Publications
Papers

Research

Math

TeX

Programs

Jobs

English

Russian

Latvian

Download

URL List

Home
Bullet A list of my publications in a BiBTeX format, and its dvi output

My recent preprints are also available from the following sources:

Bullet Dave Benson's Representations and Cohomology Preprint Archive
Bullet Hypatia Electronic Library
Bullet XXX Preprint Server
Bullet Front for the XXX Mathematics Archives

Some of my papers are available from

Bullet Axel Kohnert's Preprint Server

Research

Bullet My Maple Programs for Binary Tensor Invariants and Outerplanar Graphs. Here is the Maple Worksheet (3.1 MB) and the zipped Maple worksheet (540 KB).
Bullet Wave Graph Bases of Tensor Invariants, 31 slides from my February 12, 2002 talk at Duke's Algebraic Geometry Seminar.
The abstract is here.
Bullet The Orbit Method for Finite Groups of Nilpotency Class Two of Odd Order (dvi), 16 pages, January 16, 2000.
First, I construct an isomorphism between the categories of (topological) groups of nilpotency class 2 with 2-divisible center and (topological) Lie rings of nilpotency class 2 with 2-divisible center. That isomorphism allows us to construct adjoint and coadjoint representations as usual. For a finite group G of nilpotency class 2 of odd order, I construct a basis in its group algebra C[G], parameterized by elements of g* so that the elements of coadjoint orbits form bases of simple two-side ideals of C[G]. That construction gives us a one-to-one correspondence between G-orbits in g* and classes of equivalence of irreducible unitary representations of G, implying a very simple character formula. The properties of that correspondence are similar to the properties of the analogous correspondence given by Kirillov's orbit method for nilpotent connected and simply connected Lie groups. The diagram method introduced in my thesis, gives us a convenient way to study normal forms on the orbits and corresponding representations.
Bullet The Orbit Method for Finite Groups (October 5, 1999). Was presented on Wednesday, January 19, 2000, 5:30 pm at the AMS Session on Group Theory at the Joint AMS/MAA Meetings in Washington, DC, January 19–22, 2000. For a finite nilpotent group G, I define (by an inductive procedure) a finite G-module g* such that there is a one-to-one correspondence between G-orbits in g* and classes of equivalence of irreducible unitary representations of G. The properties of that correspondence are similar to the properties of the analogous correspondence given by Kirillov's orbit method. For instance, if G is the group of the upper-triangular unipotent n×n matrices with coefficients from a finite ring A, one can choose the additive group of lower-triangular nilpotent n×n A-matrices as g*, defining the action of G in g* the same way as Kirillov did in his classical article. Using diagram method introduced in my thesis, one can classify normal forms on the orbits and corresponding representations. I discuss the generalization of that construction for other classes of groups as well. In particular, for symmetric groups all the theory looks completely different from the standard approach.
Bullet Wave graph bases of tensor invariants of classical Lie groups and algebras (abstract, dvi, pdf, October 1, 1998). Was presented at the Joint AMS/MAA Meetings in San Antonio, Texas, January 13–16, 1999 (January 16, 9:15 a.m., AMS Session on Topological Groups and Lie Groups.)
Introduced in author's thesis, wave graphs give us a new combinatorial approach to the representations and invariants of classical Lie groups and algebras. For example, wave graphs
Wave graphs
correspond to the following invariants of $ SL(3)$ and $ Sp(4)$:
$\displaystyle \left( (x\wedge y\wedge z)\otimes(x\wedge y\wedge z)\right)^{(34)}, \quad \left( (\omega\wedge\omega)\otimes\omega\right)^{(465)}$    
where $ \omega=p_1\wedge q_1 + p_2\wedge q_2$.
Bullet A combinatorial approach to representations of Lie groups and algebras, PhD Thesis, University of Pennsylvania (1998), 134+vii pages
First I describe the invariants and decompositions of tensor products of polynomial representations of SL(2) in the terms of outerplanar graphs, i.e. graphs with the vertices 0, 1, ..., m, the edges of which can be drawn in the upper half-plane without intersections. Then I use wave graphs introduced here to give an analogous description for tensor invariants of SL(n) and Sp(2n). I use quite a different combinatorial approach to the description of the representations of unipotent groups of Lie type, through 'diagrams of representations'. This approach leads to various applications, including explicit formulas for fractional residues (i.e. invariants of some generalizations of differential forms). Conversely, these combinatorial approaches to representations allow us to use known representation-theoretical results to get the explicit formulas for the enumeration of the corresponding combinatorial objects, like walks on lattices, or the counting of some specific graphs.
Bullet Enumeration of walks on lattices. I (dvi, ps), 37 pages (March 25, 1998)
This work develops a methodical approach to counting of walks on cartesian products, biproducts, symmetric and exterior powers and bipowers, Schur operations, coverings and semicoverings of weighted graphs. For weight and root lattices of semisimple Lie algebras, this approach allows us to compute various combinatorial and representation-theoretical constants, in particular, the number of plane symplectic wave graphs with given number of vertices.
Bullet Symplectic tensor invariants, wave graphs and S-tris (dvi, ps, pdf), 16 pages (March 23, 1998)
The spaces of invariants of tensor powers of the defining representation of Sp(2n) are provided with the bases parametrized by symplectic wave graphs introduced here especially for this purpose. The proof utilizes a game similar to Tetris, named here S-tris. This work continues my previous work on the tensor invariants of SL(n), wave graphs and L-tris.
BulletDiagrams of representations (dvi, ps, pdf), 19 pages (March 14, 1998)
For a representation of a Lie algebra, one can construct a diagram of the representation, i. e. a directed graph with edges labeled by matrix elements of the representation. This article explains how to use these diagrams to describe normal forms, orbits and invariants of the representation, especially for the case of nilpotent Lie algebras.
BulletFractional residues (dvi, ps, pdf), 17 pages (March 5, 1998)
Invariants of generalized tensor fields on a line are classified using special polynomials Pmk(-1/l) introduced here for this purpose. For the case of positive characteristic, a new invariant of formal power series, a width, is defined. Some applications to the geometric quantization of a line and conformal quantum field theory are discussed as well.
BulletTensor invariants of SL(n), wave graphs and L-tris (dvi, ps, pdf), 8 pages (March 2, 1998)
The space of invariants of a tensor product of representations of SL(n) is provided with the basis parametrized by wave graphs introduced here especially for this purpose. The proof utilizes a game similar to Tetris, named here L-tris.
BulletA brief outline of my current and intended research, available also in dvi, ps, and pdf formats, 2 pages (December 14, 1997)
BulletTensor decompositions for SL(2) and outerplanar graphs (dvi, ps, pdf), 21 pages (November 24, 1997) Submitted to the Journal of Combinatorial Theory, Series A.
The main result of this article is the decomposition of tensor products of representations of SL(2) in the sum of irreducible representations parametrized by outerplanar graphs. An outerplanar graph is a graph with the vertices 0, 1, 2, ..., m, edges of which can be drawn in the upper half-plane without intersections. I allow for a graph to have multiple edges, but don't allow loops.
BulletTensor invariants of SL(2) and outerplanar graphs (dvi, ps), 7 pages (May 4, 1997)
The space of invariants of a tensor product of representations of SL(2) is provided with the basis parametrized by outerplanar graphs.

