Mathematicians have long used representation theory to study linear symmetries and algebraic structures, now researchers are developing new theoretical tools to approach the subject and gain new insights. We spoke to Dr Dmitry Gourevitch about his work in relative representation theory, which brings together elements of several branches of mathematics
Background images describe the action of the group of 3x3 matrices with unit determinant over real and p-adic local fields on its quotient by its maximal compact subgroup, courtesy of Prof. A. Aizenbud and his students Y. Hendel and I. Glazer.
Branching out across maths disciplines Many objects in mathematics, physics, and other sciences possess natural symmetries, and over the course of history a number of theories and conceptual frameworks have been developed to study them. Based at the Weizmann Institute of
of permutations of a deck of cards – the set of all the possible ways to reorder it. There are all kinds of groups – like symmetries, they usually act on something,” outlines Dr Gourevitch. “Then there are also subgroups to consider.”
infinite-dimensional representations, functional analysis comes in. Then, when we study relative representation theory, geometry When we study
also comes in. So it’s at the crossroads of three major parts of mathematics – analysis, geometry and algebra Science in Israel, Dr Dmitry Gourevitch specialises in a field called representation theory, which concerns the study of linear symmetries. “There are lots of symmetrical figures, like circles, squares, cubes, while many molecules are quite symmetrical for example,” he says. A key topic in Dr Gourevitch’s research is the study of groups, a specific type of algebraic structure which is closely related to the idea of symmetry. “A group is a set with an invertible operation, such as multiplication or addition. Numbers (integral or real) are one example of a group, while another example is a group
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Representation theory Researchers are investigating this topic further in the RelRepDist project, an ERCbacked initiative which aims to develop new theoretical tools to describe and study these groups in greater depth. “It’s difficult to compute something that is non-linear, yet the study of linear models should be much easier, and indeed it is,” continues Dr Gourevitch. “Group representation theory is the study of linear symmetries. A representation of a group is a way to present its quotient as a group of symmetries of a linear space.”
The study of representations of finite groups started around 120 years ago. All the simple finite groups were classified around 10 years ago, and in some sense also all of their representations. The next step is the study of infinite compact Lie groups. For example, the group of all possible rotations in the space is a compact Lie group since it is closed, bounded and smooth. By now, this subject had also become quite classical. The majority of the groups Dr Gourevitch and his colleagues are interested in are infinite and non-compact however, like the group of all invertible 3 by 3 matrices. They consider representations of such groups in symmetries of infinitedimensional spaces. This field has been intensively studied over the years, and significant progress has been made. “On one hand, the irreducible representations have been classified to some extent by Langlands,” says Dr Gourevitch. These answers are not perfect however, as the description is very complicated and implicit. “The existing classification describes some representations as small parts of huge spaces.”
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