[Paper Review] A Little bijection for affine Stanley symmetric functions

Thomas Lam, Mark Shimozono|ArXiv.org|Jan 19, 2006

Advanced Combinatorial Mathematics6 references3 citations

TL;DR

This paper generalizes Little's combinatorial bijection for Stanley symmetric functions to the affine setting, introducing an affine Little bijection that establishes a signed involution on reduced words of affine permutations. The key contribution is an affine analogue of Little's identity, proving that the sum of affine Stanley symmetric functions over right $ r $-covers equals the sum over left $ r $-covers, which provides a combinatorial foundation for the Lascoux-Schützenberger tree in the affine case.

ABSTRACT

David Little developed a combinatorial algorithm to study the Schur-positivity of Stanley symmetric functions and the Lascoux-Schützenberger tree. We generalize this algorithm to affine Stanley symmetric functions.

Motivation & Objective

• To extend Little's combinatorial bijection from finite to affine symmetric groups, specifically for affine Stanley symmetric functions.
• To establish an affine analogue of Little's identity relating sums over right and left covers in the strong order on affine permutations.
• To provide a combinatorial framework for understanding the structure of affine Stanley symmetric functions, analogous to the Lascoux-Schützenberger tree in the finite case.
• To explore connections between the affine Little bijection and the affine Chevalley formula, though full positivity in affine Schur functions is not proven here.

Proposed method

• Generalizes Little's insertion-based bijection to the affine setting using reduced words in the affine symmetric group $\tilde{S}_n$.
• Defines right $ r $-covers and left $ r $-covers of a permutation $ v $ via multiplication by transpositions $ t_{r,s} $ with $ r < s $ or $ s < r $, respectively.
• Constructs a signed involution on reduced words that interchanges right and left $ r $-covers, preserving the sum of affine Stanley symmetric functions.
• Uses the structure of cyclically decreasing words and their factorization into maximal cyclic intervals to define canonical reduced words for affine permutations.
• Applies combinatorial lemmas on length reduction and reflection factorizations to prove the existence and uniqueness of critical reflection pairs in non-reduced products.
• Relies on the strong Bruhat order and reflection relations in $\tilde{S}_n$ to define the covering relations and ensure the bijection respects the poset structure.

Experimental results

Research questions

• RQ1 Can Little's finite symmetric group bijection be extended to the affine symmetric group $\tilde{S}_n$?
• RQ2 Does an affine analogue of Little's identity hold, equating sums of affine Stanley symmetric functions over right and left $ r $-covers?
• RQ3 How does the affine Little bijection relate to the affine Chevalley formula and geometric structures in the cohomology of the affine Grassmannian?
• RQ4 Can this bijection be used to prove the Schur-positivity of affine Stanley symmetric functions in the affine Schur basis?
• RQ5 What is the role of cyclically decreasing reduced words in constructing canonical representatives for affine permutations?

Key findings

• An affine Little bijection is constructed as a signed involution on reduced words that interchanges right and left $ r $-covers of a given affine permutation.
• The paper proves an affine analogue of Little's identity: $\sum_{w \in \Psi^{+}_r(v)} \tilde{F}_w = \sum_{w \in \Psi^{-}_r(v)} \tilde{F}_w $, which holds for all $ v \in \tilde{S}_n $ and $ r \in \mathbb{Z}/n\mathbb{Z} $.
• The bijection is shown to be closely related to the affine Chevalley formula, suggesting deeper connections to the geometry of the affine Grassmannian.
• The canonical reduced words for cyclically decreasing elements are characterized as shuffles of words from maximal cyclic intervals, and all reduced words for such elements are cyclically decreasing.
• The uniqueness of reduced words for cyclically decreasing elements is established: every reduced word for $ w(A) $ is cyclically decreasing with underlying set $ A $.
• The paper shows that $ \ell(xy) < \ell(x) + \ell(y) $ if and only if there exists a reflection $ t $ such that $ xt < x $ and $ ty < y $, which is used to characterize non-reduced products of reflections.