Lectures in algebraic combinatorics. Young's construction, seminormal representations, \(\mathfrak{sl}(2)\) representations, heaps, basics on finite fields
This is a collection of notes based on lectures given by Adriano Garsia at the University of California, San Diego. The overriding theme is the relationship between combinatorics and algebraic results, with combinatorial objects used to develop results in representation theory and other areas of algebra. The general approach is algebraic. For example, when working in the linear span of Young tableaux, the lectures usually discuss the image of a linear operator acting on a tableau, rather than the equivalent view of a weighted sum of tableaux. The first lecture presents Young's natural form for construction of the irreducible representations of the symmetric group, with simplifications due to von Neumann. The second lecture is a natural continuation of the first, developing properties of these representations. The Murphy elements of the group algebra of the symmetric group are used to give an explicit construction of Young's seminormal form. There are symmetric polynomials which can be evaluated to give the characters of the representations. The third lecture begins with abstractly defined operators, then shows how they act on \(\mathfrak{sl}(2)\) strings in order to get a matrix representation. This leads to an algebraic proof that the poset of Ferrers diagrams in an \(m\times n\) rectangle is Sperner. Respresentations of \(\mathfrak{sl}(2)\) are used to decompose the poset algebra (and thus the underlying poset) into chains. The fourth lecture develops relations between many different combinatorial and algebraic objects. The basic combinatorial object is a monomer-dimer heap, a series of needles which are stacked with either single balls or two balls on adjacent needles joined by a bar. These correspond to Motzkin paths (paths with steps \((1,1), (1,0), (1,-1)\)). There is also a connection with continued fractions, and the expansions of these continued fractions correspond to several classical families of orthogonal polynomials. The fifth lecture, while not combinatorial, is in the same style. It develops the basic theory of finite fields, with the intended audience the students in a cryptography class.
scientific article; zbMATH DE number 7258507