Related papers: Explicit sharbly cycles at the virtual cohomologic…
Let St denote the Steinberg module of $SL_n(Q)$ tensored with Q. Let Sh denote the sharbly resolution of St. By Borel-Serre duality, $H^{n(n-1)/2-i}(SL_n(Z),Q)$ is isomorphic to $H_i(SL_n(Z),St)$. The latter is isomorphic to the homology of…
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…
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…
We prove that H^{d-1}(SL_n Z; Q) = 0, where d = n-choose-2 is the cohomological dimension of SL_n Z, and similarly for GL_n Z. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the…
For a number ring $\mathcal{O}$, Borel and Serre proved that $\text{SL}_n(\mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that $\text{GL}_n(\mathcal{O})$ is a virtual duality group.…
We construct an explicit canonical cycle in the top-dimensional homology of the Voronoi complex associated with an arithmetic group. This cycle relates to the cohomology of SL$_n(\mathbb{Z})$ with rational coefficients at the virtual…
We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral…
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 $\text{St}^\omega_{2n}(K)$ is…
We give several resolutions of the Steinberg representation St_n for the general linear group over a principal ideal domain, in particular over Z. We compare them, and use these results to prove that the computations in [AGM4] are…
A "squarefree module" over a polynomial ring $S = k[x_1, .., x_n]$ is a generalization of a Stanley-Reisner ring, and allows us to apply homological methods to the study of monomial ideals systematically. Let $Sq$ be the category of…
We compute the cohomological Hall algebra of zero-dimensional sheaves on an arbitrary smooth quasi-projective surface $S$ with pure cohomology, deriving an explicit presentation by generators and relations. When $S$ has trivial canonical…
We investigate the homology of a congruence subgroup Gamma of SL_3(Z) with coefficients in the Steinberg modules St(Q^3) and St(E^3), where E is a real quadratic field and the coefficients are Q. By Borel-Serre duality, H_0(Gamma, St(Q^3))…
We discuss several topics of homological algebra for the Lie superalgebra osp(1|2n). First we focus on Bott-Kostant cohomology, which yields classical results although the cohomology is not given by the kernel of the Kostant quabla…
In \cite{FMX19}, it is proved that the convolution algebra of top Borel-Moore homology on Steinberg variety of type $B/C$ realizes $U(sl_n^{\theta})$, where $sl_{n}^{\theta}$ is the fixed point subalgebra of involution on $sl_n$. So top…
We determine the graded composition multiplicity in the symmetric algebra S(V) of the natural GL_n(q)-module V, or equivalently in the coinvariant algebra of V, for a large class of irreducible modules around the Steinberg module. This was…
The quantum Euclidean spheres, $S_q^{N-1}$, are (noncommutative) homogeneous spaces of quantum orthogonal groups, $\SO_q(N)$. The *-algebra $A(S^{N-1}_q)$ of polynomial functions on each of these is given by generators and relations which…
The purpose of this paper is to study categorifications of tensor products of finite dimensional modules for the quantum group for sl(2). The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie…
Let $SL_2$ be the rank one simple algebraic group defined over an algebraically closed field $k$ of characteristic $p>0$. The paper presents a new method for computing the dimension of the cohomology spaces $\text{H}^n(SL_2,V(m))$ for Weyl…
We construct an injective algebra homomorphism of the quantum group $U_q(\mathfrak{sl}_{n+1})$ into a quantum cluster algebra $\mathbf{L}_n$ associated to the moduli space of framed $PGL_{n+1}$-local systems on a marked punctured disk. We…
We give an explicit formula for the correspondence between simple Yetter-Drinfeld modules for certain finite-dimensional pointed Hopf algebras $H$ and those for cocycle twists $H^{\sigma}$ of $H$. This implies an equivalence between modules…