GL07: Titles and Abstracts

On a conjecture by Dipper, James and Murphy

Susumu Ariki (Research Institute for Mathematical Sciences, Kyoto University)

Abstract: In 1995, Dipper, James and Murphy made a precise conjecture for when the bilinear form of a cell module of the Hecke algebra of type B vanishes. Their aim to classify simple modules was achieved by a different approach, but the conjecture has remained open. Based on jointwork with Kreiman and Tsuchioka, Jacon and I have settled the conjecture affirmatively. The proof is achieved by first converting the original problem to a crystal theoretic problem, then by inventing a non-crystal theoretic method to control its path model in the sense of Littelmann.

Cellular structures on Hecke algebras of type B

Cedric Bonnafé (Université de Franche-Comté)

Abstract: Joint work with N. Jacon: Let Wn denote the Weyl group of type Bn and let A = Z[Q,Q-1, q, q-1]. We denote by Hn the Hecke algebra of Wn with parameters Q and q. A cellular basis on Hn has been constructed by Graham and Lehrer using Kazhdan-Lusztig basis for the Hecke algebra of the symmetric group Sn. Other cellular bases have also been constructed independently by Dipper-James-Mathas and Du-Scott by purely combinatorial methods, and have been used to construct a Schur algebra of type Bn. We shall show in this talk that there are considerable evidences that the Kazhdan-Lusztig theory with unequal parameters should provide several cellular bases (at least n in the generic case) and that this should give rise to the construction of n different Schur algebras in type Bn. Such constructions (up to Morita equivalences) are also provided by the theory of Cherednik algebras. We shall also review in this talk the connections between these questions and the question of classifying simple modules of specializations of Hn and computing the decomposition matrix (Ariki┐s Theorem, Uglov multipartitions, work of Geck-Jacon, Yvonne┐s conjecture...).

Finding and resolving the maximal singularities of Schubert varieties

Jim Carrell (University of British Columbia)

Abstract: Let G be a semi-simple algebraic group and B a Borel subgroup of G. The closure of a B-orbit in the flag variety G/B is called a Schubert variety. Every Schubert variety X is a projective variety. By the celebrated work of Kazhdan and Lusztig, the singular locus of X in the sense of rational smoothness determines where certain Kazhdan-Lusztig polynomials are nontrivial and hence is of great importance in representation theory. On the other hand, the singular locus of X in the sense of algebraic geometry is also of interest. In this talk, we will survey how the singular loci are determined and how they are related. We will also describe a partial desingularization via blowing up in the rationally smooth setting.

On the Bruhat decomposition and structure constants of Hecke algebras of finite Chevalley groups

Charlie Curtis (University of Oregon)

Abstract: The main result is the calculation of the intersection BwByBx−1B, for w, x, y in the Weyl group, in terms of the root structure. The set was originally used by Iwahori to obtain the structure constants of the Iwahori Hecke algebra. As applications a formula is obtained for the structure constants of the Hecke algebras of Gelfand-Graev representations, and a new formula for the polynomials Rx,y of Kazhdan and Lusztig.

From Broué's conjectures to questions in braid group theory

François Digne (Université de Picardie)

Abstract: Broué's conjectures on blocks with abelian defect imply, in the case of a finite group of Lie type, that the Hecke algebra of some reflection group W should act on the cohomology of a Deligne-Lusztig variety. One tries to define such an action by making the braid group of W act on the variety itself. This leads to some interesting conjectures in braid group theory, involving Lehrer-Springer generalization of regular elements in reflection groups.

Some of these conjectures have been recently solved.

Orbital convolutions: a new approach to the Kirillov theory

Tony Dooley (University of New South Wales)

Abstract:

The homology of the Steinberg variety and Weyl group coinvariants

Matt Douglass (University of North Texas)

Abstract: The Steinberg variety, Z, of a reductive complex algebraic group, G, is a variety that has been used to understand representations of reductive groups of the same type as G. The first part of this talk will give a quick survey some applications and generalizations of the Steinberg variety. The second part of the talk will be devoted to computing the total Borel-Moore homology of Z.

