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Summary.
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The goal of this course is to
offer an elementary and effective presentation
of the theory of symmetric polynomials in the
light of new methodologies allowing a more
flexible use of the formalism and applications
to multilinear algebra, combinatorics, algebra,
representation theory, mathematical physics
(Bose-Fermi correspondence and integrable
systems), algebraic geometry. It also allow to
revisit the Grassmannian Varieties already dealt
with in the Intensive
Course No. 1.
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Programme.
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- Action of groups on a set, fixed loci,
action of the symmetric group on polynomials
- Symmetric polynomials and vector space
generated by partitions; Symmetric polynomials
and exterior algebras;
- Projective limits of symmetric polynomials:
symmetric functions. Basic notions of
symmetric functions. The Schur basis;
- Relations with other areas of mathematics:
linear differential equations, representation
of Lie algebras of endomorphisms, relations
with mathematical physics;
- Open problem session.
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Prerequisites.
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Basics of linear
algebra and differential and integral calculus 1
and 2. Principles of abstract algebra. Definition
of a group, a ring, a module over a ring R, an
algebra over a ring R, a derivation of an
R-algebra. Combinatorial elements; binomial
coefficients, Pascal's triangle.
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References.
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Lecture notes will be prepared
along the course delivering. The course will be a
proper subset of the content of the books
- I.G. Macdonald, Symmetric functions
and Hall polynomials, Oxf. Class.
Texts Phys. Sci., The Clarendon Press,
Oxford University Press, New York, 2015;
- H. S. Wilf, Generatingfunctionology,
A K Peters, Ltd., Wellesley, MA, 2006;
- S. Amukugo, E. Lazarus, J. Lichela, G.
Marelli, M. Mugochi, The LMS-MARM
Lectures on the algebraic perspective of
Linear ODE, 2025, forthcoming.
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