arXiv subjects

Simon Gritschacher

Publications and source records attributed to Simon Gritschacher.

15 recordsLinked to original sources

arXiv2026

Torsion primes for spaces of commuting elements in Lie groups

Let $G$ be a complex reductive group and $C_m(G)_1$ the identity component of the space of $m$-tuples of commuting elements in $G$. We prove that for every $m\geq 2$, the integral singular cohomology of $C_m(G)_1$ has torsion precisely at those primes which divide the order of the Weyl group of $G$. We deduce the analogous statement for compact Lie groups, settling a conjecture of Kishimoto and Takeda, and prove the corresponding result for compact Lie algebras. We prove these results using Smith theory for the conjugation action by semisimple elements of prime power order.

math.RT↗

arXiv2026

The equivariant cohomology ring of the representation variety $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C}))$

We give a presentation of the $\mathrm{GL}_n(\mathbb{C})$-equivariant cohomology ring with $\mathbb{Z}$-coefficients of the variety $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C})) \subseteq \mathrm{GL}_n(\mathbb{C})^2$ for any $n$. It is torsion free and minimally generated as a $H^\ast B\mathrm{GL}_n(\mathbb{C})$-algebra by $3n$ elements. The ideal of relations is the saturation of an $n$-generator ideal by even powers of the Vandermonde polynomial. For coefficients in a field whose characteristic does not divide $n!$, we also give a presentation of the non-equivariant cohomology ring of $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C}))$.

math.AT↗

arXiv2025

Spaces of homomorphisms, formality and Hochschild homology

Let $G$ be a discrete group. The topological category of finite dimensional unitary representations of $G$ is symmetric monoidal under direct sum and has an associated $\mathbb{E}_\infty$-space $\mathcal{K}^{\mathrm{def}}(G)$. We show that if $G$ and $A$ are finitely generated groups and $A$ is abelian, then $\mathcal{K}^{\mathrm{def}}(G\times A)\simeq \mathcal{K}^{\mathrm{def}}(G)\otimes \widehat{A}$ as $\mathbb{E}_\infty$-spaces, where $\widehat{A}$ is the Pontryagin dual of $A$. We deduce a homology stability result for the homomorphism varieties $\mathrm{Hom}(G\times \mathbb{Z}^r,U(n))$ using the local-to-global principle for homology stability of Kupers--Miller. For a finitely generated free group $F$ and a field $k$ of characteristic zero, we show that the singular $k$-chains in $\mathcal{K}^{\mathrm{def}}(F)$ are formal as an $\mathbb{E}_\infty$-$k$-algebra. Using this we describe the equivariant homology of $\mathrm{Hom}(F \times A,U(n))$ for every $n$ in terms of higher Hochschild homology of an explicitly determined commutative $k$-algebra. As an example we show that $\mathrm{Hom}(F\times \mathbb{Z}^r,U(2))$ is $U(2)$-equivariantly formal for every $r$ and we compute the Poincar{é} polynomial.

math.AT↗

arXiv2025

A stable splitting for spaces of commuting elements in unitary groups

We prove an analogue of Miller's stable splitting of the unitary group $U(m)$ for spaces of commuting elements in $U(m)$. After inverting $m!$, the space $\text{Hom}(\mathbb{Z}^n,U(m))$ splits stably as a wedge of Thom-like spaces of bundles of commuting varieties over certain partial flag manifolds. Using Steenrod operations we prove that our splitting does not hold integrally. Analogous decompositions for symplectic and orthogonal groups as well as homological results for the one-point compactification of the commuting variety in a Lie algebra are also provided.

math.AT↗

arXiv2024

Commuting varieties and the rank filtration of topological K-theory

We consider the space of $n$-tuples of pairwise commuting elements in the Lie algebra of $U(m)$. We relate its one-point compactification to the subquotients of certain rank filtrations of connective complex $K$-theory. We also describe the variant for connective real $K$-theory.