Quantum gl, infinite q-Schur algebras and their representations

Jie Du (University of New South Wales)

Abstract: Almost at the same time as C.M. Ringel discovered the Hall algebra realization of the positive part of the quantum enveloping algebras associated with semisimple complex Lie algebras, A.A. Beilinson, R. MacPherson and G. Lustig discovered a realization for the entire quantum gln via a geometric setting of quantum Schur algebras (or q-Schur algebras). This remarkable work has many applications. For example, it provides a crude model for the introduction of modified quantum groups, it leads to the settlement of the integral Schur-Weyl reciprocity and, hence, the reciprocity at any root of unity, and it has also provided a geometric approach to study quantum affine sln. The BLM work has also been used to investigate the presentations of q-Schur algebras, infinitesimal quantum gln and their associated little q-Schur algebras. In this talk, I will focus on the latest developments of the Beilinson-Lusztig-MacPherson approach in the study of quantum gl, infinite q-Schur algebras and their representations.

This is joint work with Qiang Fu.

Universal root systems

Matthew Dyer (University of Notre Dame)

Abstract: We consider a certain category, the objects of which are Coxeter systems with a given "reflection representation" over a possibly non-commutative coefficient ring. The full subcategory of objects associated to a fixed Coxeter system has (trivially) a universal object. A basic observation used to establish favorable properties of the corresponding "universal coefficent rings" and "universal root systems" is that the universal objects behave functorially with respect to morphisms of Coxeter systems which are injective on the simple reflections.

The X-ray transform on complex projective space

Michael Eastwood (University of Adelaide)

Abstract: The classical Radon transform takes a function on the plane and integrates it over the straight lines in the plane. Its invertibility provides the mathematical basis of modern medical imaging techniques. The X-ray transform takes a function in three-space and integrates it over the straight lines, the terminology being motivated by medical imaging. As one might expect, both of these transforms are best viewed on real projective space. In this talk, I shall discuss what happens on complex projective space where the straight lines are the Fubini-Study geodesics. This is joint work with Hubert Goldschmidt.

An extension of James's Conjecture

Matt Fayers (Queen Mary)

Abstract: Let Hn denote the Iwahori-Hecke algebra of the symmetric group over a field F; we are interested in computing the decomposition numbers for Hn. If F has infinite characteristic, then there is a recursive algorithm for doing this. If F has finite characteristic, then the decomposition matrix can be obtained from the corresponding matrix in infinite characteristic by post-multiplying by an adjustment matrix, but the adjustment matrices remain very mysterious. Restricting attention to individual blocks of Hn James's Conjecture suggests a sufficient criterion for the adjustment matrix of a block to be the identity matrix. We extend this to give a conjectured necessary and sufficient criterion, and prove that our criterion is necessary.

Free fermion and plane partitions

Omar Foda (University of Melbourne)

Abstract: Elements from classical integrable systems show up with increasing regularity in recent developments in mathematical physics.

The Kyoto school approach to classical integrable systems can be used to classify and extend these results.

I wish to introduce (and give some examples of) a recently discovered connection between plane partitions and free fermion vertex operators.

The talk should be (hopefully) introductory and the example(s) to be discussed are meant to be quite explicit.

Joint work with M Wheeler (Melbourne).

Langlands duality for quantum groups

Dennis Gaitsgory (Harvard University)

Abstract: The geometric Satake equivalence states that one can realize the Langlands dual group LG, or rather the category of its representations Rep(LG), as the category of G[[t]]-equivariant perverse sheaves on the affine Grassmannian GrG:=G((t))/G[[t]].

The category Rep(LG) has a natural 1-parameter deformation to Rep(Uq(LG))—the category of representations of the quantum group. Can this category be realized geometrically? It is clear that the answer has to do with replacing the loop group G((t)) by its Kac-Moody extension, but naive attemps at q-deforming the Satake equivalence have so far failed.

In the talk, I'll discuss a solution to this problem, suggested recently by Jacob Lurie. Its essence is that instead of the G[[t]]-equivariance condition on a perverse sheaf, we should impose a Whittaker condition. This allows for the desired q-deformation.

If time permits, we will also discuss the implications of the above construction to both local and global quantum Langlands duality.

Cherednik algebras and Character sheaves

Victor Ginzburg (University of Chicago)

Abstract: We introduce a class of holonomic D-modules on GLn× Cn. The corresponding perverse sheaves are reminiscent of (and include as special cases) Lusztig's character sheaves.

Counting conjugacy classes in algebraic groups

Simon Goodwin (University of Birmingham)

Abstract: Let G be a linear algebraic group defined over Fp and let X be a G-variety defined over Fp. For q a power of p, we write G(q) for the group of Fq-rational points of G, and X(q) for the set of Fq-rational points of X. We consider questions about uniformity in q of the number k(G(q),X(q)) of G(q)-orbits in X(q). We are mainly concerned with the case where X is a normal subgroup or overgroup of G, and G is acting by conjugation.

