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For an infinite field $F$, we study the cokernel of the map of homology groups $H_{n+1}(mathrm{GL}_{n-1}(F),mathbb{k}) to H_{n+1}(mathrm{GL}_{n}(F),mathbb{k})$, where $mathbb{k}$ is a field such that $(n-2)!in mathbb{k}^times$, and the kernel of the natural map $H_{n}big(mathrm{GL}_{n-1}(F),mathbb{Z}big[frac{1}{(n-2)!} big] big) to H_{n}big(mathrm{GL}_{n}(F),mathbb{Z}big[frac{1}{(n-2)!} big]big)$. We give conjectural estimates of these cokernel and kernel and prove our conjectures for $nleq 4$.
For a commutative ring R with many units, we describe the kernel of H_3(inc): H_3(GL_2(R), Z) --> H_3(GL_3(R), Z). Moreover we show that the elements of this kernel are of order at most two. As an application we study the indecomposable part of K_3(R).
We study multiplicities of unipotent characters in tensor products of unipotent characters of GL(n,q). We prove that these multiplicities are polynomials in q with non-negative integer coefficients. We study the degree of these polynomials and give a
We survey recent progress in computing with finitely generated linear groups over infinite fields, describing the mathematical background of a methodology applied to design practical algorithms for these groups. Implementations of the algorithms have
In this article we study the homology of nilpotent groups. In particular a certain vanishing result for the homology and cohomology of nilpotent groups is proved.
We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over $mathbb Z,$ and study connections between them.