Affine Stanley symmetric functions

By Thomas Lam. 2005

112 citations recorded by the source corpus.

We define a new family [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i"/] of generating functions for w ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i"/] which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions including symmetry, dominance and conjugation. We conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine Schur functions. As applications, we show how affine Stanley symmetric functions generalize the (dual of the) k -Schur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov. Conjecturally, affine Stanley symmetric functions should be related to the cohomology of the affine flag variety.

Subfields: Discrete Mathematics and Combinatorics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/312f7f71462bb75446bf0401f30105ee309f073c

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In Discrete Mathematics and Combinatorics

1. Thomas Lam
2. Mark Shimozono
3. Anne Schilling
4. Jennifer Morse
5. Luc Lapointe
6. Mike Zabrocki
7. James Haglund
8. Anders Skovsted Buch

In Molecular Biology

1. Jennifer Morse
2. Luc Lapointe
3. Mark Shimozono
4. Anne Schilling
5. Mike Zabrocki

Citations in the corpus

• Schubert polynomials for the affine Grassmannian (2006, methodology): “In the cohomology case, these polynomials were introduced by Lapointe and Morse in [15] where they were called dual k-Schur functions and also studied by myself in [10] where they were called affine Schur functions.”
• K-theory Schubert calculus of the affine Grassmannian (2009, methodology): “In [ Lam06 ], the affine stable Grothendieck polynomialsGw(x) were introduced, where w ∈ Waf is an affine permutation.”
• Affine Insertion and Pieri Rules for the Affine Grassmannian (2006, methodology): “Coincidence with RSK as n→ ∞ 70 References 71”
• QUANTUM COHOMOLOGY AND THE k-SCHUR BASIS (2005, methodology): “Details of a connection between the dual k-Schur functions and the cylindric Schur functions of [25] is also carried out in [ 17 ].”
• k-Schur Functions and Affine Schubert Calculus (2013, methodology): “(321-avoiding affine permutations [78]) Extend the results in Section 1.”