arXiv subjects

Urshita Pal

Publications and source records attributed to Urshita Pal.

4 recordsLinked to original sources

arXiv2026

The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide

Two recent constructions independently give bigraded commutative Hopf algebra structures on the homology of $\mathrm{GL}(\mathbb{Z})$ with rational Steinberg coefficients. One, by Ash--Miller--Patzt, uses the sharbly resolution. The other, by Brown--Chan--Galatius--Payne, uses the Quillen spectral sequence and rank-filtered maps on $BK(\mathbb{Z})$. We prove that the two structures coincide. Both are induced by join and parabolic restriction operations on Steinberg modules.

math.AT↗

arXiv2026

The top cohomology of principal congruence subgroups of special linear groups over Euclidean number rings

For $R$ a Euclidean number ring, and let $Γ_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(R)$. Borel--Serre showed that the cohomology of $Γ_n(p)$ vanishes above a degree $ν$ that is quadratic in $n$. Let $K$ be the fraction field of $R$, and $\mathcal{T}_n(K)$ the Tits building of $\text{SL}_n(K)$. For $R=\mathbb{Z}$, Lee--Szczarba asked when $\text{H}^ν(Γ_n(p))$ is isomorphic to $\widetilde{\text{H}}_{n-2}(\mathcal{T}_n(K)/Γ_n(p))$, which was answered by Miller--Patzt--Putman. We study a generalized version of Lee--Szczarba's question. We prove that for a prime $p$ in a Euclidean number ring $R$ with fraction field $K$, that a natural map $\text{H}^ν(Γ_n(p)) \to \widetilde{\text{H}}_{n-2}(\mathcal{T}_n(K)/Γ_n(p))$ is always surjective, and give a sufficent set of conditions on $p \in R$ that guarantee when this map is an isomorphism.

math.NT↗

arXiv2026

A projective resolution of the symplectic Steinberg module

Borel--Serre proved that for a number ring $R$ with fraction field $K$, the symplectic group $\text{Sp}_{2n}(R)$ is a virtual duality group of degree quadratic in $n$, and that the symplectic Steinberg module $\text{St}^ω_{2n}(K)$ is its dualizing module. We construct a projective resolution of this symplectic Steinberg module as an $\text{Sp}_{2n}(R)$-representation, that is similar in form to a resolution of Lee--Szczarba for the special linear group, but whose construction is more involved. When $R$ is a Euclidean number ring, we use this resolution to compute the top degree cohomology of principal level-$p$ congruence subgroups of $\text{Sp}_{2n}(R)$, for primes $p \in R$ such that the natural map $R^\times \to (R/(p))^\times$ is surjective.

math.AT↗

arXiv2026

Representation stability in the (co)homology of vertical configuration spaces

In this paper, we study sequences of topological spaces called "vertical configuration spaces" of points in Euclidean space. We apply the theory of FI$_G$-modules, and results of Bianchi-Kranhold, to show that their (co)homology groups are "representation stable" with respect to natural actions of wreath products $S_k \wr S_n$. In particular, we show that in each (co)homological degree, the (co)homology groups (viewed as $S_k \wr S_n$-representations) can be expressed as induced representations of a specific form. Consequently, the characters of their rational (co)homology groups, and the patterns of irreducible $S_k \wr S_n$-representation constituents of these groups, stabilize in a strong sense. In addition, we give a new proof of rational (co)homological stability for unordered vertical configuration spaces, with an improved stable range.

math.AT↗