Teaching

I have not updated links here for a long time. See a series of new links on my starting page here and at Shepherd, and on my Maple page.
Bullet Elementary Differential Equations (Fall 1998):
Bullet Calculus I (Fall 1998):
Bullet Calculus II (Fall 1998):
Bullet Calculus III (Summer 1998):
BulletMy teaching philosophy (dvi, ps, pdf), 2 pages (January 16, 1998)
BulletWeights and roots (dvi), 5 pages (March 31, 1997)
A survey of the basic constructions used to classify and study representations and invariants of semisimple Lie groups and algebras.
BulletThe solutions of some problems from An Introduction to the Theory of Numbers / Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery (Fall, 1996)
BulletFirst steps in number theory (September 15, 1996). Translations:

On paper

BulletThe Petrovsky numbers and multiplicities of representations,
Master's Thesis, University of Latvia, Riga, 1995 (in Russian).
BulletLucky tickets and the Petrovsky numbers,
Bachelor's Thesis, University of Latvia, Riga, 1995 (in Russian).
BulletOn the log-concavity, Kvant, 11/12, 1993, p. 1--9 (in Russian).
BulletSome other notes, preprints and manuscripts.

Ruler

Updated on December 20, 2002.

Ruler

Copyright © 1996--2002 Aleksandrs Mihailovs Email