Finite, Tame, and Wild Actions of Parabolic Subgroups in GL(V) on Certain Unipotent Subgroups

2007

Transformation Groups

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Let k be an algebraically closed field. Let B be the Borel subgroup of GLn(k) consisting of nonsingular upper triangular matrices. Let b = Lie B be the Lie algebra of upper triangular n × n matrices and u the Lie subalgebra of b consisting of strictly upper triangular matrices. We classify all Lie ideals n of b, satisfying u ′ ⊆ n ⊆ u, such that B acts (by conjugation) on n with a dense orbit. Further, in case B does not act with a dense orbit, we give the minimal codimension of a B-orbit in n. This can be viewed as a first step towards the difficult open problem of classifying of all ideals n ⊆ u such that B acts on n with a dense orbit.The proofs of our main results require a translation into the representation theory of a certain quasi-hereditary algebra At,1. In this setting we find the minimal dimension of Ext 1 A t,1 (M, M ) for a ∆-good At,1-module of certain fixed ∆-dimension vectors.

Section: Basic Notation and Referencesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Prehomogeneous spaces for Borel subgroups of general linear groups

Goodwin

2007

Transformation Groups

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“…In a series of papers [HR1], [HR2] and [BH1], recently all instances of parabolic subgroups R in GL n (k) acting with a finite number of orbits on r (l) u were classified. One main step in the proof of these results relates (by an ad hoc construction) the orbits of the action of R on r (l) u to a classification problem of ∆-filtered modules over a certain quasi-hereditary algebra A, where A depends only on the number of blocks of R/R u and on l.…”

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confidence: 99%

“…In the present note, we give a general approach and thus explain the occurrence of quasi-hereditary algebras in [HR1], [HR2] and [BH1]. Moreover, this approach allows results concerning ∆-filtered modules over quasihereditary algebras (such as the existence of almost split sequences, properties of the Euler form, degeneration of modules) to be applied to the various categories mat B stemming from subspace categories of directed vector space categories, orbits of parabolic groups R on some unipotent normal subgroup U and other upper triangular situations.…”

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confidence: 99%

Matrices over upper triangular bimodules and Δ-filtered modules over quasi-hereditary algebras

Brüstle

2000

Colloq. Math.

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Abstract. Let Λ be a directed finite-dimensional algebra over a field k, and let B be an upper triangular bimodule over Λ. Then we show that the category of B-matrices mat B admits a projective generator P whose endomorphism algebra End P is quasihereditary. If A denotes the opposite algebra of End P , then the functor Hom(P, −) induces an equivalence between mat B and the category F(∆) of ∆-filtered A-modules. Moreover, any quasi-hereditary algebra whose category of ∆-filtered modules is equivalent to mat B is Morita equivalent to A.1. Introduction. The aim of this note is to interpret matrices over upper triangular bimodules as ∆-filtered modules over certain quasi-hereditary algebras. We therefore fix a finite-dimensional algebra Λ over a field k, and consider a finite-dimensional Λ-Λ-bimodule B.The category mat B of matrices over B can be defined as follows: Let 1 = e 1 + . . . + e t be a decomposition of the unit element of Λ into pairwise orthogonal primitive idempotents. Then the bimodule B decomposes as k-vector space into a direct sum B = i,j e i Be j . A matrix over B is a pair (d, M ) wheret is a dimension vector and M = (M ij ) i,j∈{1,...,t} is a (square) block matrix whose blocks M ij are matrices of size d j × d i with entries in e i Be j .A morphism in mat B from (d, M ) to (d , M ) is a block matrix H = (H ij ) i,j∈{1,...,t} whose blocks H ij are matrices of size d j × d i with entries in e i Λe j such that HM = M H.In our main result, we require that the algebra Λ is directed and the bimodule B is upper triangular over the directed algebra Λ, i.e. there is an ordering of the idempotents e 1 , . . . , e t such that e i (rad Λ)e j = 0 and e i Be j = 0 whenever i ≥ j. Here we denote by rad Λ the Jacobson radical of Λ.

The Bruhat–Chevalley Order of Parabolic Group Actions in General Linear Groups and Degeneration for Δ-Filtered Modules

Brüstle

Advances in Mathematics

Röhrle

et al. 1999

dedicated to helmut lenzing on the occasion of his 60th birthday Let k be an algebraically closed field and V a finite dimensional k-space. Let GL(V ) be the general linear group of V and P a parabolic subgroup of GL(V ). Now P acts on its unipotent radical P u and on p u =Lie P u , the Lie algebra of P u , via the adjoint action. More generally, we consider the action of P on the l th member of the descending central series of p u denoted by p (l ) u . All instances when P acts on p (l ) u for l 0 with a finite number of orbits are known. In this note, we give a complete combinatorial description of the closure relation on the set of P-orbits on p (l ) u , i.e., the Bruhat Chevalley order, for every finite case. There is a canonical bijection between the set of P-orbits on p (l ) u and the set of isomorphism classes of 2-filtered modules of a particular dimension vector e of a certain quasi-hereditary algebra A(t, l ). These isomorphism classes in turn are given by the orbits of a reductive group G(e) on the variety R(2)(e) of all A(t, l )-modules with 2-filtration Article ID aima.1999.1851, available online at http:ÂÂwww.idealibrary.com on 203 and dimension vector e. The subcategory of A(t, l )-mod of all 2-filtered A(t, l )modules of dimension vector e is denoted by F(2)(e). In our chief result, Theorem 1.1, we show that provided there is only a finite number of isomorphism classes of indecomposable modules in F(2) the following three posets coincide:(1) the Bruhat Chevalley order on the set of P-orbits on p (l ) u ; (2) the Bruhat Chevalley order on the set of G(e)-orbits on R(2)(e);(3) the poset opposite to the so called hom-order on the set of isomorphism classes of F(2)(e).The advantage of this hom-order is that it is given purely by discrete invariants and that it can be computed explicitly for any given finite case. Theorem 1.1 then in turn allows us to explicitly determine the closure relations for the P-orbits on p (l ) u with the aid of this hom-order. We present some examples of Hasse diagrams in an Appendix. Academic PressAlthough the motivation and the goal for this work is the combinatorial description of the Bruhat Chevalley order of the parabolic group actions described above, the representation theoretic methods developed for this purpose, especially the description of degenerations of 2-filtered modules in the 2-finite setting in terms of the so called hom-order, seems to be of interest in its own right. The development of the relevant theory in Sections 3, 4, and 5, carried out for an arbitrary 2-finite quasi-hereditary algebra, may be viewed and read independently from the other parts of this work. 204BRU STLE ET AL.