math.AT↗

arXiv2021

On families of nilpotent subgroups and associated coset posets

We study some properties of the coset poset associated with the family of subgroups of class $\leq 2$ of a nilpotent group of class $\leq 3$. We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is $2$-Engel, and $2$-connected if and only if the group is nilpotent of class $2$ or less. We determine the homotopy type of the coset poset for the group of $4\times 4$ upper unitriangular matrices over $\mathbb{F}_p$, and for the Burnside groups of exponent $3$.

math.GR↗

arXiv2021

On the second homotopy group of spaces of commuting elements in Lie groups

Let $G$ be a compact connected Lie group and $n\geqslant 1$ an integer. Consider the space of ordered commuting $n$-tuples in $G$, $Hom(\mathbb{Z}^n,G)$, and its quotient under the adjoint action, $Rep(\mathbb{Z}^n,G):=Hom(\mathbb{Z}^n,G)/G$. In this article we study and in many cases compute the homotopy groups $π_2(Hom(\mathbb{Z}^n,G))$. For $G$ simply--connected and simple we show that $π_2(Hom(\mathbb{Z}^2,G))\cong \mathbb{Z}$ and $π_2(Rep(\mathbb{Z}^2,G))\cong \mathbb{Z}$, and that on these groups the quotient map $Hom(\mathbb{Z}^2,G)\to Rep(\mathbb{Z}^2,G)$ induces multiplication by the Dynkin index of $G$. More generally we show that if $G$ is simple and $Hom(\mathbb{Z}^2,G)_{1}\subseteq Hom(\mathbb{Z}^2,G)$ is the path--component of the trivial homomorphism, then $H_2(Hom(\mathbb{Z}^2,G)_{1};\mathbb{Z})$ is an extension of the Schur multiplier of $π_1(G)^2$ by $\mathbb{Z}$. We apply our computations to prove that if $B_{com}G_{1}$ is the classifying space for commutativity at the identity component, then $π_4(B_{com}G_{1})\cong \mathbb{Z}\oplus \mathbb{Z}$, and we construct examples of non-trivial transitionally commutative structures on the trivial principal $G$-bundle over the sphere $\mathbb{S}^{4}$.

math.AT↗

arXiv2020

Higher generation by abelian subgroups in Lie groups

To a compact Lie group $G$ one can associate a space $E(2,G)$ akin to the poset of cosets of abelian subgroups of a discrete group. The space $E(2,G)$ was introduced by Adem, F. Cohen and Torres-Giese, and subsequently studied by Adem and Gómez, and other authors. In this short note, we prove that $G$ is abelian if and only if $π_i(E(2,G))=0$ for $i=1,2,4$. This is a Lie group analogue of the fact that the poset of cosets of abelian subgroups of a discrete group is simply--connected if and only if the group is abelian.

math.AT↗

arXiv2019

Classifying spaces for commutativity of low-dimensional Lie groups

For each of the groups $G = O(2), SU(2), U(2)$, we compute the integral and $\mathbb{F}_2$-cohomology rings of $B_\text{com} G$ (the classifying space for commutativity of $G$), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of $E_\text{com} G$ (the homotopy fiber of the inclusion $B_\text{com} G \to BG$), and some low-dimensional homotopy groups of $B_\text{com} G$.

math.AT↗

arXiv2019

Commuting matrices and Atiyah's Real K-theory

We describe the $C_2$-equivariant homotopy type of the space of commuting n-tuples in the stable unitary group in terms of Real K-theory. The result is used to give a complete calculation of the homotopy groups of the space of commuting n-tuples in the stable orthogonal group, as well as of the coefficient ring for commutative orthogonal K-theory.

math.AT↗

arXiv2017

A remark on the group-completion theorem

Suppose that $M$ is a topological monoid satisfying $π_0M=\mathbb{N}$ to which the McDuff-Segal group-completion theorem applies. This implies that a certain map $f: \mathbb{M}_{\infty}\rightarrow ΩBM$ defined on an infinite mapping telescope is a homology equivalence with integer coefficients. In this short note we give an elementary proof of the result that if left- and right-stabilisation commute on $H_1(M)$, then the "McDuff-Segal comparison map" $f$ is acyclic. For example, this always holds if $π_0M$ lies in the centre of the Pontryagin ring $H_{\ast}(M)$. As an application we describe conditions on a commutative $\mathbb{I}$-monoid $X$ under which $\text{hocolim}_{\mathbb{I}}X$ can be identified with a Quillen plus-construction.