Affine and cyclotomic Birman–Wenzl–Murakami algebras

Fred Goodman (University of Iowa)

Abstract: Affine and cyclotomic BMW algebras are BMW analogues of affine and cyclotomic Hecke algebras. I will survey what I know about these algebras and describe some difficulties that arise in the cyclotomic case. This is mostly joint work with Holly Hauschild Mosley.

In search of a character

John Graham

(Joint work with Richard Green.)

Abstract: The Hecke algebra of the symmetric group has a quotient called the Temperley Lieb algebra which is often defined using certain planar diagrams. We extend this diagram calculus to an analogous quotient of the Hecke algebra of any Weyl group in a uniform manner. We also discuss the affine case.

TBA

Ian Grojnowski (University of Cambridge)

Abstract:

Modular forms and derivations of the free Lie algebra of rank 2

Dick Hain (Duke University)

Abstract:

BMW algebra, quantized coordinate algebra and type C Schur-Weyl duality

Jun Hu (Beijing Institute of Technology)

Abstract: We prove an integral version of Schur-Weyl duality between the specialized Birman-Murakami-Wenzl algebra Bn(−q2m+1,q) and the quantum algebra associated to the symplectic Lie algebra sp2m. In particular, we deduce that this Schur-Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer-Zhang [Strongly multiplicity free modules for Lie algebras and quantum groups, J. Algebra (306) 2006, 138-174] in the symplectic case.

Reduction modulo p of Deligne-Lusztig characters: revisited

Jens Jantzen (Aarhus Universitet)

Abstract:

Representations of cyclotomic Hecke algebras and W-algebras

Alexander Kleshchev (University of Oregon)

Abstract: In this talk we will explain an attempt to define Verma modules for finite W-algebras of any type (by Brundan-Goodwin-Kleshchev). This is a step toward understanding finite dimensional modules over W-algebra, at least in the standard Levy type. We then explain how everything works nicely in type A, where a complete and satisfactory theory is available. We will also talk about applications to (degenerate) cyclotomic Hecke algebras via a Schur-Weyl type duality between these algebras and finite W-algebras in type A.

A Galois correspondence for reductive groups

Hanspeter Kraft (University of Basel, Switzerland)

Abstract: Given a selfdual faithful representation V of a reductive group G (in characteristic zero) we obtain a beautiful Galois correspondence between closed reductive subgroups of G and certain subalgebras of the tensor algebra T(V), namely those which are graded, contraction-closed and non-degenerate. (These notions will be explained in the talk.) As a consequence, one gets – in a unified way – the well-known First Fundamental Theorems for the classical groups and – in addition – some new FFTs for other groups. (This is based on some unpublished work of Lex Schrijver in the case of compact groups.)

Bivariant K-theory, noncommutative correspondences and duality

Mathai Varghese (University of Adelaide)

Abstract: I will give new decription of bivariant K-theory in terms of noncommutative correspondences, a diagram calculus for bivariant K-theory to manipulate the intersection products, and the applications to a noncommutative Grothendieck-Riemann-Roch theorem and duality. This represents ongoing joint work with Brodzki, Rosenberg and Szabo.

From Broué's conjectures to cyclotomic Hecke algebras

Jean Michel (Paris VII University)

Abstract: If the l-Sylow S of G is abelian, there is a derived equivalence between the principal l-block B of G and the principal l-block b of NG(S); this refers to the blocks of the group algebras over a large enough extension Zl. By Rickard's theorem, such an equivalence is realized by a tilting complex, that is, a complex T in the derived category Db(B) such that EndDb(B)(T,T[k]) = 0 if k ≠ 0 and EndDb(B)(T,T) ≅ b. We will explain how for finite reductive groups, a tilting complex is given by the cohomology of some Deligne-Lusztig variety, and the isomorphism EndDb(B)(T,T) ≅ b involves cyclotomic Hecke algebras.

Fusion procedure for the symmetric group

Alex Molev (University of Sydney)

Abstract: A new version of Cherednik's fusion procedure will be discussed. We show that the matrix unit formulas for the symmetric group provided by this procedure can be derived from a construction of Murphy.

Volume polynomials and the moduli space of curves

Paul Norbury (University of Melbourne)

Abstract: The moduli space of genus g curves with n labeled points is a Kahler manifold of finite volume. The Kahler form can be deformed to give symplectic moduli spaces consisting of hyperbolic genus g surfaces with n geodesic boundary components of given lengths. Mirzakhani proved that the volumes of these moduli spaces are polynomials in the lengths of the n geodesic boundary components by computing the volumes recursively in g and n. The polynomials have deep properties - in particular their coefficients are rational intersection numbers on the moduli space of curves. By allowing cone angles on hyperbolic surfaces we give new recursion relations between the volume polynomials. This has interesting consequences for the geometry of the moduli space.

