Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of the natural module for $G$. Infiniteness is proven using a dimension criterion which involves root systems.
Let $(W,S)$ be a Coxeter system of finite rank and let $J,K\subset S$. We study the rationality of the Poincaré series of the set of representatives of minimal length of $(W_J,W_K)$-double cosets of $W$: we conclude that it depends mostly on the rationality of the Poincaré series of the normalizers of finite parabolic subgroups of $W$. For affine Weyl groups, we prove that all these series are rational and we give some explicit examples.
Let $H$ be a hyperbolic group, $A$ and $B$ be subgroups of $H$, and $gr(H,A,B)$ be the growth function of the double cosets $AhB, h \in H$. We prove that the behavior of $gr(H,A,B)$ splits into two different cases. If $A$ and $B$ are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as $gr(H,A,B)$. We can even take $A=B$. In contrast, for quasiconvex subgroups A and B of infinite index, $gr(H,A,B)$ is exponential. Moreover, there exists a constant $λ> 0$, such that $gr(H,A,B)(r) >λf_H(r)$ for all big enough $r$, where $f_H(r)$ is the growth function…
We derive double coset formulae for the genus and extended genus of a finitely generated nilpotent group G, using the notions of bounded and bounded above automorphisms of $\prod G_S$, which are defined relative to a fixed fracture square for G.
We develop combinatorics of parabolic double cosets in finite Coxeter groups as a follow-up of recent articles by Billey-Konvalinka-Petersen-Slofstra-Tenner and Petersen. (1) We construct a double coset system as a generalization of a two-sided analogue of a Coxeter complex and present its order structure with its local dimension function on certain connected components. As applications of double cosets to Bruhat graphs, we also prove: (2) every parabolic double coset is regular, (3) invariance of degree on Bruhat graph on lower intervals as an analogy of the one for Kazhdan-Lusztig polynomial…
Let $H$ and $K$ be subgroups of a finite group $G$. Pick $g \in G$ uniformly at random. We study the distribution induced on double cosets. Three examples are treated in detail: 1) $H = K = $ the Borel subgroup in $GL_n(\mathbb{F}_q)$. This leads to new theorems for Mallows measure on permutations and new insights into the LU matrix factorization. 2) The double cosets of the hyperoctahedral group inside $S_{2n}$, which leads to new applications of the Ewens's sampling formula of mathematical genetics. 3) Finally, if $H$ and $K$ are parabolic subgroups of $S_n$, the double cosets are `contingen…
We parametrize the space of double cosets of the group $GL(n,\Bbbk)$ with respect to two subgroups $T_-$, $T_+$ of block strictly triangular matrices. In Addendum, we consider the quasi-regular representation of $GL(n,\Bbb{C})$ in $L^2$ on $T_- \ GL(n,\Bbb{C})$, observe that it admits an additional group of symmetries, find the joint spectrum, and observe that it is multiplicity free.
This is the author's second paper treating the double coset problem for classical groups. Let $G$ be an algebraic group over an algebraically closed field $K$. The double coset problem consists of classifying the pairs $H,J$ of closed connected subgroups of $G$ with finitely many $(H,J)$-double cosets in $G$. The critical setup occurs when one of $H,J$, say $H$, is reductive, and $J$ is a parabolic subgroup. Assume that $G$ is a classical group, $H$ is simple and $J$ is a maximal parabolic $P_k$, the stabilizer of a totally singular $k$-space. Then most candidates have $k=1$ or $k=2$. The case…
Let $\mathbb F$ be a finite field. Consider a direct sum $V$ of an infinite number of copies of $\mathbb F$, consider the dual space $V^\diamond$, i.~e., the direct product of an infinite number of copies of $\mathbb F$. Consider the direct sum ${\mathbb V}=V\oplus V^\diamond$. The object of the paper is the group $\mathbf{GL}$ of continuous linear operators in $\mathbb V$. We reduce the theory of unitary representations of $\mathbf{GL}$ to projective representations of a certain category whose morphisms are linear relations in finite-dimensional linear spaces over $\mathbb F$. In fact we cons…
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences（俄罗斯科学院西伯利亚分院索博列夫数学研究所）；St. Petersburg Department of Steklov Mathematical Institute of the Russian Academy of Sciences（俄罗斯科学院斯特克洛夫数学研究所圣彼得堡分部）
Let $G$ be a totally disconnected, locally compact group and let $H$ be an equicontinuously (for example, compactly) generated group of automorphisms of $G$. We show that every distal action of $H$ on a coset space of $G$ is a SIN action, with the small invariant neighbourhoods arising from open $H$-invariant subgroups. We obtain a number of consequences for the structure of the collection of open subgroups. For example, it follows that for every compactly generated subgroup $K$ of $G$, there is a compactly generated open subgroup $E$ of $G$ such that $K \le E$ and such that every open subgrou…
Let $G$ and $H$ be quasi-isometric finitely generated groups and let $P\leq G$; is there a subgroup $Q$ (or a collection of subgroups) of $H$ whose left cosets coarsely reflect the geometry of the left cosets of $P$ in $G$? We explore sufficient conditions for a positive answer. The article consider pairs of the form $(G,\mathcal{P})$ where $G$ is a finitely generated group and $\mathcal{P}$ a finite collection of subgroups, there is a notion of quasi-isometry of pairs, and quasi-isometrically characteristic collection of subgroups. A subgroup is qi-characteristic if it belongs to a qi-charact…
We provide an algorithm to solve the word problem in all fundamental groups of closed 3-manifolds; in particular, we show that these groups are autostackable. This provides a common framework for a solution to the word problem in any closed 3-manifold group using finite state automata. We also introduce the notion of a group which is autostackable respecting a subgroup, and show that a fundamental group of a graph of groups whose vertex groups are autostackable respecting any edge group is autostackable. A group that is strongly coset automatic over an autostackable subgroup, using a prefix-cl…
Let $G$ be a finitely generated group. Cashen and Mackay proved that if the contracting boundary of $G$ with the topology of fellow travelling quasi-geodesics is compact then $G$ is a hyperbolic group. Let $\mathcal{H}$ be a finite collection of finitely generated infinite index subgroups of $G$. Let $G^h$ be the cusped space obtained by attaching combinatorial horoballs to each left cosets of elements of $\mathcal {H}$. In this article, we prove that if the combinatorial horoballs are contracting and $G^h$ has compact contracting boundary then $G$ is hyperbolic relative to $\mathcal{H}$.
This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double coset decomposition with respect to a Siegel maximal parabolic subgroup, which is important in computing infinite-dimensional representations for some algebraic groups.
We consider groups $\mathbb{I}$ of isometries of ultrametric Urysohn spaces $\mathbb{U}$. Such spaces $\mathbb{U}$ admit transparent realizations as boundaries of certain $R$-trees and the groups $\mathbb{I}$ are groups of automorphisms of these $R$-trees. Denote by $\mathbb{I}[X]\subset \mathbb{I}$ stabilizers of finite subspaces $X\subset \mathbb{U}$. Double cosets $\mathbb{I}[X]\cdot g\cdot \mathbb{I}[Y]$, where $g\in \mathbb{I}$, are enumerated by ultrametrics on union of spaces $X\cup Y$. We construct natural associative multiplications on double coset spaces $\mathbb{I}[X]\backslash \mat…
