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Ilia Smilga

Publications and source records attributed to Ilia Smilga.

17 recordsLinked to original sources
arXiv2024

Adjoints in symmetric squares of Lie algebra representations

*Caveat: we learned post-factum that most of these results are not novel. We are keeping this paper for continuity reasons.* Given finite-dimensional complex representations $V$ and $V'$ of a simply-connected semisimple compact Lie group $G$, we determine the dimension of the $G$-invariant subspace of $\mathrm{adj}(G)\otimes V\otimes V'$, of $\mathrm{adj}(G)\otimes S^2 V$, and of $\mathrm{adj}(G)\otimesΛ^2 V$, where $\mathrm{adj}(G)$ is the adjoint representation. In other words we derive the multiplicity with which summands of $\mathrm{adj}(G)$ appear in a tensor product $V \otimes V'$ or (anti)symmetric square $S^2 V$ or $Λ^2 V$. We find in particular that the dimension of the $G$-invariant subspace of $\mathrm{adj}(G)\otimes S^2 V$ is larger than (resp. smaller or equal to) that of $\mathrm{adj}(G)\otimesΛ^2 V$ for a symplectic (resp. orthogonal) representation $V$.

math.RT↗
arXiv2024

Proper affine actions: a sufficient criterion

For a semisimple real Lie group $G$ with an irreducible representation $ρ$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $ρ$ for existence of a group of affine transformations of $V$ whose linear part is Zariski-dense in $ρ(G)$ and that is free, nonabelian and acts properly discontinuously on $V$. This new criterion is more general than the one given in the author's previous paper "Proper affine actions in non-swinging representations" (submitted; available at arXiv:1605.03833), insofar as it also deals with "swinging" representations. We conjecture that it is actually a necessary and sufficient criterion, applicable to all representations.

math.GR↗
arXiv2023

Action of $w_0$ on $V^L$: the special case of $\mathfrak{so}(1,n)$

In this note, we present an algorithm that allows to answer any individual instance of the following question. Let $G_{\mathbb{R}}$ be a semisimple real Lie group, and $V$ an irreducible representation of $G_{\mathbb{R}}$. How does the longest element $w_0$ of the restricted Weyl group $W$ act on the subspace $V^L$ of $V$ formed by vectors that are invariant by $L$, the centralizer of a maximal split torus of $G_{\mathbb{R}}$? This algorithm comprises two parts. First we describe a complete answer to this question in the particular case where $G_{\mathbb{R}} = \operatorname{SO}(1,n)$ for any $n \geq 2$. Then, for an arbitrary $G_{\mathbb{R}}$, we show that it suffices to do the computation in a well-chosen subgroup $S_{\mathbb{R}} \subset G_{\mathbb{R}}$ which is (up to isogeny) the product of several groups that are either compact, abelian or isomorphic to $\operatorname{SO}(1,n)$ for some $n \geq 2$.

math.RT↗
arXiv2023

Representations having vectors fixed by a Levi subgroup

For any semisimple real Lie algebra $\mathfrak{g}_\mathbb{R}$, we classify the representations of $\mathfrak{g}_\mathbb{R}$ that have at least one nonzero vector on which the centralizer of a Cartan subspace, also known as the centralizer of a maximal split torus, acts trivially. In the process, we revisit the notion of $\mathfrak{g}$-standard Young tableaux, introduced by Lakshmibai and studied by Littelmann, that provides a combinatorial model for the characters of the irreducible representations of any classical semisimple Lie algebra $\mathfrak{g}$. We construct a new version of these objects, which differs from the old one for $\mathfrak{g} = \mathfrak{so}(2r)$ and seems, in some sense, simpler and more natural.

math.RT↗
arXiv2023

Geometrically and topologically random surfaces in a closed hyperbolic three manifold

We study the distribution of geometrically and topologically nearly geodesic random surfaces in a closed hyperbolic 3-manifold M. In particular, we describe PSL(2,R) invariant measures on the Grassmann bundle G(M) which arise as limits of random minimal surfaces. It is showed that if M contains at least one totally geodesic subsurface then every topological limiting measure is totally scarring (i.e supported on the totally geodesic locus), while we prove that geometrical limiting measures are never totally scarring.

