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Peter Patzt

Publications and source records attributed to Peter Patzt.

At least 19 recordsLinked to original sources
arXiv2025

Homological vanishing for the Steinberg representation II: reductive groups and integral conjectures

We prove that the homology groups of any connected reductive group over a field with coefficients in the Steinberg representation vanish in a range. The generalizes work of Ash-Putman-Sam on the classical split groups. We state a connectivity conjecture that would allow us to prove such a vanishing result for $SL_n(\mathbb{Z})$, as was conjectured by Church-Farb-Putman. We prove some special cases of this conjecture and use it to refine known results about the first and second of homology of $SL_n(\mathbb{Z})$ with Steinberg coefficients.

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arXiv2024

Uniform twisted homological stability

We prove a homological stability theorem for families of discrete groups (e.g. mapping class groups, automorphism groups of free groups, braid groups) with coefficients in a sequence of irreducible algebraic representations of arithmetic groups. The novelty is that the stable range is independent of the choice of representation. Combined with earlier work of Bergstr\"om--Diaconu--Petersen--Westerland this proves the Conrey--Farmer--Keating--Rubinstein--Snaith predictions for all moments of the family of quadratic $L$-functions over function fields, for sufficiently large odd prime powers.

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arXiv2023

A presentation of symplectic Steinberg modules and cohomology of $\operatorname{Sp}_{2n}(\mathbb{Z})$

Borel-Serre proved that the integral symplectic group $\operatorname{Sp}_{2n}(\mathbb{Z})$ is a virtual duality group of dimension $n^2$ and that the symplectic Steinberg module $\operatorname{St}^\omega_n(\mathbb{Q})$ is its dualising module. This module is the top-dimensional homology of the Tits building associated to $\operatorname{Sp}_{2n}(\mathbb{Q})$. We find a presentation of this Steinberg module and use it to show that the codimension-1 rational cohomology of $\operatorname{Sp}_{2n}(\mathbb{Z})$ vanishes for $n \geq 2$, $H^{n^2 -1}(\operatorname{Sp}_{2n}(\mathbb{Z});\mathbb{Q}) \cong 0$. Equivalently, the rational cohomology of the moduli stack $\mathcal{A}_n$ of principally polarised abelian varieties of dimension $2n$ vanishes in the same degree. Our findings suggest a vanishing pattern for high-dimensional cohomology in degree $n^2-i$, similar to the one conjectured by Church-Farb-Putman for special linear groups.

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arXiv2023

The homology of the partition algebras

We show that the homology of the partition algebras, interpreted as appropriate Tor-groups, is isomorphic to that of the symmetric groups in a range of degrees that increases with the number of nodes. Furthermore, we show that when the defining parameter $\delta$ of the partition algebra is invertible, the homology of the partition algebra is in fact isomorphic to the homology of the symmetric group in all degrees. These results parallel those obtained for the Brauer algebras in the authors' earlier work, but with significant differences and difficulties in the inductive resolution and high acyclicity arguments required to prove them. Our results join the growing literature on homological stability for algebras, which now encompasses the Temperley-Lieb, Brauer and partition algebras, as well as the Iwahori-Hecke algebras of types A and B.

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arXiv2023

On rank filtrations of algebraic K-theory and Steinberg modules

Motivated by his work on the stable rank filtration of algebraic K-theory spectra, Rognes defined a simplicial complex called the common basis complex and conjectured that this complex is highly connected for local rings and Euclidean domains. We prove this conjecture in the case of fields. Our methods give a novel description of this common basis complex of a PID as an iterated bar construction on an equivariant monoid built out of Tits buildings. We also identify the Koszul dual of a certain equivariant ring assembled out of Steinberg modules.

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arXiv2022

Quotients of braid groups by their congruence subgroups

The congruence subgroups of braid groups arise from a congruence condition on the integral Burau representation $B_n \to \operatorname{GL}_{n}(\mathbb Z)$. We find the image of such congruence subgroups in $\operatorname{GL}_{n}(\mathbb Z)$-an open problem posed by Dan Margalit. Additionally, we characterize the quotients of braid groups by their congruence subgroups in terms of symplectic congruence subgroups.

