Balanced Labellings and Schubert Polynomials
نویسندگان
چکیده
We study balanced labellings of diagrams representing the inversions in a permutation. These are known to be natural encodings of reduced decompositions of permutations w 2 n , and we show that they also give combinatorial descriptions of both the Stanley symmetric functions F w and and the Schubert polynomial S w associated with w. Furthermore, they lead to an explicit basis for the Schubert modules introduced by Kraskiewicz and Pragacz.
منابع مشابه
Factorizations in Schubert cells
For any reduced decomposition i = (i1, i2, . . . , iN ) of a permutation w and any ring R we construct a bijection Pi : (x1, x2, . . . , xN ) 7→ Pi1(x1)Pi2(x2) · · · PiN (xN ) from R to the Schubert cell of w, where Pi1(x1), Pi2(x2), . . . , PiN (xN ) stand for certain elementary matrices satisfying Coxeter-type relations. We show how to factor explicitly any element of a Schubert cell into a p...
متن کاملDiagrams of Affine Permutations and Their Labellings
We study affine permutation diagrams and their labellings with positive integers. Balanced labellings of a Rothe diagram of a finite permutation were defined by FominGreene-Reiner-Shimozono, and we extend this notion to affine permutations. The balanced labellings give a natural encoding of the reduced decompositions of affine permutations. We show that the sum of weight monomials of the column...
متن کاملA Unified Approach to Combinatorial Formulas for Schubert Polynomials
Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials; we also present simplifications in some of the existing approaches to this area. We designate certain line diagrams known as rc-graphs as the main structure...
متن کاملDiagrams of affine permutations , balanced labellings , and affine Stanley symmetric functions (
We study the diagrams of affine permutations and their balanced labellings. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the column strict balanced labellings is the affine Stanley symmetric function defi...
متن کاملSkew Schubert Polynomials
We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron and Sottile in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schub...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Eur. J. Comb.
دوره 18 شماره
صفحات -
تاریخ انتشار 1997