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Conjugation on Varieties of Nilpotent Matrices

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Conjugation on Varieties of Nilpotent Matrices Conjugation on varieties of nilpotent matrices Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich C der Bergischen Universität Wuppertal vorgelegt von Magdalena Boos im März 2012 Die Dissertation kann wie folgt zitiert werden: urn:nbn:de:hbz:468-20120710-103808-3 [http://nbn-resolving.de/urn/resolver.pl?urn=urn%3Anbn%3Ade%3Ahbz%3A468-20120710-103808-3] For my mum 4 Contents Introduction7 1 Theoretical background 15 1.1 Methods from Algebraic geometry and Invariant theory 15 1.1.1 Algebraic group actions 16 1.1.2 Invariants and algebraic quotients 17 1.1.3 Semi-invariants and GIT-quotients 19 1.1.4 Toric varieties 21 1.2 Representation theory of finite-dimensional algebras 22 1.2.1 Covering theory of quiver algebras 24 1.2.2 Tame and wild algebras 26 1.2.3 Degenerations 27 2 The concrete setup 29 2.1 (Oriented) Link patterns 29 2.2 Known results 31 2.2.1 Results of M. Jordan and M. Gerstenhaber 31 2.2.2 Results of A. Melnikov 32 2.2.3 Results of L. Hille and G. Röhrle 33 2.3 Representation-theoretic approach 34 3 Nilpotency degree 2 37 3.1 Classification of the orbits 37 3.1.1 Parabolic orbits 37 3.1.2 Unipotent orbits 47 3.1.3 The examples n=3 and n=4 49 3.2 Homomorphisms, extensions and their combinatorial interpretation 52 3.3 Closures of Borel orbits 56 3.3.1 Minimal, disjoint pieces of degenerations 57 3.3.2 Minimal degenerations in general 67 3.3.3 Dimensions and the open orbit 78 3.3.4 Minimal singularities 80 3.4 Closures of parabolic orbits 84 3.4.1 Minimal, disjoint degenerations 84 3.4.2 Dimensions of orbits 86 6 Contents 4 Finite classifications in higher nilpotency degrees 89 4.1 Maximal parabolic action for x=3 89 4.1.1 Classification of the orbits 89 4.1.2 Orbit closures 98 4.1.3 Dimensions and the open orbit 100 4.1.4 The parabolic subgroup of block sizes (2,2) 104 4.2 A finiteness criterion 106 4.3 A wildness criterion 113 4.3.1 The representation type of the corresponding algebras 113 4.3.2 Concrete 2-parameter families via tree modules 114 5 Generic classification in the nilpotent cone 119 5.1 Generic normal forms 119 5.2 (Semi-) Invariants 127 5.3 Generation of semi-invariant rings 129 6 Towards an algebraic U-quotient of the nilpotent cone 135 6.1 A quotient criterion for unipotent actions 135 6.1.1 The examples n=2 and n=3 137 6.2 Toric invariants 140 6.2.1 Reductions 141 6.2.2 General description of toric invariants 148 6.3 Generic separation of the U-orbits 155 6.4 The toric variety X 157 6.4.1 Toric operation(s) on the U-invariant ring 157 6.4.2 Explicit description of toric invariants 158 6.4.3 Interrelation between N//U and X 159 6.5 The case n=4 161 7 Towards a GIT-quotient for the Borel action 167 7.1 The examples n=2 and n=3 167 7.2 Generic separation of the same weight 168 7.3 Translation to the language of quiver moduli 173 A Appendix 177 A.1 Singular computations 177 A.2 The parabolic subgroup of block sizes (4,3) 179 Bibliography 181 Introduction The study of algebraic group actions on affine varieties, especially the "vertical" study of orbits and their closures, and the "horizontal" study of parametric families of orbits and quotients, are a common topic in algebraic Lie theory. A well-known example is the study of the adjoint action of a reductive algebraic group on its Lie algebra and numerous variants thereof, in particular the conjugacy classes of com- plex (nilpotent) square matrices. In 1870, the classification of the orbits by so-called Jordan normal forms was described by M. Jordan [Jordan, 1989, Jordan, 1871]. Their closures were described by M. Gersten- haber [Gerstenhaber, 1959] and W. Hesselink [Hesselink, 1976] in the second half of the twentieth century in terms of partitions and visualized by combinatorial objects named Young Diagrams. Algebraic group actions of reductive groups have particularly been discussed elaborately in connection with orbit spaces and more generally algebraic quotients, even though their application to concrete examples is far from being trivial. In case of a non-reductive group, even most of these results fail to hold true immediately. For example, Hilbert’s theorem [Hilbert, 1890] yields that for reductive groups, the invari- ant ring is finitely generated; and a criterion for algebraic quotients is valid [Kraft, 1984]. In 1958, though, M. Nagata [Nagata, 1960, Nagata, 1959] constructed a counterexample of a not finitely generated invariant ring corresponding to a non-reductive algebraic group action, which answered Hilbert’s fourteenth problem in the negative. One exception are algebraic actions