The symmetric functions catalog

An overview of symmetric functions and related topics

2020-10-08

Catalan symmetric functions

Catalan symmetric functions were introduced in [Che10] and [Pan10]. These functions are $GL_\ell$-equivariant Euler characteristics of vector bundles on the flag variety. The Catalan symmetric functions specialize to $k$-Schur functions, see [BMPS18].

Definition

Let $R_{ij}$ be the raising operators on Schur functions, so that

\[ R_{ij} \schurS_{\alpha} = \schurS_{\alpha+\varepsilon_i-\varepsilon_j}. \]

Here, $\alpha$ can be a composition, and we then evaluate the Schur function using the Jacobi–Trudi formula.

A root ideal $\Phi$ is a an upper order ideal in

\[ \Delta_\ell^+ \coloneqq \{ (i,j) : 1 \leq i \lt j \leq \ell \} \]

with the partial order relation $(a,b) \leq (c,d)$ when $a\geq c$ and $b \leq d.$ There are $\catalan(\ell)$ different such order ideals, thus explaining the name.

Example.

For example $\{ 15, 25, 35, 14, 24 \}$ is an order ideal, as they are entries above a Dyck path in the diagram:

$ \mathbf{15}$$ \mathbf{25}$$ \mathbf{35}$$45$$5$
$ \mathbf{14}$$ \mathbf{24}$$34$$4$ 
$13$$23$$3$  
$12$$2$   
$1$    

See also how area sequences of length $\ell$ are related to unit interval graphs.

Let $\gamma \in \setZ^\ell$ and let $\Phi$ be a root ideal. The Catalan symmetric function is then defined as

\[ \catalanH_{\Phi,\gamma}(\xvec;t) \coloneqq \prod_{(i,j)\in \Phi} (1-t R_{ij})^{-1}\schurS_{\gamma}(\xvec). \]

Note that

\[ (1-rR_{ij})^{-1} = 1 + tR_{ij} + t^2 (R_{ij})^2 + t^3 (R_{ij})^3 + \dotsb \]

but one only has to apply a finite number of these as $(R_{ij})^k$ kills any Schur function for sufficiently large $k.$ By construction, $\catalanH_{\Phi,\gamma}(\xvec;t)$ is a symmetric function.

The Catalan symmetric functions generalize the transformed Hall–Littlewood polynomials. We have that if $\mu$ is a partition of $n,$ then we use the full set of roots and have

\[ \hallLittlewoodT_{\mu}(\xvec;q) = \catalanH_{\Delta_n^+,\mu}(\xvec;q). \]
Example.

For example, $\Phi = \{ 15, 25, 35, 14, 24 \}$ and $\gamma=(4,2,1,1,0)$ gives

\[ \catalanH_{\Phi,\gamma}(\xvec;t) = \schurS_{4211} + t \schurS_{431} + t \schurS_{521} \]

In [Che10], it was conjectured that $\catalanH_{\Phi,\mu}(\xvec;t)$ is Schur-positive for any $\Phi$ and partition $\mu.$ This conjecture is resolved by J. Blasiak, J. Morse and A. Pun in [BMP20]. They introduce a larger family of non-symmetric Catalan functions, the tame non-symmetric Catalan functions, $\catalanH_{\Phi,\mu,w}(\xvec;t),$ which depend on an additional parameter $w \in \symS_n.$ It is then proved that $\catalanH_{\Phi,\mu,w}(\xvec;t)$ are key positive, which then implies the Schur positivity for $\catalanH_{\Phi,\mu}(\xvec;t).$

Katalan symmetric functions

In [BMS20], the authors consider a $K$-theoretic version of the Catalan symmetric functions, named Katalan symmetric functions and show that this family includes the $K$-$k$-Schur functions, and the usual Catalan symmetric functions.

$k$-Schur polynomials

The $k$-Schur functions were introduced in [LLM03] (under a different name and the notation $A_\mu^{(k)}[\xvec;t]$ was used). The motivation was to provide a strong refinement of the Schur positivity conjecture of the modified Macdonald polynomials. Several alternative definitions of $k$-Schur functions has since then surfaced, and not all have been proved to be equivalent. For a thorough reference, see the book [LLMSSZ14].

For each integer $k,$ we have the family of $k$-Schur functions $\{ \kSchur^{(k)}_\lambda(\xvec) \}$ where the $\lambda$ are $k$-bounded partitions, meaning that $\lambda_1 \leq k.$ The $k$-Schur functions form a basis in the subring of symmetric functions, spanned by $\completeH_1,\dotsc,\completeH_k,$ the complete homogeneous symmetric functions.

