Department of Mathematics
National University of Singapore

Algebra & Number Theory Seminar
Semester 1, 2005/6

Venue: Colloquium Room B (S14-03-09).  Time: 2-3 pm
(unless otherwise stated)

Abstracts

Some characteristic classes in number theory
Lim Meng Fai

The talk is based on a paper by Max Karoubi and Thierry Lambre, titled “Quelques classes caractéristiques en théorie des nombres” (Some characteristic classes in number theory), which appears in J. Reine. Angew. Math. 543 (2002), 169-186.  The Dennis trace map mod n is defined from K₁(A ; Z/n) to the Ω_dR(A)/(n) where Ω_dR(A) is the Kahler de Rham module of differentials in A. In the talk, we will be concerned with the case when A is the ring of integers of the cyclotomic field, giving a proof of the first case of Fermat’s Last Theorem for regular primes using the trace map. After which we will also give a sufficient condition in terms of the number of roots of Mirimanoff polynomials for the first case of Fermat’s Last Theorem to hold.

The partition function, p(n), counts the number of ways a positive integer n can be expressed as a sum of nonincreasing positive integers. This talk is aimed at giving a brief introduction to some open problems on the partition function and recent developments. We will also discuss other open problems and functions related to the partition function.

Invariant theory and computer vision
Gregor Kempor, Technische Universität München

This talk is about joint work with Mireille Boutin, which originated in the following question: Is a configuration of n points in Euclidean space uniquely determined (up to rigid motions and renumbering of the points) by the distribution of distances between pairs of points? As we will see by examples, the answer is NO in general. However, the good news is that point configurations which are not reconstructible from their distribution of distances are rare, in the sense that the reconstructible configurations form a Zariski-dense subset. This result has obvious applications in computer vision. In the talk the question and its answer will be put into the context of invariant theory and computer algebra. Apart from the Euclidean group, we shall consider some other groups, particularly the projective group, which are relevant in computer vision.

Let G be the general linear group or the symplectic group over the complex number field, and U be its maximal unipotent subgroup. We study standard monomial theory of the ring of regular functions on G/U, called the flag algebra, using the philosophy of Groebner bases and SAGBI bases combined with classical invariant theory. We describe the flag algebra in terms of Gelfand-Tsetlin patterns. In particular, we show that the flag algebra is a flat deformation of a toric variety.

Shuffling and r-quasisymmetric polynomials
Nolan Wallach

In this lecture we will see how the spectral decomposition of a specific element of the symmetric algebra (the sum of all to down shuffles) is intimately related with the structure of the ring of r-quasisymmetric polynomials. In particular we will show that these algebras are free over the symmetric polynomials. This work is joint with Adriano Garsia.

The direct problem in representation theory is to describe the category of representations of a given group. The inverse problem is concerned with extracting information about a group from information about its representations. I will describe a few different approaches to these problems in this talk.

The Fock space representation of the quantum affine algebra of sl_n
Tan Kai Meng

The Fock space representation of the quantum affine algebra of sl_n has two well-known bases as a vector space over the field C(q): the standard basis and the canonical basis, both of which are indexed by the set of all partitions of negative integers.  The coefficient d_λμ(q) of the standard basis element s(λ) when the canonical basis element G(μ) is written in terms of the standard basis is found to enjoy remarkable properties.  For example, d_λλ(q) = 1, and if d_λμ(q) ≠ 0 with λ ≠ μ, then d_λμ(q) is a polynomial in q with nonnegative integer coefficients and no constant term.  These polynomials are known to be parabolic Kazhdan-Lusztig polynomials, and is proved by Varagnolo-Vasserot and Ariki that when evaluated at q = 1, they give the decomposition numbers of v-Schur algebras and Iwahori-Hecke algebras at complex n-th root of unity. While there are algorithms to compute these polynomials, they are inherently recursive, and even with the help of computers, it is impractical to do so when λ and μ are partitions of say 100.  In this talk, we will present some known closed formulas for these polynomials.

Extending π-systems to bases of root systems
Helmer Aslaksen

Let R be an indecomposable root system. It is well known that any root is part of a basis B of R. But when can you extend a set, C, of two or more roots to a basis B of R? A π-system is a linearly independent set of roots such that if α and β are in C, then α − β is not a root. We will use results of Dynkin and Bourbaki to show that with two exceptions, A₃ ≤ B_n and A₇ ≤ E₈, an indecomposable π-system whose Dynkin diagram is a subdiagram of the Dynkin diagrams of R can always be extended to a basis of R.