math.GR↗
arXiv2022

On the codimension-two cohomology of $\mathrm{SL}_n(\mathbb{Z})$

Borel-Serre proved that $\mathrm{SL}_n(\mathbb{Z})$ is a virtual duality group of dimension $n \choose 2$ and the Steinberg module $\mathrm{St}_n(\mathbb{Q})$ is its dualizing module. This module is the top-dimensional homology group of the Tits building associated to $\mathrm{SL}_n(\mathbb{Q})$. We determine the "relations among the relations" of this Steinberg module. That is, we construct an explicit partial resolution of length two of the $\mathrm{SL}_n(\mathbb{Z})$-module $\mathrm{St}_n(\mathbb{Q})$. We use this partial resolution to show the codimension-2 rational cohomology group $H^{{n \choose 2} -2}(\mathrm{SL}_n(\mathbb{Z});\mathbb{Q})$ of $\mathrm{SL}_n(\mathbb{Z})$ vanishes for $n \geq 3$. This resolves a case of a conjecture of Church-Farb-Putman. We also produce lower bounds for the codimension-1 cohomology of certain congruence subgroups of $\mathrm{SL}_n(\mathbb{Z})$.

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arXiv2021

Representation stability, secondary stability, and polynomial functors

We prove a general representation stability result for polynomial coefficient systems which lets us prove representation stability and secondary homological stability for many families of groups with polynomial coefficients. This gives two generalizations of classical homological stability theorems with twisted coefficients. We apply our results to prove homological stability for hyperelliptic mapping class groups with twisted coefficients, prove new representation stability results for congruence subgroups, establish secondary homological stability for groups of diffeomorphisms of surfaces viewed as discrete groups, and improve the known stable range for homological stability for general linear groups of the sphere spectrum.

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arXiv2020

Representation stability for diagram algebras

We introduce stability categories for diagram algebras---analogues to Randal-Williams and Wahl's homogeneous categories. We use these to study representation stability properties of the Temperley--Lieb algebras, the Brauer algebras, and the partition algebras.

math.RT↗
arXiv2020

The homology of the Brauer algebras

This paper investigates the homology of the Brauer algebras, interpreted as appropriate Tor-groups, and shows that it is closely related to the homology of the symmetric group. Our main results show that when the defining parameter of the Brauer algebra is invertible, then the homology of the Brauer algebra is isomorphic to the homology of the symmetric group, and that when the parameter is not invertible, this isomorphism still holds in a range of degrees that increases with n.

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arXiv2020

On the generalized Bykovskii presentation of Steinberg modules

We study presentations of the virtual dualizing modules of special linear groups of number rings, the Steinberg modules. Bykovskii gave a presentation for the Steinberg modules of the integers, and our main result is a generalization of this presentation to the Gaussian integers and the Eisenstein integers. We also show that this generalization does not give a presentation for the Steinberg modules of several Euclidean number rings.

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arXiv2020

On the top dimensional cohomology groups of congruence subgroups of $\text{SL}_n(\mathbb{Z})$

Let $Γ_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(\mathbb{Z})$. Borel-Serre proved that the cohomology of $Γ_n(p)$ vanishes above degree $\binom{n}{2}$. We study the cohomology in this top degree $\binom{n}{2}$. Let $\mathcal{T}_n(\mathbb{Q})$ denote the Tits building of $\text{SL}_n(\mathbb{Q})$. Lee-Szczarba conjectured that $H^{\binom{n}{2}}(Γ_n(p))$ is isomorphic to $\widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/Γ_n(p))$ and proved that this holds for $p=3$. We partially prove and partially disprove this conjecture by showing that a natural map $H^{\binom{n}{2}}(Γ_n(p)) \rightarrow \widetilde{H}_{n-2}(\mathcal{T}_n(\mathbb{Q})/Γ_n(p))$ is always surjective, but is only injective for $p \leq 5$. In particular, we completely calculate $H^{\binom{n}{2}}(Γ_n(5))$ and improve known lower bounds for the ranks of $H^{\binom{n}{2}}(Γ_n(p))$ for $p \geq 5$.

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arXiv2020

Stability in the high-dimensional cohomology of congruence subgroups

We prove a representation stability result for the codimension-one cohomology of the level three congruence subgroup of $\mathbf{SL}_n(\mathbb{Z})$. This is a special case of a question of Church-Farb-Putman which we make more precise. Our methods involve proving several finiteness properties of the Steinberg module for the group $\mathbf{SL}_n(K)$ for $K$ a field. This also lets us give a new proof of Ash-Putman-Sam's homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church-Putman's homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_n(\mathbb{Z})$.

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arXiv2019

Non-integrality of some Steinberg modules

We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartments. Assuming the generalized Riemann hypothesis, this shows that the Steinberg module of a number ring is generated by integral apartments if and only if the ring is Euclidean. We also construct new cohomology classes in the top dimensional cohomology group of the special linear group of some quadratic imaginary number rings.

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