. If $ab>4$, then $\mathfrak{g}$ is of hyperbolic type. Explicit knowledge of the root strings is needed, as well as a characterization of the pairs of real roots whose sums are real. We prove that if $a$ and $b$ are both greater than one, then no sum of real roots can be a real root. We determine the root strings between real roots $\beta,\gamma$ in $H(a,1)$, $a\geq 5$ and we determine the sets $(\mathbb{Z}_{\geq 0}\alpha+\mathbb{Z}_{\geq 0}\beta)\cap\Delta^{\text{re}}(H(a,b))$. One of our tools is a characterization of the root subsystems generated by a subset of roots. We classify these subsystems in rank 2 Kac--Moody root systems. We prove that every rank two infinite root system contains an infinite family of non-isomorphic symmetric rank 2 hyperbolic root subsystems $H(k,k)$ for certain $k\geq 3$, generated by either two short or two long simple roots. We also prove that a non-symmetric hyperbolic root systems $H(a,b)$ with $a\ne b$ and $ab>5$ also contains an infinite family of non-isomorphic non-symmetric rank 2 hyperbolic root subsystems $H(a\ell,b\ell)$, for certain positive integers $\ell$., Comment: 23 pages, 5 figures, includes appendix. arXiv admin note: text overlap with arXiv:1506.05405

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: Proposition 6.4 fails for imaginary roots. This error propagates through other results and the main theorem cannot be proven without Proposition 6.4. A partial proof using a different method can be found here arXiv:2210.01644

Let $\Delta$ be a rank 2 hyperbolic root system. Then $\Delta$ has generalized Cartan matrix $H(a,b)= \left(\begin{smallmatrix} ~2 & -b\\ -a & ~2 \end{smallmatrix}\right)$ indexed by $a,b\in\mathbb{Z}$ with $ab\geq 5$. If $a\neq b$, then $\Delta$ is non-symmetric and is generated by one long simple root and one short simple root; whereas if $a= b$, $\Delta$ is symmetric and is generated by two long simple roots. We prove that if $a\neq b$, then $\Delta$ contains an infinite family of symmetric rank 2 hyperbolic root subsystems $H(k,k)$ for certain $k\geq 3$, generated by either two short or two long simple roots. We also prove that $\Delta$ contains non-symmetric rank 2 hyperbolic root subsystems $H(a',b')$, for certain $a',b'\in\mathbb{Z}$ with $a'b'\geq 5$. One of our tools is a characterization of the types of root subsystems that are generated by a subset of roots. We classify these types of subsystems in rank 2 hyperbolic root systems., Comment: This replaces the previous preprint: Lisa Carbone, Scott H. Murray and Sowmya Srinivasan, "Geometry of rank 2 hyperbolic root systems"

For $G(\mathbb{R})$ a split, simply connected, semisimple Lie group of rank $n$ and $K$ the maximal compact subgroup of $G$, we give a method for computing Iwasawa coordinates of $G/K$ using the Chevalley generators and the Steinberg presentation. When $G/K$ is a scalar coset for a supergravity theory in dimensions $\geq 3$, we determine the action of the integral form $G(\mathbb{Z})$ on $G/K$. We give explicit results for the action of the discrete $U$--duality groups $SL_2(\mathbb{Z})$ and $E_7(\mathbb{Z})$ on the scalar cosets $SL_2(\mathbb{R})/SO_2(\mathbb{R})$ and $E_{7(+7)}(\mathbb{R})/[SU(8,\mathbb{R})/\{\pm Id\}]$ for type IIB supergravity in ten dimensions and 11--dimensional supergravity in $D=4$ dimensions, respectively. For the former, we use this to determine the discrete U--duality transformations on the scalar sector in the Borel gauge and we describe the discrete symmetries of the dyonic charge lattice. We determine the spectrum--generating symmetry group for fundamental BPS solitons of type IIB supergravity in $D=10$ dimensions at the classical level and we propose an analog of this symmetry at the quantum level. We indicate how our methods can be used to study the orbits of discrete U--duality groups in general.

Examining costs and savings associated with moving every Canadian with a Literacy Level 1 or 2 (on the international literacy scale) to Level 3, this analysis is based upon statistically matched data from the "2003 International Adult Literacy and Skills Survey and the 2005-2009 Surveys of Labour and Income Dynamics." The methods provide a mechanism to explore the relationships between benefit receipt and literacy skill. As champions for literacy and essential skills, we know that literacy is a key factor in determining an individual's labour market outcomes--and we set out to prove it by looking at linkages between benefit programs (Employment Insurance, Workers' Compensation and Social Assistance) and literacy levels. Key findings are: (1) National and international literature confirms the correlation between literacy proficiency and individual earnings, as is the common belief within the field, and that the impact of literacy on earnings varies from country to country yet is strongest in North America; (2) Those Canadians who do not complete high school are more than three and a half times as likely to be on social assistance; (3) Social Assistance rolls across Canada would be reduced by 84,000 people, if Canadians' literacy level was increased to OECD recommended standards for internationally competitive productivity; (4) Direct annual savings resulting from an assumed one-time $29 billion investment in upskilling Canadian adults 16 and older would be $2.92 billion: $330 million on Employment Insurance, $2.1 billion on Social Assistance, and $490 million on Workers Compensation; (5) Investment in upskilling Canadians would increase earnings by $85.25 billion or an average of $3,244/worker annually; and (6) Tax revenue and program savings would increase annual fiscal return for governments in Canada by 425%. Three annexes are included: (1) References to Research; (2) Statistical Tables; and (3) Methods Employed to Link the Survey of Labour and Income Dynamics (SLID) and International Adult Literacy and Skills Survey (IALSS) (IALSS) Datasets.

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.

The research described in this note aims at solving the constructive membership problem for the class of quasisimple classical groups. Our algorithms are developed in the black-box group model; that is, they do not require specific characteristics of the representations in which the input groups are given. The elements of a black-box group are represented, not necessarily uniquely, as bit strings of uniform length. We assume the existence of oracles to compute the product of two elements, the inverse of an element, and to test if two strings represent the same element. Solving the constructive membership problem for a black-box group $G$ requires to write every element of $G$ as a word in a given generating set. In practice we write the elements of $G$ as straight-line programs (SLPs) which can be viewed as a compact way of writing words.

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.

The minimal faithful permutation degree of a finite group $G$, denote by $\mu(G)$ is the least non-negative integer $n$ such that $G$ embeds inside the symmetric group $\Sym(n)$. In this paper, we outline a Magma proof that 10 is the smallest degree for which there are groups $G$ and $H$ such that $\mu(G \times H) < \mu(G)+ \mu(H)$., Comment: 4 pages