Edinburgh Algebra Seminars

F Planas-Vilanova: The first part of the talk will be devoted to review some results on the uniform Artin-Rees properties. In the second part we will present a negative answer to the problem, open for twenty years, as to whether the full uniform Artin-Rees property holds on the prime spectrum of an excellent ring (it was known to hold locally on the prime spectrum of such a ring). This is joint work with Liam O'Carroll.

S Rees: I shall discuss the word problem for groups, introduced a century ago by Dehn, and solved by him for surface groups. I am interested in how the structure of the set of words representing the identity element of a group (this set is called the `word problem' for the group) is related to the structure of the whole group. In the late 1980's Muller and Schupp proved that the word problem for a group can be recognised using a pushdown automaton (a machine using a simple stack memory) precisely when the group is virtually free. In this case it is Dehn's algorithm that can be programmed on the pushdown automaton. A set of words that can be recognised using a pushdown automaton can be constructed using a context-free grammar, and conversely. The proof of Muller and Schupp's result is based on the realisation that the underlying context-free grammar of the word problem puts a restriction on the geometry of the group. There's a correspondence between the types of machines that recognise sets of strings and the grammars that build them, but this result suggests that the grammar constructing the word problem of a group is more clearly related to the structure of the group than is the machine that recognises it. I shall report on my recent joint work with Holt and Shapiro. This examines the grammar associated with word problems that can be solved using a generalisation of Dehn's algorithms developed by Goodman and Shapiro; we see that in this case the grammar is always `growing context-sensitive'. We extend Goodman and Shapiro's work, and find a host of examples of groups with word problems that are context-sensitive but not growing context-sensitive. Hence we can answer questions of Kambites and Otto, who found the first example of a word problem in that category.

S Goodwin: To each nilpotent element e in a complex semisimple Lie algebra \g, one can associate a finite W-algebra denoted by U(\g,e). This algebra can be viewed as the enveloping of the Slodowy slice through the adjoint orbit of e, and has many connections to other areas of Lie theory. After presenting some history and motivation we will present an approach, due to Brundan, Kleshchev and the author, to highest weight representation theory of finite W-algebras. There is not a natural comultiplication on finite W-algebras; however, it is possible to give the tensor product of a U(\g,e)-module with a finite dimensional U(\g)-module the structure of a U(\g,e)-module. We will discuss properties of these tensor products, which are expected to be of importance in understanding the representation theory of U(\g,e).

J Stokman: One of the Macdonald conjectures is the duality -by now- theorem. The duality theorem points out the bispectral nature of the Macdonald polynomials. It was proven by Cherednik using the double affine Hecke algebra. In this talk I will establish the interplay between the double affine Hecke algebra and bispectrality on a more fundamental level. It leads to a bispectral version of the quantum Knizhnik-Zamolodchikov equations and to an integrable bispectral problem associated to the Macdonald operators. This is joint work with Michel van Meer.

P Turner: Starting with a directed graph, I will describe the construction of a homology theory for algebras, related to the Khovanov homology of graphs. When the graph is the n-gon, this homology agrees with Hochschild homology up to degree n.

R Tange: Let U be the universal enveloping algebra of the Lie algebra g of a reductive group G over an algebraically closed field of characteristic p and let Z be the centre of U. The algebraic variety corresponding to Z is called the Zassenhaus variety of g. Unlike in characteristic 0, Z is not a polynomial ring, in fact the Zassenhaus variety is not smooth. I will show (under certain mild assumptions) that Z is a unique factorisation domain and that its field of fractions is purely transcendental over k (i.e. the Zassenhaus variety is rational). If time allows I will indicate the relevance of the Zassenhaus variety for the representation theory of g and a relation with the Gelfand-Kirillov conjecture.

S Schleimer: We will sketch a proof that Aut(G) has polynomial-time word problem when G is a word hyperbolic group. The heart of the argument is the idea from computer science; straight-line programs are widely studied in the field of data compression. As it so happens, they are also well suited for analyzing group automorphisms.

P Burillo: Higher-dimensional analogs of Thompson's group V have been introduced recently by Brin. We will recall their definition and find the analog of the standard interpretation of Thompson's groups by tree pair diagrams. We will use this interpretation to give presentations for them (both finite and infinite), and to find estimates for the word metric of these groups in terms of the number of carets in the tree pair diagram. Finally, we will show that the inclusion of F, T and V in the higher-dimensional groups is exponentially distorted.

A Ram: The double affine Hecke algebra (DAHA) of type C has special properties (6 parameters!) and distinguished quotients. The Macdonald polynomials for this Hecke algebra are the Koornwinder polynomials and the Askey-Wilson polynomials. One interesting quotient of the DAHA is the two boundary Temperley-Lieb algebra. The 2 boundary Temperley-Lieb algebra points the way to a family of centralizer algebras which includes the 2 boundary BMW (Birman-Murakami-Wenzl) algebras. This talk will a medley of vignettes around double affine type C braid groups and quotient algebras.

P Achar: Perverse sheaves, introduced around 1980, have many remarkable properties, involving such notions as Poincare-Verdier duality, weight filtrations and "purity," and the celebrated Decomposition Theorem. These properties have made perverse sheaves into an incredibly powerful tool, especially for applications in representation theory. "Staggered sheaves" are a new attempt to duplicate some of these properties in the setting of vector bundles and coherent sheaves. I will discuss the ingredients that go into defining staggered sheaves, state the main results that are known so far, and perhaps speculate on potential applications. This will be an introductory talk: I will not assume any familiarity with perverse or staggered sheaves, and I will try to focus on examples on A^1 or P^1. Some of the results on staggered sheaves are joint work with D. Sage and D. Treumann.

