The symmetric functions catalog

An overview of symmetric functions and related topics

2024-03-13

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.

The flagged skew Schur polynomials, $\schurS_{\lambda/\mu,a,b}(\avec,\bvec)$ are now defined as the sum over all skew semistandard Young tableaux of shape $\lambda/\mu$ such that entries in row $i$ must be from the interval $\{a_i,a_i+1,\dotsb,b_i\}.$ We still require that the corresponding flags are weakly increasing.

With $\completeH_m(a,b) \coloneqq \completeH_m(x_a,x_{a+1},\dotsc,x_b),$ one can show the Jacobi–Trudi type formula

\begin{equation} \schurS_{\lambda/\mu}(\avec,\bvec) = \left| h_{\lambda_i-\mu_j - i + j}(a_j,b_i) \right|_{1 \leq i,j \leq \ell}. \end{equation}

This was proved by I. Gessel (see [GV89]) and later in [Wac85]. An alternative proof is given in [Chu92], which relies on a recursion.

Unlike the classical Jacobi–Trudi identities, exchanging $\completeH \leftrightarrow \elementaryE$ does not in general yield the same determinant. Rather, the corresponding determinant of elementary symmetric polynomials is equal to a column-flagged Schur polynomial (where the tableaux in the generating set have bounds on the columns rather than the rows); see [Wac85]. An identity between determinants of this type is given in [McD23], thus giving a duality for row- and column-flagged Schur polynomials under certain conditions.

Kostka coefficients

The flagged Kostka coefficients are defined as the coefficients in the monomial expansion of the flagged Schur functions. Given a skew shape $\lambda/\mu$ and a vector $(\nu_1,\dotsc,\nu_\ell)$ with $\nu_i \geq 0,$ we define the flagged Kostka coefficient $K_{\lambda/\mu,\nu}(\avec,\bvec)$ via the expansion

\[ \schurS_{\lambda/\mu}(\avec,\bvec) = \sum_{\nu} K_{\lambda/\mu,\nu}(\avec,\bvec) \cdot \xvec^\nu. \]

In [AO23], we prove the following results:

Theorem (Alexandersson–Kantarci-Oğuz (2023)).

We have that for any integer $k \geq 1,$

\[ K_{\lambda/\mu,\nu}(\avec,\bvec) \gt 0 \iff K_{k\lambda/k\mu,k\nu}(\avec,\bvec) \gt 0 \]

where multiplication of a partition by $k$ is done elementwise. This is a generalization of saturation for Gelfand–Tsetlin polytopes.

Moreover, the map

\[ k \mapsto K_{k\lambda/k\mu,k\nu}(\avec,\bvec) \]

is a polynomial in $k.$

Note that it is relatively easy to see that the map

\[ k \mapsto K_{k\lambda/k\mu,k\nu}(\avec,\bvec) \]

is a quasi-polynomial as this is the Ehrhart function for a face of a (rational) Gelfand–Tsetlin polytope.

The following conjecture is then a flagged generalization of a conjecture by King, Tollu and Toumazet, [KTT04].

Conjecture (Alexandersson–Kantarci-Oğuz (2023)).

The Ehrhart polynomial

\[ k \mapsto K_{k\lambda/k\mu,k\nu}(\avec,\bvec) \]

has non-negative coefficients.

Key expansion

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

Twist by roots of unity

In [Kum23] the author generalizes a previous result by Kumari [Kum22], and [Thm. 16, LO22].

References