In [LM07], it is shown that whenever the hook-length of $\lambda$ is no larger than $k,$ we have the identity

\[ \kSchur^{(k)}_\lambda(\xvec) = \schurS_\lambda(\xvec). \]

Hence, as $k\to \infty,$ the $k$-Schur functions reduce to the usual Schur functions.

Definition

Note that there are several different, but conjectured equivalent definitions of $k$-Schur functions. We use the definition in [BMPS18], which defines the $k$-Schur functions as Catalan symmetric functions for special root ideals. Let $\mu$ be a partition with at most $\ell$ parts and $\mu_1 \leq k.$

Let $\Phi_\mu \coloneqq \{ (i,j) \in \Delta_\ell^+ : k-\mu_i+i \lt j \}$ and define the $k$-Schur functions as

\[ \kSchur^{(k)}_\mu(\xvec;t) \coloneqq \catalanH_{\Phi_\mu,\mu}(\xvec;t) = \prod_{i=1}^\ell \prod_{j=k+1-\mu_i+i}^\ell (1-tR_{ij})^{-1} \schurS_\mu. \]

The specialization $t=1$ are also called $k$-Schur functions,

\[ \kSchur^{(k)}_\mu(\xvec) \coloneqq \kSchur^{(k)}_\mu(\xvec;1). \]

Most results so far concern this specialization.

Example.

We have the following Schur expansions of some $k$-Schur functions:

\[ \kSchur^{(2)}_{221}(\xvec) = \schurS_{221}(\xvec)+\schurS_{311}(\xvec)+2\schurS_{320}(\xvec)+2\schurS_{410}(\xvec)+\schurS_{500}(\xvec) \] \[ \kSchur^{(3)}_{221}(\xvec) = \schurS_{221}(\xvec)+\schurS_{320}(\xvec)\quad \text{ and } \quad \kSchur^{(4)}_{221}(\xvec) = \schurS_{221}(\xvec) \]

Relation with affine Stanley symmetric functions

In [Lam06a] it is shown that the $k$-Schur functions are dual to the affine Schur functions.

Pieri rule

L. Lapointe and J. Morse proved the following Pieri rule.

Theorem (See [Thm. 29, LM07]).

Let $\nu$ be a $k$-bounded partition and $r \leq k.$ Then

\[ \completeH_r \kSchur^{(k)}_\nu = \sum_{\mu \in H^k_{\nu,r}} \kSchur^{(k)}_\mu \]

where $H^k_{\nu,r}$ is a certain subset of partitions formed by adding horizontal $r$-strips to $\lambda.$ To be more precise

\[ H^k_{\nu,r} = \{ \mu : \mu/\nu \text{ is a horizontal strip and } \mu^{(k)}/\nu^{(k)} \text{ is a vertical $r$-strip} \}. \]

Here, $\mu^{(k)}$ denotes the $k$-conjugate of $\mu.$

Murnaghan–Nakayama rule

J. Bandlow, A. Schilling and M. Zabrocki prove the following analogue of the Murnaghan–Nakayama rule.

Theorem (See [BSZ11]).

For $1\leq r \leq k$ and $\lambda$ being a $k$-bounded partition,

\[ \powerSum_r \kSchur^{(k)}_\lambda = \sum_{\mu} (-1)^{\ht(\mu/\lambda)} \kSchur^{(k)}_\mu \]

where the sum is over all $k$-bounded partitions $\mu$ such that $\mu/\lambda$ is a $k$-ribbon of size $r.$ The definition of a $k$-ribbon is somewhat involved, see [BSZ11] for details.

An alternative proof is given by Lee in [Thm. 10.1, Lee15].

Littlewood–Richardson rule

It is conjectured that the coefficients $c^{\nu,k}_{\lambda\mu}$ in

\[ \kSchur^{(k)}_\lambda \kSchur^{(k)}_\mu = \sum_{\nu : \nu_1 \leq k} c^{\nu,k}_{\lambda\mu} \kSchur^{(k)}_\nu \]

are all non-negative. These coefficients 3-point Gromow–Witten invariants, see [LM08], and thus sometimes proved to be non-negative.

Schur expansion

Conjecture (See [LM07]).

It is conjectured that

\[ \kSchur^{(k)}_\lambda(\xvec) = \schurS_\lambda + \sum_{\mu \triangleright \lambda } d^{k}_{\lambda\mu} \schurS_{\mu}(\xvec) \]

with non-negative $d^{k}_{\lambda\mu}.$

This conjecture was recently proved, see this preprint.

References