Schubert polynomials and the nilCoxeter algebra
type: Item
instance of: scholarly article
title: Schubert polynomials and the nilCoxeter algebra
zbMATH Open document ID: 0809.05091
author: Richard P. Stanley
author: Sergey Fomin
published in: Advances in Mathematics
publication date: 1994-09-27T00:00:00Z
review text: Schubert polynomials \({\mathfrak S}_ \sigma(x_ 1,x_ 2,\dots)\) indexed by permutations have been introduced and investigated by \textit{I. N. Bernstein}, \textit{I. M. Gel'fand} and \textit{S. I. Gel'fand} [Russ. Math. Surveys 28, No. 3, 1-26 (1973; Zbl 0286.57025)], \textit{M. Demazure} [Ann. Sci. École Norm. Sup., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009)], and by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031)]; see also their paper [Symmetry and flag manifolds, Lect. Notes in Math. 996, 118-144 (1983; Zbl 0542.14031)]. In this paper the theory of Schubert polynomials is recovered using the nilCoxeter algebra \({\mathfrak C}_ n\) with the identity element \(e\), given by its generators and defining relations as the \(K\)-algebra \[ \begin{multlined} {\mathfrak C}_ n=\Bigl\langle u_ 1,\dots, u_{n-1}\mid u^ 2_ i= 0\;(i\in I_{n-1}),\;u_ i u_ j= u_ j u_ i\\ (\text{for }| i- j|\geq 2),\text{ and } u_ i u_{i+1} u_ i= u_{i+1} u_ i u_{i+1} (\text{for } i\in I_{n-2})\Bigr\rangle\end{multlined} \] over any commutative ring \(K\); here \(I_ n= \{1,2,\dots, n\}\). This algebra can be faithfully represented by the algebra of operators generated by \(\Phi_ i\) \((i\in I_{n-1})\), \[ \Phi_ i(\sigma)= \begin{cases} \sigma\tau_ i &\text{if } \ell(\sigma\tau_ i)= \ell(\sigma)+1;\\ 0 & \text{otherwise}.\end{cases} \] Here, \(\sigma\) is any permutation in the symmetric group \({\mathcal S}_ n\) defined on \(I_ n\), \(\tau_ i\) \((i\in I_{n-1})\) is the `adjacent' transposition \((i,i+1)\), and \(\ell(\sigma)\) is the length of \(\sigma\in {\mathcal S}_ n\) defined as the minimal \(p\) such that \(\sigma= \tau_{a_ 1}\cdot\tau_{a_ 2}\cdot\dots\cdot \tau_{a_ p}\) for some \(a_ j\in I_{n-1}\). A sequence \(a= (a_ 1,\dots, a_ p)\), \(a_ j\in I_{n-1}\) is called a reduced decomposition of \(\sigma\) if \(p= \ell(\sigma)\). \(R(\sigma)\) denote the set of all reduced decompositions for \(\sigma\). For any reduced d…
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 611215
zbMATH Keywords: Schubert polynomials
zbMATH Keywords: nilCoxeter algebra
zbMATH Keywords: reduced decomposition
Wikidata QID: Q56625076
reviewed by: U. Kaljulaid
MaRDI profile type: Publication
full work available at URL: aima.1994.1009
OpenAlex ID: W2080720229
DOI: 10.1006/AIMA.1994.1009
description: scientific article; zbMATH DE number 611215