86 research outputs found
Two-row Springer fibers, foams and arc algebras of type D
No full textThis thesis is concerned with the geometry, topology and combinatorics arising in the study of arc algebras of types Bm-1 and Dm. These algebras are closely related to infinite-dimensional representation theory of Lie algebras, the geometry of perverse sheaves on isotropic Grassmannians and the representation theory of non-semisimple Brauer algebras. Results of Ehrig and Stroppel show that the center of the arc algebra of type Dm (which is isomorphic to the arc algebra of type Bm-1) is isomorphic to the cohomology ring of a certain two-row Springer fiber of type Dm.In the first part of this thesis we combinatorially construct an explicit topological model for every two-row Springer fiber of type Dm and prove that the respective topological model is homeomorphic to its corresponding Springer fiber. In doing so, we confirm a conjecture by Ehrig and Stroppel concerning the topology of the equal-row Springer fiber for type Dm. Moreover, we show that every two-row Springer fiber of type Cm-1 is homeomorphic (even isomorphic as an algebraic variety) to a certain two-row Springer fiber of type Dm. This unexpected isomorphism of algebraic varieties can be interpreted as Langlands dual to the known isomorphism between the arc algebras of type Bm-1 and Dm.In the second part we explain an elementary, topological construction of the Springer representation on the homology of (topological) Springer fibers of types Cm-1 and Dm in the two-row case. The Weyl group action and the component group action admit a diagrammatic description in terms of cup diagrams which appear in the definition of the arc algebras of types Bm-1 and Dm. We determine the decomposition of the representations into irreducibles and relate our construction to classical Springer theory. In addition to that, we give an explicit presentation of the cohomology rings of all two-row Springer fibers in types Cm-1 and Dm.In the third part we describe a low-dimensional topology approach to understanding the arc algebras of types Bm-1 and Dm using two-dimensional surfaces and TQFTs. More precisely, we combinatorially describe the 2-category of singular cobordisms, called (rank one) foams, which governs the functorial version of (type A) Khovanov homology. As an application we use this singular cobordism construction to realize the arc algebras of type Bm-1 and Dm topologically by establishing an explicit isomorphism to a web algebra arising from foams. This result reduces the proof of the associativity of the arc algebras of type Bm-1 and Dm (which requires hard combinatorial work and a cumbersome amount of computations) to certain obvious topological equivalences between two-dimensional surfaces. Moreover, it shows how to remove the technical and unnatural condition of having to choose a certain admissible order of surgery moves in order for the multiplication to be well-defined in the original definition of the arc algebra
On functors associated to a simple root
No full textAssociated to a simple root of a finite-dimensional complex semisimple Lie algebra, there are several endofunctors (defined by Arkhipov, Enright, Frenkel, Irving, Jantzen, Joseph, Mathieu, Vogan and Zuckerman) on the BGG category O. We study their relations, compute cohomologies of their derived functors and describe the monoid generated by Arkhipov's and Joseph's functors and the monoid generated by Irving's functors. It turns out that the endomorphism rings of all elements in these monoids are isomorphic. We prove that the functors give rise to an action of the singular braid monoid on the bounded derived category of Oo. We also use Arkhipov's, Joseph's and Irving's functors to produce new generalized tilting modules
On finitistic dimension of stratified algebras
No full textIn this survey we discuss the results on the finitistic dimension of various stratified algebras. We describe what is
already known, present some recent estimates, and list some open
problems.The author acknowledges the supports of The Swedish Research Council,
The Royal Swedish Academy of Sciences, and The Swedish Foundation for
International Cooperation in Research and Higher Education (STINT).
The author thanks Anders Frisk and Catharina Stroppel for their comments on the paper
On finitistic dimension of stratified algebras
No full textIn this survey we discuss the results on the finitistic dimension of various stratified algebras. We describe what is
already known, present some recent estimates, and list some open
problems.The author acknowledges the supports of The Swedish Research Council,
The Royal Swedish Academy of Sciences, and The Swedish Foundation for
International Cooperation in Research and Higher Education (STINT).
