Schubert polynomials, Kazhdan-Lusztig basis and characters
type: Item
instance of: scholarly article
title: Schubert polynomials, Kazhdan-Lusztig basis and characters
zbMATH Open document ID: 0969.05066
DOI: 10.1016/S0012-365X(99)00272-1
author: Yuval Roichman
published in: Discrete Mathematics
publication date: 2001-10-03T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 20B30
zbMATH DE Number: 1453361
zbMATH Keywords: coinvariant algebra
zbMATH Keywords: Kazhdan-Lusztig cells
zbMATH Keywords: Schubert polynomials
zbMATH Keywords: character formula
zbMATH Keywords: irreducible representations
zbMATH Keywords: symmetric group
reviewed by: Stephanie van Willigenburg
MaRDI profile type: Publication
Wikidata QID: Q127349588
description: scientific article; zbMATH DE number 1453361