math.AT↗

arXiv2017

The spectrum for commutative complex $K$-theory

We study commutative complex $K$-theory, a generalised cohomology theory built from spaces of ordered commuting tuples in the unitary groups. We show that the spectrum for commutative complex $K$-theory is stably equivalent to the $ku$-group ring of $BU(1)$ and thus obtain a splitting of its representing space $B_{com}U$ as a product of all the terms in the Whitehead tower for $BU$, $B_{com}U\simeq BU\times BU\langle 4\rangle \times BU\langle 6\rangle \times \dots .$ As a consequence of the spectrum level identification we obtain the ring of coefficients for this theory. Using the rational Hopf ring for $B_{com}U$ we describe the relationship of our results with a previous computation of the rational cohomology algebra of $B_{com}U$. This gives an essentially complete description of the space $B_{com}U$ introduced by A. Adem and J. Gómez.

math.AT↗

arXiv2016

Variable Flavor Number Scheme for Final State Jets in Thrust

We present results for mass effects coming from secondary radiation of heavy quark pairs related to gluon splitting in the thrust distribution for e+e- collisions. The results are given in the dijet limit where the hard interaction scale and the scales related to collinear and soft radiation are widely separated. We account for the corresponding fixed-order corrections at O(alpha_s^2) and the summation of all logarithmic terms related to the hard, collinear and soft scales as well as the quark mass at N3LL order. We also remove the O(Lambda_QCD) renormalon in the partonic soft function leading to an infrared evolution equation with a matching condition related to the massive quark threshold. The quark mass can be arbitrary, ranging from the infinitely heavy case, where decoupling takes place, down to the massless limit where the results smoothly merge into the well known predictions for massless quarks. Our results are formulated in the framework of factorization theorems for e+e- dijet production and provide universal threshold corrections for the renormalization group evolution of the hard current, the jet and soft functions at the scale where the massive quarks are integrated out. The results represent a first explicit realization of a variable flavor number scheme for final state jets along the lines of the well known flavor number dependent evolution of the strong coupling alpha_s and the parton distribution functions.

hep-ph↗

arXiv2014

Two loop soft function for secondary massive quarks

We present the calculation of the $\mathcal{O}(α_s^2 C_F T_F)$ massive quark corrections to the soft function for the double hemisphere jet mass distribution in $e^+ e^-$ collisions, a necessary ingredient for the calculation of several event shape distributions at N${}^3$LL order. Using the mass as an infrared regulator allows us to derive an expression for the massless momentum space limit, which has not been given so far in the literature. Furthermore, we compute the corresponding corrections in the soft function for thrust, the most prominent projection of the double hemisphere mass distribution. Finally we give expressions for the corresponding renormalon subtractions in the gap scheme.

hep-ph↗

arXiv2013

Secondary Heavy Quark Production in Jets through Mass Modes

We present an effective field theory method to determine secondary massive quark effects in jet production taking the thrust distribution for e+ e- collisions in the dijet limit as a concrete example. The method is based on the field theoretic treatment of collinear and soft mass modes which have to be separated coherently from the collinear and ultrasoft modes related to massless quarks and gluons. For thrust the structure of the conceptual setup is closely related to the production of massive gauge bosons and involves four different effective field theories to describe all possible kinematic situations. The effective field theories merge into each other continuously and thus allow for a continuous description from infinitely heavy to arbitrarily small masses keeping the exact mass dependence of the most singular terms treated through factorization. The mass mode field theory method we present here is in the spirit of the variable fermion number scheme originally proposed by Aivazis, Collins, Olness and Tung and can also be applied in hadron collisions.

hep-ph↗