Schubert polynomials, Kazhdan-Lusztig basis and characters
This fine paper delivers a combinatorial formula for the characters of the homogeneous components of the coinvariant algebra---\(\mathbb{Q}[x_1,\dots, x_n]\) factored out by those polynomials invariant under the action of the symmetric group with no constant term. It begins with a clear and concise exposition of all the ingredients needed including Schubert polynomials, Monk's formula, and Kazhdan-Lusztig cells. From here the derivation of the action of the symmetric group on Schubert polynomials and some related inner product results lead to a swift proof of the aforementioned character formula, which is also shown to be equivalent to the decomposition of homogeneous components of the coinvariant algebra into irreducible representations. Finally we are given a taster for two subsequent works ``Deformation of the coinvariant algebra'' and ``Major index of shuffles and restriction of representations'', the former of which yields a \(q\)-analogue of the character formula in this paper, the latter an algebraic interpretation of the set of all permutations of length \(k\) in the symmetric group.
scientific article; zbMATH DE number 1453361