Noncommutative Schubert polynomials
type: Item
instance of: scholarly article
title: Noncommutative Schubert polynomials
zbMATH Open document ID: 0717.20017
DOI: 10.1007/BF01079531
publication date: 1989-01-01T00:00:00Z
review text: The remarkable role of Schur functions and Schubert polynomials is known in the group representation theory; [\textit{I. N. Bernstein}, \textit{I. M. Gelfand}, \textit{S. I. Gelfand}, Usp. Mat. Nauk 28, No.3, 3-26 (1973; Zbl 0286.57025)] and [\textit{W. Kraskiewicz}, \textit{P. Pragacz}, C. R. Acad. Sci., Paris, Sér. I 304, 209-211 (1987; Zbl 0642.13011)]. This important paper is a concise and readable summary (the proofs are not given) of a ``noncommutative lifting'' of these notions and the corresponding basic facts. In this lifting the notions `tableau' and `nilplaxtic relation' are central. Let us fix some (ordered) alphabet \(X=\{x_ 1<x_ 2<...\}\). In the free monoid \(X^*\) a word \(w=x_{i_ 1}x_{i_ 2}...x_{i_ k}\) with \(x_{i_ 1}>x_{i_ 2}>...>x_{i_ k}\) is called column; its length \(| w|\) is k and its contents C(w) is the set \(\{x_{i_ 1},...,x_{i_ k}\}\). If for two given columns v and w there exists a nonincreasing injection C(v)\(\to C(w)\), then it is said that w majorizes v (denoted by \(w\gg v)\). A finite product \(t=v_ 1v_ 2..\). of columns such that \(v_ 1\gg v_ 2\gg..\). is called a tableau; its shape is the partition \((| v_ 1|,| v_ 2|,...)\) of \(| t|\). A tableau t is called standard if its shape is a permutation on the set \(\{\) 1,2,...,\(| t| \}\); t is called key if for all k, \(v_{k+1}\) is a subword of \(v_ k\). On \(X^*\) there exist two important equivalences: the plaxtic (\(\equiv)\) and nilplaxtic (\(\cong)\) relation. The first is generated by elementary relations (PL1) \(a_ ka_ ia_ j\equiv a_ ia_ ka_ j\), \(a_ ja_ ia_ k\equiv a_ ja_ ka_ i\) and (PL2) \(a_ ja_ ia_ j\equiv a_ ja_ ja_ i\), \(a_ ja_ ia_ i\equiv a_ ia_ ja_ i\), (PL1) and (PL2) both in the case \(i<j<k\). The second is given also by (PL1) and (PL2) in all cases excluding that of i, j with \(j=i+1\)- then (PL2) is replaced by (Nil PL): \(a_ ia_{i+1}a_ i\cong a_{i+1}a_ ia_{i+1}\), \(a_ ia_ i=0\). The Schensted construction is primarily concerned with the plaxtic relation, it…
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 4181625
zbMATH Keywords: Schur functions
zbMATH Keywords: Schubert polynomials
zbMATH Keywords: nilplaxtic relation
zbMATH Keywords: alphabet
zbMATH Keywords: monoid
zbMATH Keywords: word
zbMATH Keywords: Schensted construction
zbMATH Keywords: Coxeter relations
zbMATH Keywords: transpositions
zbMATH Keywords: nilplaxtic classes
zbMATH Keywords: standard tableaux
zbMATH Keywords: sums of tableaux
zbMATH Keywords: reduced decompositions
author: Alain Lascoux
author: Marcel Paul Schützenberger
reviewed by: U. Kaljulaid
MaRDI profile type: Publication
cites work: Balanced tableaux
cites work: Sur la topologie de certains espaces homogenes
cites work: Geometry of G/P. V
cites work: Pseudocharacters on free products of semigroups
full work available at URL: https://doi.org/10.1007/bf01079531
OpenAlex ID: W1990770892
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4181625