Seminar/Bachelorarbeit Algebra
Winter Semester 2024/25
Moderator: Prof. Dr. William Crawley-Boevey
Entry in ekvv
Topic and Literature
Students may suggest their own topic, subject to agreement. Alternatively, students can choose a topic in Classical Representation Theory and Invariant Theory, which includes the representation theory of the symmetric group and of the general linear group and the invariant theory of binary forms.
Some possible topics are as follows:
- Representation theory of the symmetric group.
- Character formulas for the symmetric group.
- Schur-Weyl duality.
- The rational representations of GL(n,C).
- The First Fundamental Theorem of invariant theory for GL(n,C).
- Invariants and covariants of binary forms.
The listed topics are perhaps suitable for a lecture by the student. For the Bachelor Thesis, I would expect the student to write about this topic and also further developments contained in one or more research papers.
Recommended previous knowledge. Familiarity with categories and functors, rings and modules, multilinear algebra, group representation theory and a little commutative algebra. (All of these topics were covered in my Algebra II course in Summer Semester 2024.)
Some references:
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W. Crawley-Boevey, Lectures on representation theory and invariant theory. A graduate course given in 1989/90 at Bielefeld University. List of corrections.
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I. Dolgachev, Lectures on invariant theory, CUP 2003, E-book.
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W. Fulton, Young tableaux : with applications to representation theory and geometry, CUP 1997. E-book.
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J. A. Green, Polynomial representations of GLn, 2nd edition, Springer 2007, E-book.
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R. M. Howe, An Invitation to Representation Theory, Springer 2022. E-book.
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G. James and A. Kerber, The representation theory of the symmetric group, CUP 1984, E-book.
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H. Kraft and C. Procesi, Classical Invariant Theory: A Primer, Draft 1996.
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J. Kung and G.-C. Rota, The invariant theory of binary forms, Bull. Amer. Math. Soc. (N.S.) 10(1984), no.1, 27–85.
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P.-L. Meliot, Representation Theory of Symmetric Groups, CRC Press 2017, E-book.
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P. J. Olver, Classical invariant theory, CUP 2003. E-book.
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C. Procesi, Lie Groups: An Approach through Invariants and Representations, Springer 2007. E-book.
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T. A. Springer, Invariant Theory, Springer 1977, E-book.
Notes
Students may also be interested in my Masters Sequence which begins with a course on Homological Algebra.