math.GT↗
arXiv2022

Action of $w_0$ on $V^L$ for orthogonal and exceptional groups

In this note, we present some results that partially answer the following question. Let $G$ be a simple real Lie group; what is the set of representations $V$ of $G$ in which the longest element $w_0$ of the restricted Weyl group $W$ acts nontrivially on the subspace $V^L$ of $V$ formed by vectors that are invariant by $L$, the centralizer of a maximal split torus of $G$? We give a conjectural answer to that question, as well as the experimental results that back this conjecture, when $G$ is either an orthogonal group (real form of $\operatorname{SO}_n(\mathbb{C})$ for some $n$) or an exceptional group.

math.RT↗
arXiv2021

New sequences of non-free rational points

We exhibit some new infinite families of rational values of $τ$, some of them squares of rationals, for which the group or even the semigroup generated by the matrices $\left( \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 1 & 0 \\ τ& 1 \end{smallmatrix} \right)$ is not free.

math.GR↗
arXiv2020

Action of the restricted Weyl group on the $L$-invariant vectors of a representation

This note constitutes a brief survey of our recent work on the problem of determining, for a given real Lie group~$G$, the set of representations~$V$ in which the longest element~$w_0$ of the restricted Weyl group~$W$ acts nontrivially on the subspace~$V^L$ of~$V$ formed by vectors that are invariant by~$L$, the centralizer of a maximal split torus of~$G$.

math.GR↗
arXiv2020

Proper actions of discrete groups of affine transformations

In the early 1980's Margulis startled the world by showing the existence of proper affine actions of free groups on 3-space, answering a provocative and suggestive question Milnor posed in 1977. In this paper we discuss the historical background motivating this question, recent progress on this subject, and future directions inspired by this discovery.

math.GR↗
arXiv2019

Action of Weyl group on zero weight space

For any simple complex Lie group we classify irreducible finite-dimensional representations $ρ$ for which the longest element $w_0$ of the Weyl group acts nontrivially on the zero weight space. Among irreducible representations that have zero among their weights, $w_0$ acts by $\pm$Id if and only if the highest weight of $ρ$ is a multiple of a fundamental weight, with a coefficient less than a bound that depends on the group and on the fundamental weight.

math.RT↗
arXiv2019

Construction of Milnorian representations

We prove a partial converse to the main theorem of the author's previous paper "Proper affine actions: a sufficient criterion" (submitted; available at arXiv:1612.08942). More precisely, let $G$ be a semisimple real Lie group with a representation $ρ$ on a finite-dimensional real vector space $V$, that does not satisfy the criterion from the previous paper. Assuming that $ρ$ is irreducible and under some additional assumptions on $G$ and $ρ$, we then prove that there does not exist a group of affine transformations acting properly discontinuously on $V$ whose linear part is Zariski-dense in $ρ(G)$.

math.GR↗
arXiv2018

Proper affine actions in non-swinging representations

For a semisimple real Lie group $G$ with an irreducible representation $ρ$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $ρ$ for existence of a group of affine transformations of $V$ whose linear part is Zariski-dense in $ρ(G)$ and that is free, nonabelian and acts properly discontinuously on $V$.

math.GR↗
arXiv2016

Proper affine actions on semisimple Lie algebras

For any noncompact semisimple real Lie group $G$, we construct a group of affine transformations of its Lie algebra $\mathfrak{g}$ whose linear part is Zariski-dense in $\operatorname{Ad} G$ and which is free, nonabelian and acts properly discontinuously on $\mathfrak{g}$.

math.GR↗
arXiv2013

Fundamental domains for properly discontinuous affine groups

We construct a fundamental region for the action on the $2d+1$-dimensional affine space of some free, discrete, properly discontinuous groups of affine transformations preserving a quadratic form of signature $(d+1, d)$, where $d$ is any odd positive integer.

math.GR↗
arXiv2012

Harmonic functions on the Sierpinski triangle

In this paper, we give a few results on the local behavior of harmonic functions on the Sierpinski triangle - more precisely, of their restriction to a side of the triangle. First we present a general formula that gives the Hölder exponent of such a function in a given point. From this formula, we deduce an explicit algorithm to calculate this exponent in any rational point, and the fact that the derivative of such a function is always equal to 0, infinity or undefined.

math.DS↗