Finding involutions in finite Lie type groups of odd characteristic

Cheryl Praeger (University of Western Australia)

Abstract: The fundamental problem that prompted the research I will report on was that of estimating the proportion of involutions in a finite classical group that have a "large but not too large" fixed point space. Such involutions were required by Leedham-Green and O'Brien for a black-box classical group recognition algorithm. Applying a theorem of Niemeyer and mine enabled Leedham-Green and O'Brien to construct such an involution, with high probability, by examining O(n) random elements, where n is the dimension of the classical group. In joint work with Niemeyer and Lubeck, we have reduced the number of random elements needed to O(log n). Our methods exploit the link between F-stable tori and Weyl group elements in classical groups, and also apply results about the probability distribution of elements in finite symmetric groups.

Polytopes and box-splines

Claudio Procesi (Universita di Roma, La Sapienza)

Abstract:

Combinatorics in affine flag varieties

Arun Ram (University of Wisconsin, Madison)

Abstract: This talk is about the combinatorics of indexing points in affine flag varieties. It is possible to make choices so that the points are indexed by a refinement of Littelmann's path model in such a way that the Schubert cell and the Mirkovic-Vilonen slice are easily read off the "path" indexing of the point. From this, the relations for the affine Hecke algebra can be derived, both in the Iwahori-Matsumoto and in the Bernstien generators. If time permits I will discuss the action of the "root operators" on points, and/or the relation to the Kamnitzer and Baumann-Gaussent indexings of Mirkovic-Vilonen cycles.

A totally disconnected perspective on Kac-Moody groups

Jacqui Ramagge (University of Wollongong)

Abstract: In 1994 Willis developed the start of a structure theory for totally disconnected locally compact groups. This has initiated a program to classify the topologically simple groups in the class.

I will explain why such groups are interesting, give an introduction to the structure theory and discuss some results, with a particular focus on Kac-Moody groups.

Mistake bounds and compression in learning theory

Hyam Rubinstein (University of Melbourne)

Abstract: This is joint work with B. Rubinstein and P. Bartlett. Machine learning combines ideas from computer science, statistics and functional analysis. However there are also geometric and topological aspects. PAC (probably approximately correct) learning was introduced by Valiant in the early 1980s. As larger sample sizes are given, then a class of concepts can be learnt more and more accurately, (called learnability of the class) if and only if the VC dimension is finite. It is well known that a mistake bound for prediction is given by the one inclusion graph density, which is bounded by the VC dimension. We prove that a better mistake bound, found experimentally by Kuzmin and Warmuth, is valid. We also extend the classical bounds to the case of multiclasses, where a finite number of possibilities occur for each sample choice, rather than two. Finally we establish the peeling conjecture of Kuzmin and Warmuth.This is that maximum classes of given VC dimension can be ordered so that one concept vertex can be removed at a time, where the vertex has degree at most the VC dimension. As corollaries, this explains why maximum classes have many compression schemes and why it is difficult to compress maximal classes by embedding them into maximum ones of the same VC dimension. This is related to the important problem, whether compressibility is equivalent to finite VC dimension (and hence learnability).

Cyclotomic q-Schur algebras associated to the Ariki-Koike algebra

Toshiaki Shoji (Graduate School of Mathematics, Nagoya University)

Abstract: Let Hn,r be the Ariki-Koike algebra associated to the complex reflection group Sn |× (Z/rZ)n, and S be the cyclotomic q-Schur algebra associated to Hn,r introduced by Dipper-James-Mathas. For each p = (r_1, ..., rg) such that r1 + ... + rg = r, we define a subalgebra S p of S and its quotient algebra S p. It is shown that S p is a standardly based algebra and S p is a cellular algebra. By making use of these algebras, we show that certain decomposition numbers for S can be expressed as a product of decomposition numbers for cyclotomic q-Schur algebras associated to smaller Ariki-Koike algebras Hnk,rk.

Kato's exotic nilpotent cone

T. A. Springer

Abstract: Let G = Sp2n(C), acting in V = C2n. Put V = V ⊕ Λ2 V. The exotic nilcone is the nilcone (set of unstable points) in V, relative to the action of G. It is used by Syu Kato in his geometric approach to the representation theory of the multi-parameter Hecke affine algebra associated to G.