R Thomas: We are interested in the notion of computing in structures. One approach would be to take a general model of computation such as a Turing machine. A structure would then be said to be computable if its domain can represented by a set which is accepted by a Turing machine and if there are decision-making Turing machines for each of its relations. However, there have been various ideas put forward to restrict the model of computation used; whilst the range of possible structures decreases, certain properties of the structures may become decidable. One interesting approach was introduced by Khoussainov and Nerode who considered structures whose domain and relations can be checked by finite automata; such a structure is said to be FA-presentable. This was inspired, in part, by the theory of automatic groups; however, the definitions are somewhat different. We will survey some of what is known about FA-presentable structures, contrasting it with the theory of automatic groups and posing some open questions. The talk is intended to be self-contained, in that no prior knowledge of these topics is assumed.

M Neunhoeffer: In the context of the Matrix Group Recognition Project the following is an important task: Given G=< g_1, ..., g_k >, find a non-trivial element x in G that is contained in a proper normal subgroup or fail if G is simple. In this talk I explain why this problem is important and present some ideas how to tackle it.

S Cooper: Certain data about a finite set of distinct, reduced points in projective space can be obtained from its Hilbert function. It is well known what these Hilbert functions look like, and it is natural to try to generalize this characterization to non-reduced schemes. In particular, we consider a fat point scheme determined by a set of distinct points (called the support) and non-negative integers (called the multiplicities). In general, it is not yet known what the Hilbert functions are for fat points with fixed multiplicities as the support points vary. However, if the points are in projective 2-space and the number of support points is 8 or less, we can write down all of the possible Hilbert functions for any given set of multiplicities (due to Guardo-Harbourne and Geramita-Harbourne-Migliore). In this talk we focus on what can be said, in projective 2-space, given information about what collinearities occur among the support points. Using this information we measure how far related sequences can be from being exact on global sections. Doing so, we obtain upper and lower bounds for the Hilbert function of the fat point scheme. Moreover, we give a simple criterion for when the bounds coincide yielding a precise calculation of the Hilbert function in this case. This is joint work with B. Harbourne and Z. Teitler.

C Hajarnavis: A commutative Noetherian ring of finite global dimension is a direct sum of integral domains (including fields). In the dimension 1 case (i.e. hereditary rings) these are Dedekind domains. In the non-commutative case there is an extensive theory of hereditary rings showing a much more complex situation. In this survey talk we look at the situation for dimension 2 and higher and also mention some recent work.

E Letellier: We conjecture a formula for the mixed Hodge polynomials of representations varieties of the fundamental group of punctured Riemann surfaces in terms of Macdonald polynomials. In this talk we will bring evidences for this conjecture and see some applications in the representation theory of quivers.

V Dotsenko: The goal of this talk is to discuss several series of graded vectors spaces whose series of dimensions include the series of Catalan numbers (and their generalisations), and the sequence (n+1)^{n-1} of "parking functions numbers". First of all, we show how these vector spaces arise from representation-theoretical constructions for some associative algebras. Another way to construct vector spaces with same dimensions and graded characters is to consider spaces of global sections of certain vector bundles on (zero fibres of) Hilbert schemes (for the latter the dimension and character formulae were obtained by Haiman in his works on diagonal harmonics and the "n! conjecture"). I shall formulate a conjecture relating these two constructions and try to explain some reasons for this conjecture to be true.

S Lyle: In the representation theory of the symmetric group, an important open problem is to determine the structure of certain objects known as Specht modules. I will talk about a method of constructing homomorphisms between pairs of Specht modules using the Jucys-Murphy elements. This is joint work with Andrew Mathas.

S Sierra: Let B = B(X, L, f) be the twisted homogeneous coordinate ring associated to a complex projective variety X, an automorphism f of X, and an appropriately ample invertible sheaf L. We study the primitive spectrum of B, and show that there is an intriguing relationship between primitivity of B and the dynamics of the automorphism f. In many cases Dixmier and Moeglin's characterization of primitive ideals in enveloping algebras generalizes to B; in particular, this holds if X is a surface. This is joint work with J. Bell and D. Rogalski.

J Heinloth: To study the spaces of bundles on a Riemann surface (or algebraic curves), the so called uniformization theorem has been a very useful tool. This result says that these spaces can be viewed as a quotient of the space of all maps of the circle into a Lie group. A similar result has been conjectured by Pappas and Rapoport for spaces of bundles equipped with different types of extra structure. I would like to explain, what these are, how they are related to twisted loop groups and why the general setup allows to give a short proof of the conjecture. At the end of the talk I will try to indicate an application of this to point counting arguments for these moduli spaces.

V Reiner: (This is joint work with B. Broer, L. Smith, and P. Webb.) The theorems in the title are classical results in the invariant theory of finite subgroups of GL_n(C) generated by reflections. After reviewing these results, we show how to extend them in several directions, removing many of their hypotheses. In particular, our results work over an arbitrary field k rather than the complex numbers. Also our version of the Chevalley-Shephard-Todd theorem applies to any finite subgroup of GL_n(k), not just reflection groups. If time permits, we will mention the combinatorial applications in characteristic p that motivated us.

For further information, contact

Stefan Kolb
Department of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Mayfield Road, Edinburgh EH9 3JZ.

Richard Weidmann
School of Mathematical and Computer Sciences, Scott Russell Building, Heriot-Watt University, Edinburgh EH14 4AS.

In search of lost time

In the archives, you can find lists of speakers from earlier sessions:

1998-99 1999-00 2000-01 2001-02 2002-03 2003-04 2004-05 2005-06 2006-07 2007-08

The Maxwell Institute for the Mathematical Sciences is part of the
Edinburgh Research Partnership in Engineering and Mathematics.