of unipotent subgroups that are induced by reductive groups, since the corresponding invariant ring is always finitely generated [Kraft, 1984]. Our main attention in this work is turned towards algebraic non-reductive group actions that are induced by the conjugation action of the general linear group GLn over C. For ex- ample, the standard parabolic subgroups P (and, therefore, the Borel subgroup B) and the unipotent subgroup U of GLn are not reductive. It suggests itself to consider their action (x) on the variety Nn of x-nilpotent matrices of square size n via conjugation. We discuss this setup in detail and, thereby, generalize certain known results. (x) The examination of a group action of P on Nn can be refined if we consider the P-action on a single nilpotent GLn-orbit O. This setup generalizes to arbitrary reductive groups G, where the classification of the orbits is equally interesting. Our considerations are, how- ever, restricted to the group GLn. 8 Introduction Recent studies A recent development in this field is A. Melnikov’s study of the B-action on the variety of upper-triangular 2-nilpotent matrices via conjugation [Melnikov, 2000, Melnikov, 2006, Melnikov, 2007] motivated by Springer Theory. The detailed description of the orbits and their closures is given in terms of so-called link patterns; these are combinatorial objects visualizing the set of involutions in the symmetric group S n. The Ph.D. thesis of B. D. Rothbach yields a description of the B-orbits and equations for their closures in the variety of 2-nilpotent matrices. There seems to be a desingularization of the B-orbit closures as well. However, this work was not available to the author of the present thesis, except for parts of it (see [Rothbach, 2009]). N(3) In her Bachelor thesis [Halbach, 2009], B. Halbach describes the B-action on 3 in all detail. The B-orbits as well as their closures, their minimal degenerations and their singularities are explicity given and a generic normal form is introduced. She general- izes the latter, obtaining a large set of pairwise non-B-conjugate matrices. These yield a (n) generic normal form in the nilpotent cone N B Nn for arbitrary n which is proven in [Boos and Reineke, 2011]. Another recent outcome is L. Hille’s and G. Röhrle’s study of an arbitrary parabolic ac- tion on the variety n of upper-triangular nilpotent matrices [Hille and Röhrle, 1999]. They obtain a criterion which determines whether the group action admits a finite or infinite number of orbits. Content of this work We translate the group action of P to a certain group action in the representation theory of finite dimensional algebras via an associated fibre product. In more detail, we “reduce” the classification of orbits of the action to the knowledge of certain isomorphism classes of representations of a quiver Q given by a linearly oriented quiver of Dynkin type An with a loop at the sink and an admissible ideal I given by just one relation. The quiver Q and the ideal I depend on the parabolic subgroup P and on the nilpotency degree x. Although in general, the classification of these isomorphism classes is far from well- known, there are several cases in which the algebra KQ=I is representation-finite and the representations can be classified using the decomposition theorem of W. Krull, R. Remak and O. Schmidt. Our first aim is to classify those cases in which a finite group action arises. The most obvi- ous case is an arbitrary parabolic action on the variety of 2-nilpotent matrices; the covering quiver of Q is of Dynkin type A2n, then. We classify the orbits in terms of “(enhanced) oriented link patterns”, a natural generaliza- tion of A. Melnikov’s link patterns, and provide the concrete structure of the orbit closures: By translating a description of orbit closures given by G. Zwara [Zwara, 1999], a natural generalization of the description A. Melnikov obtains in [Melnikov, 2006] is deduced. 9 By calculating the dimension of the stabilizer of a 2-nilpotent matrix (with respect to the chosen parabolic subgroup), we are able to compute the dimension of every orbit (closure); the description of the open orbit follows naturally. In case of the B-action, we give an explicit description of the minimal degenerations. Therefore, we describe all minimal, disjoint degenerations by using several results of K. Bongartz [Bongartz, 1994] and generalize them to arbitrary minimal degenerations after- wards. In this way, the precise degeneration diagram is obtained; this yields a concrete algorithm to derive a set-theoretic description of the orbit closure of a given matrix by turning around the arrows in the corresponding oriented link pattern.
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