The symmetric functions catalog

An overview of symmetric functions and related topics

2019-05-20

Schur polynomials (flagged)

The flagged Schur polynomials generalize the classical Schur polynomials and these are no longer symmetric in general. They were first considered in [LS82a], in the study of Schubert polynomials.

We follow the introduction given in [MS15b].

Let $m \leq n$ be fixed integers and let $\lambda$ be a partition with $m$ parts and let $1 \leq b_1 \leq b_2 \leq \dotsb \leq b_\ell = n$ be a sequence of integers. Let $T(\lambda,b)$ denote the set of semi-standard tableaux of shape $\lambda$ such that the entries in row $i$ do not exceed $b_i.$

Then the flagged Schur polynomial is defined as

$\schurS_{\lambda,b}(x_1,\dotsc,x_n) \coloneqq \sum_{T \in T(\lambda,b)} \xvec^T.$

Note that if $b_1=\dotsb = b_m = n,$ then $\schurS_{\lambda,b}(\xvec)$ is the usual Schur polynomial in $n$ variables.

$h$-flagged Schur polynomials

Whenever $b = (h,h+1,\dotsc,h+m)$ for some $h,$ we say that $\schurS_{\lambda,b}(\xvec)$ is $h$-flagged, and use the shorthand notation $\schurS^{(h)}_{\lambda}(\xvec).$

Divided differences

In [Wac85], it is proven that flagged Schur polynomials can be obtained via divided difference operators:

\begin{align*} \schurS_{\lambda,b}(\xvec) = \partial_w( x_1^{a_1} \dotsm x_m^{a_m}) \end{align*}

where $a_i = \lambda_i + b_i -i$ and

$w = (m,m+1,\dotsc, b_m-1,\; m-1,m,\dotsc, b_{m-1} - 1, \dotsc 1,2,\dotsc, b_1 -1 )$

and the entries in the word are applied from left to right.

Jacobi-Trudi identity

Define the complete homogeneous polynomials in $k$ variables as

$\completeH_d(\xvec_k) \coloneqq \sum_{i_1 \leq \dotsb \leq i_d} x_{i_1}\dotsm x_{i_k}.$

Then, using the Lindström–Gessel–Viennot lemma, one can prove that

$\schurS_{\lambda,b}(\xvec) = \det( \completeH_{\lambda_i -i+j}(\xvec_{b_i}) )_{1\leq i,j \leq m}.$

See also [Wac85]. This generalizes the classical Jacobi–Trudi identity for Schur polynomials.

Key expansion

Flagged skew Schur polynomials may be defined as

$\schurS_{\lambda/\mu,b}(\xvec) = \det( \completeH_{\lambda_i-\mu_j -i+j}(\xvec_{b_i}) )_{1\leq i,j \leq m}.$

The flagged skew Schur polynomials expand positively into key polynomials, see [RS95].