Noncommutative Schubert polynomials
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 extends to the nilplaxtic case (Th. 1). The nilplaxtic relations contain Coxeter relations for transpositions. So the set of reduced decompositions of any permutation \(\mu\) is a disjoint union of nilplaxtic classes. Any such class (not containing 0) contains exactly one tableau t and its elements are in 1-1 correspondence with standard tableaux of the same shape as t. Using these remarkable facts the paper describes the way how general (noncommutative) Schubert polynomials \(X_{\mu}\) can be introduced as elements of \({\mathbb{Z}}<X>\) and how they can be found as certain sums of tableaux connected with the reduced decompositions of \(\mu\).
scientific article; zbMATH DE number 4181625