ISSN:
1572-9192
Keywords:
divided difference operator
;
Schubert polynomial
;
reduced decomposition
;
Edelman-Greene correspondence
;
321-avoiding permutation
;
flag skew Schur function
;
principal specialization
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Schubert polynomials were introduced by Bernstein et al. and Demazure, and were extensively developed by Lascoux, Schützenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polynomial $$\mathfrak{S}_\omega $$ in terms of the reduced decompositions of the permutation w. Using this result, a variation of Schensted's correspondence due to Edelman and Greene allows one to associate in a natural way a certain set $$\mathcal{M}_\omega $$ of tableaux with w, each tableau contributing a single term to $$\mathfrak{S}_\omega $$ . This correspondence leads to many problems and conjectures, whose interrelation is investigated. In Section 2 we consider permutations with no decreasing subsequence of length three (or 321-avoiding permutations). We show for such permutations that $$\mathfrak{S}_\omega $$ is a flag skew Schur function. In Section 3 we use this result to obtain some interesting properties of the rational function $$8_{\lambda /\mu } (1,q,q^2 , \ldots )$$ , where $$8_{\lambda /\mu } $$ denotes a skew Schur function.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1022419800503
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