Mathematics

We show that each orbit of a Borel subgroup B n?? of GL(n-1) (respectively SO(n-1)) on the flag variety of GL(n) (respectively of SO(n)) is a bundle over a B n?? -orbit on a generalized flag variety of GL(n-1) (respectively SO(n-1)), with fiber isomorphic to an orbit of an analogous subgroup on a smaller flag variety. As a consequence, we develop an inductive procedure to classify B n?? -orbits on the flag variety. Our method is essentially uniform in the two cases. As further consequences, in the sequel to this paper we give an explicit combinatorial classification of orbits and determine completely the closure relation between orbit closures. This further develops work of Hashimoto in the general linear group case.

The geometric Satake correspondence can be regarded as a geometric construction of the rational representations of a complex connected reductive group G. In their study of this correspondence, Mirković and Vilonen introduced algebraic cycles that provide a linear basis in each irreducible representation. Generalizing this construction, Goncharov and Shen define a linear basis in each tensor product of irreducible representations. We investigate these bases and show that they share many properties with the dual canonical bases of Lusztig.

We show that bigrassmannian permutations determine the socle of the cokernel of an inclusion of Verma modules in type A . All such socular constituents turn out to be indexed by Weyl group elements from the penultimate two-sided cell. Combinatorially, the socular constituents in the cokernel of the inclusion of a Verma module indexed by w∈ S n into the dominant Verma module are shown to be determined by the essential set of w and their degrees in the graded picture are shown to be computable in terms of the associated rank function. As an application, we compute the first extension from a simple module to a Verma module.

We introduce a method that produces a bijection between the posets silt?�A and silt?�B formed by the isomorphism classes of basic silting complexes over finite-dimensional k -algebras A and B , by lifting A and B to two k[[X]] -orders which are isomorphic as rings. We apply this to a class of algebras generalising Brauer graph and weighted surface algebras, showing that their silting posets are multiplicity-independent in most cases. Under stronger hypotheses we also prove the existence of large multiplicity-independent subgroups in their derived Picard groups as well as multiplicity-invariance of TrPicent . As an application to the modular representation theory of finite groups we show that if B and C are blocks with |IBr(B)|=|IBr(C)| whose defect groups are either both cyclic, both dihedral or both quaternion, then the posets tilt?�B and tilt?�C are isomorphic (except, possibly, in the quaternion case with |IBr(B)|=2 ) and TrPicent(B)?�TrPicent(C) (except, possibly, in the quaternion and dihedral cases with |IBr(B)|=2 ).

As a sequel of \cite{Ou}, in this shot note, we investigate block basis for coinvariants of finite modular pseudo-reflection groups. We are particularly interested in the case where G is a subgroup of the parabolic subgroups of G L n (q) which generalizes the Weyl groups of restricted Cartan type Lie algebra.

Let F be a non-Archimedean locally compact field of residue characteristic p, let G be an inner form of GL(n,F) with n>0, and let l be a prime number different from p. We describe the block decomposition of the category of finite length smooth representations of G with coefficients in an algebraically closed field of characteristic l. Unlike the case of complex representations of an arbitrary p-adic reductive group and that of l-modular representations of GL(n,F), several non-isomorphic supercuspidal supports may correspond to the same block. We describe the (finitely many) supercuspidal supports corresponding to a given block. We also prove that a supercuspidal block is equivalent to the principal (that is, the one which contains the trivial character) block of the multiplicative group of a suitable division algebra, and we determine those irreducible representations having a nontrivial extension with a given supercuspidal representation of G.

Linckelmann and Murphy have classified the Morita equivalence classes of p-blocks of finite groups whose basic algebra has dimension at most 12. We extend their classification to dimension 13 and 14. As predicted by Donovan's Conjecture, we obtain only finitely many such Morita equivalence classes.

Motivated by the reduction techniques involving character triples for the local-global conjectures, we show that a blockwise relation between module triples is a consequence of a derived equivalence with additional properties. Moreover, we show that this relation is compatible with wreath products.

We describe the fermionic and bosonic Fock representation of the Lie super-algebra of endomorphisms of the exterior algebra of the Q -vector space of infinite countable dimension, vanishing at all but finitely many basis elements. We achieve the goal by exploiting the extension of the Schubert derivations to the Fermionic Fock space.

The homological theory of Auslander-Platzeck-Todorov on idempotent ideals laid much of the groundwork for higher Auslander-Reiten theory, providing the key technical lemmas for both higher Auslander correspondence as well as the construction of higher Nakayama algebras, among other results. Given a finite-dimensional algebra A and idempotent e , we expand on a criterion of Jasso-Külshammer in order to determine a correspondence between the 2 -precluster-tilting subcategories of mod(A) and mod(A/?�e?? . This is then applied in the context of generalising dimer algebras on surfaces with boundary idempotent.