The symmetric functions catalog

An overview of symmetric functions and related topics


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].