My talk will be quite elementary. I shall discuss the basic properties of V.

Categorical self duality and quantum groupoids

Ross Street (Macquarie University)

(Joint work with Craig Pastro.)

Abstract: Given a monoidal comonad G on a monoidal category C, Moerdijk has observed that the category CG of Eilenberg-Moore G-coalgebras (better called G-comodules) is monoidal via a lifting of the tensor product of C. Recall that the left internal hom BD of two objects B and D in C, when it exists, is defined by a natural bijection between morphisms ABD and morphisms ABD. We look into the structure required on G to lift left internal homs to CG. The case where C is autonomous (also called compact or rigid, and meaning the existence of all dual objects) was studied by Bruguières-Virelizier. The monoidal category C is *-autonomous (in the sense of M. Barr) when there is an object D such that BD exists for all B and the functor (-)D: CopC is an equivalence of categories. This implies C is self-dual; such a $D$ is called a dualizing object. We apply our results to determine when C is *-autonomous given that C is. The definition of Hopf algebra in a monoidal category C depends on a braiding for C, however, some constructions of invariants of 3-manifolds have motivated the quest for a generalization of Hopf algebra to non-braided contexts. Our results can be viewed as providing this generalization. Alternatively, and this was our main motivation, there is the observation by K. Szlachányi that a notion dual to one of M. Takeuchi is a monoidal comonad in an appropriate context. B. Day and the speaker renamed that notion quantum category and claimed that *-autonomy arises when we want the quantum category to be a quantum groupoid.

Weight 2 blocks of q-Schur algebras

Kai Meng Tan (National University of Singapore)

Abstract:

Finite dimensional representations of Cherednik algebras

Michela Varagnolo

(Joint work with E. Vasserot.)

Abstract: We study the finite dimensional representations of Cherednik algebras. The representation theory of these algebras splits into two case: the one of DAHA (double affine Hecke algebras) and the one of rational DAHA (a degeneration of the first one). The main result is the classification of all finite dimensional spherical simple modules for the rational DAHA. In order to do so we embeds the category of finite dimensional modules for the rational DAHA into the category of finite dimensional modules for the DAHA. Then we use a geometric construction of simple modules for the DAHA (proved by Vasserot) via the homology of the affine Springer fibers.

Cellular algebras

Changchang Xi (Beijing Normal University)

Abstract: In this talk I shall first report some results on cellular algebras introduced by Graham and Lehrer, and then introduce the so-called affine cellular algebras, a generalization of cellular algebras, and survey the general representation theory and structural theory of affine cellular algebras. Typical examples of affine cellular algebras are the affine Temperley-Lieb algebras studied by Graham and Lehrer, and others, and the affine Hecke algebras of type A studied by Lusztig, Nanhua Xi, and others.

The contents of my talk are taken from joint works with Steffen Koenig.

Some maximal elements in a baby Verma module

Nanhua Xi (Chinese Academy of Sciences)

Abstract: Some maximal elements in a baby Verma module are constructed. The elements should be helpful to understand the structure of baby Verma modules.

Twisted deformation quantization of algebraic varieties

Amnon Yekutieli (Ben Gurion University)

Abstract: Let X be a smooth algebraic variety over a field of characteristic 0, endowed with a Poisson bracket. A quantization of this Poisson bracket is a formal associative deformation of the structure sheaf OX, which realizes the Poisson bracket as its first order commutator. More generally one can consider Poisson deformations of OX and their quantizations.

I will explain what these deformations are. Then I'll state a theorem which says that under certain cohomological conditions on X, there is a canonical quantization map (up to gauge equivalence). This is an algebro-geometric analogue of the celebrated result of Kontsevich (which talks about differentiable manifolds).

It appears that in general, without these cohomological conditions, the quantization will not be a sheaf of algebras, but rather a stack of algebroids, otherwise called a twisted associative deformation of OX.

In the second half of the talk I'll talk about twisted deformations and twisted quantization, finishing with a conjecture.

The work is joint with F. Leitner (BGU).

A Temperley-Lieb analogue for the Birman-Wenzl-Murakami algebra

Ruibin Zhang (University of Sydney)

(Joint work with Gus Lehrer.)

Abstract: The Temperley-Lieb algebra may be thought of as a quotient of the Hecke algebra of type A, acting on tensor space as the commutant of the natural action of Uq(sl2) on (C(q)2)n. We define and study a quotient of the Birman-Wenzl-Murakami algebra, which plays an analogous role for the 3-dimensional irreducible representation of Uq(sl2).