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Symmetrizing and Markov Traces on Algebras
Research supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the Basic Research Financing (Horizontal support for all Sciences), National Recovery and Resilience Plan (Greece 2.0), Project Number: 15659, Project Acronym: SYMATRAL.
Mission of the project
Mission of the project
The trace of a matrix is a particular case of a symmetrizing trace, which also exists on finite group algebras and several other algebras of finite type. A symmetrizing trace on an algebra is a powerful tool for understanding its structure and representation theory; hence its construction on important families of algebras is a high priority goal in Algebra. Moreover, Markov traces, which generalize symmetrizing traces, have applications in knot theory, because they are used in the definition of knot invariants. Knot theory in turn has numerous applications to other sciences, from the study of electric circuits to the study of proteins and DNA structure.
The aim of this project is the construction of symmetrizing and Markov traces on two very important families of algebras, Hecke algebras and Brauer algebras.
Work Package 1
Work Package 1
Hecke algebras appear naturally in the study of the representation theory of finite reductive groups, but they can also be defined independently as deformations of the group algebra of the corresponding Weyl groups. Complex reflection groups are natural generalizations of Weyl groups. Our aim is to define a symmetrizing trace on Hecke algebras of complex reflection groups, thus proving some fundamental conjectures on the structure of these algebras that are open for more than 20 years.
Work Package 2
Work Package 2
We wish to construct a symmetrizing trace on the Brauer-Chen algebra, defined by Chen 10 years ago as a generalization to all complex reflection groups of the classical Brauer algebra, which in turn generalizes the symmetric group, in terms of its group algebra, but also of its role in the celebrated Schur-Weyl duality.
Work Package 3
Work Package 3
We will explore several interesting examples in the universe of Hecke and Brauer algebras regarding the definition of Markov traces and their use in the construction of knot invariants.
The project will involve methods from representation theory and group theory, computational algebra and symbolic computation, algebraic combinatorics and knot theory.
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