Classifying representations by way of Grassmannians

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $\Lambda$ with fixed dimension $d$ and fixed squarefree top $T$. Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations. In case of existence of a moduli space -- unexpectedly frequent in light of the stringency of fine classification -- this space is always projective and, in fact, arises as a closed subvariety ${\mathfrak{Grass}}^T_d$ of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety ${\mathfrak{Grass}}^T_d$ is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained; in this case of `finite local representation type at a given simple $T$', the radical layering $\bigl( J^lM/ J^{l+1}M \bigr)_{l \ge 0}$ is shown to be a classifying invariant for the modules with top $T$. This relies on the following general fact obtained as a byproduct: Proper degenerations of a local module $M$ never have the same radical layering as $M$

Finiteness results for Abelian tree models

Equivariant tree models are statistical models used in the reconstruction of phylogenetic trees from genetic data. Here equivariant refers to a symmetry group imposed on the root distribution and on the transition matrices in the model. We prove that if that symmetry group is Abelian, then the Zariski closures of these models are defined by polynomial equations of bounded degree, independent of the tree. Moreover, we show that there exists a polynomial-time membership test for that Zariski closure. This generalises earlier results on tensors of bounded rank, which correspond to the case where the group is trivial, and implies a qualitative variant of a quantitative conjecture by Sturmfels and Sullivant in the case where the group and the alphabet coincide. Our proofs exploit the symmetries of an infinite-dimensional projective limit of Abelian star models.Comment: 27 pages. arXiv admin note: substantial text overlap with arXiv:1103.533

On Parabolic Subgroups and Hecke Algebras of Some Fractal Groups

We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representations. The Hecke algebras associated to these parabolic subgroups are commutative, so the decomposition in irreducible components of the finite quasi-regular representations is given by the double cosets of the parabolic subgroup. Since our results derive from considerations on finite-index subgroups, they also hold for the profinite completions $\hat G$ of the groups G. The representations involved have interesting spectral properties investigated in math.GR/9910102. This paper serves as a group-theoretic counterpart to the studies in the mentionned paper. We study more carefully a few examples of fractal groups, and in doing so exhibit the first example of a torsion-free branch just-infinite group. We also produce a new example of branch just-infinite group of intermediate growth, and provide for it an L-type presentation by generators and relators.Comment: complement to math.GR/991010