A tableau formula for eta polynomials

TL;DR

This paper develops a tableau formula for eta polynomials and Stanley symmetric functions related to Grassmannian elements in type D Weyl groups, using Pieri, Giambelli formulas, and raising operators.

Contribution

It introduces a new tableau formula for eta polynomials and Stanley symmetric functions in type D, linking reduced words to standard typed tableaux on skew shapes.

Findings

Established a bijection between reduced words and standard typed tableaux.

Derived a tableau formula for eta polynomials of type D.

Connected eta polynomials to Stanley symmetric functions.

Abstract

We use the Pieri and Giambelli formulas of arXiv:0809.4966 and arXiv:1109.6669 and the calculus of raising operators developed in arXiv:0811.2781 and arXiv:0812.0639 to prove a tableau formula for eta polynomials of arXiv:1109.6669 and the Stanley symmetric functions which correspond to Grassmannian elements of the Weyl group $W_n$ of type $D_n$. We define the skew elements of $W_n$ and exhibit a bijection between the set of reduced words for any skew element $w$ in $W_n$ and a set of certain standard typed tableaux on a skew shape $\lambda/\mu$ associated to $w$.