The author thanks Anders Frisk and Catharina Stroppel for their comments on the paper
Hodge theoretic aspects of Soergel bimodules and representation theory
No full textIn the last years, methods coming from Hodge theory have proven to be fruitful in representation theory, most remarkably leading to a new algebraic proof of the Kazhdan-Lusztig conjectures based on the Hodge theory of Soergel bimodules. In this thesis we study several aspects of connection between Hodge theory and representation theory, following several directions
G-theory of group rings for finite groups
No full textIn this thesis we investigate Quillen's G-theory of group rings mostly focusing on the case of finite groups. We study the Hambleton-Taylor-Williams decomposition conjecture for G-theory of the integral group rings. The conjecture expresses n-th G-group of the integral group ring as a direct sum of G-groups of maximal orders in the simple components of QG with certain integers inverted. The HTW-conjecture generalizes the results of Lenstra and Webb on abelian groups. Webb and Yao found a counterexample to the HTW-decomposition in degree 1 but nevertheless they still expected the conjecture to hold for solvable groups. Using the results of Keating we show that the solvable group SL(2, F3) is a counterexample to the conjectured decomposition. Using the methods from modular representation theory we prove useful inequality for ranks of G-groups in degree 1. It is also shown that the HTW-decomposition gives a correct prediction for the torsion subgroup in degree 1 for all finite groups G. Furthermore, we prove that the ranks of G-groups in degree n agree with the prediction of the conjecture in all degrees apart from the degree n = 1
<em>p</em>-Kazhdan-Lusztig Theory
No full textWe describe a positive characteristic analogue of the Kazhdan-Lusztig basis for the Hecke algebra of a crystallographic Coxeter system, called the p-canonical basis. Using Soergel calculus, we present an algorithm to calculate this basis. The p-canonical basis shares strong positivity properties with the Kazhdan-Lusztig basis (similar to the ones described by the Kazhdan-Lusztig positivity conjectures), but it loses many of its combinatorial properties. For this reason, it is much harder to compute the p-canonical basis which is only known in small examples. Even without explicit knowledge of the p-canonical basis, one may obtain a first approximate description of the multiplicative structure by studying the left, right or two-sided cell preorder with respect to the p-canonical basis. The equivalence classes with respect to these cell preorders lead to the notion of p-cells. Parallel to the very rich theory of Kazhdan-Lusztig cells in characteristic 0, we try to build a similar theory in positive characteristic. The first properties of p-cells that we prove are the following: Left and right p-cells are related by taking inverses, just like for Kazhdan-Lusztig cells. The set of elements with a fixed left descent set decomposes into right p-cells. The right p-cells satisfy a similar parabolic compatibility as Kazhdan-Lusztig right cells. We show that any right p-cell preorder relation in a finite, standard parabolic subgroup WI- induces right p-cell preorder relations in each right WI-coset. In an attempt to explicitly describe p-cells in finite type A, we study the consequences of the Kazhdan-Lusztig star-operations for the p-canonical basis. We deduce many interesting relations for the structure coefficients of the p-canonical basis and for the base change coefficients between the p-canonical and the Kazhdan-Lusztig basis. These allow us to show that the right star-operations preserve the left cell preorder. Moreover, we explicitly describe the p-cells in finite type A via the Robinson-Schensted correspondence and show that they coincide with Kazhdan-Lusztig cells for all primes p. A central question is whether Kazhdan-Lusztig cells decompose into p-cells. Based on the star-operations, we can show that the equivalence classes with respect to Vogan's generalized tau-invariant decompose into left p-cells. Garfinkle showed that Vogan's generalized tau-invariant gives a complete invariant of Kazhdan-Lusztig left cells in finite types B and C. From this, we deduce that Kazhdan-Lusztig left cells in finite types B and C decompose into left p-cells for p > 2. We show that in type C3 for p=2, Kazhdan-Lusztig right (resp. two-sided) cells do not decompose into right (resp. two-sided) p-cells. Moreover, we give a criterion that reduces the question about the decomposition of Kazhdan-Lusztig cells to the minimal elements with respect to the weak right Bruhat order. Recently, Achar, Makisumi, Riche and Williamson proved character formulas for the indecomposable tilting modules of a reductive algebraic group in terms of the p-canonical basis. This further fuels interest in the p-canonical basis because the determination of the tilting characters is a long-standing open problem in modular representation theory. These new character formulas and the geometric Satake equivalence provide two connections between right p-cells in affine Weyl groups and tensor ideals of tilting modules. For this reason, affine Weyl groups of small rank provide intriguing examples of p-cells. We explicitly determine the right p-cell structure in affine types Ã1, Ã2 and partly in C˜2
Categorification of tensor powers of the vector representation of Uq(gl(1|1))
No full textWe consider the monoidal subcategory of finite-dimensional representations of Uq(gl(1|1)) generated by the vector representation, and we provide a graphical calculus for the intertwining operators, which enables to compute explicitly the canonical basis, as well as the action of Uq(gl(1|1)). We construct a categorification using graded subquotient categories of the BGG category O(gln) and graded functors between them (translation, Zuckermann’s and coapproximation functors). We describe then the regular blocks of these categories as modules over explicit diagram algebras, which are defined using Soergel modules and combinatorics of symmetric polynomials. We construct diagrammatically standard and proper standard modules for the properly stratified structure of these algebras
Quantum cluster algebras and the dual canonical basis
No full textLet Q be either a Dynkin quiver of type A with alternating orientation or the Kronecker quiver. With the indecomposable injective modules over the path algebra of Q and their Auslander-Reiten translates we associate an element w in the Weyl group of corresponding type. The thesis verifies that the subalgebra Uv(w) of the quantized universal enveloping algebra attached to w carries the structure of a quantum cluster algebra in the sense of Berenstein-Zelevinsky. The quantum cluster algebra is a v-deformation of the cluster algebra A(w) Geiß-Leclerc-Schröer attached to w. Furthermore, we show that all quantum cluster variables are elements in the dual of Lusztig's canonical basis (up to a power of v)
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