Title: Lecture on the various types of symmetric functions
Abstract: A short introduction is given to the theory of symmetric functions, quasi-symmetric functions and noncommutative symmetric functions. The theorem of Radford is mentioned. An example is given that the monomial quasi-symmetric functions M(3,6) is a counterexample to the so called Ditters conjecture A (short) proof is given, theorem 2.2 below, that the ring QSym of quasi-symmetric functions is a free polynomial algebra, generated freely by the set W of Lyndon-Witt functions. The set W is canonical in a sense, made precise in cor 2.4. These Lyndon-Witt functions were defined in [Di02a] and [Di02b]. Author’s email adress is [email protected]. based on an oral exposition 2 but not by myself 1 Symmetric functions 1.1 Generalities Let n be a positive natural number and put B = Z[x1, . . . , xn]. If Sn is the symmetric group of permutations of n obects, we may let act this group on B, by permutining the indices via σ ∗ (xi) := xσ(i) and a polynomial f ∈ B is called a symmetric polynomial if f does not change under permutations of these indices. For example n = 2 and f = x1 + x2 or f = x1x2. Generic examples are
Publication Year: 2002
Publication Date: 2002-01-01
Language: en
Type: article
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