The symmetric functions catalog

An overview of symmetric functions and related topics


See the cyclic sieving phenomenon page for the definition and related theorems.


Root posets

Let $\Phi$ be a root system and $\Phi^+$ its poset of positive roots. Armstong–Stump–Thomas [AST13] studied the action of the rowmotion operator (a.k.a. the 'Fon-Der-Flaass/Panyushev map') on the antichains of $\Phi^+.$ Addressing conjectures of Panyushev and Bessis–Reiner, they showed that this action is in equivariant bijection with the Kreweras complementation action on the $\Phi$-noncrossing partition lattice. Moreover, they showed that the $q$-$\Phi$-Catalan polynomial is a cyclic sieving polynomial for the action of rowmotion on the antichains of $\Phi^+$ (equivalently, for the Kreweras complement acting on $\Phi$-noncrossing partitions). Note that the Type $A$ case of this CSP is equivalent to the Non-crossing matchings CSP.

Rowmotion can be realized as a certain composition of toggles. By making these toggles piecewise-linear, one obtains a piecewise-linear action of rowmotion on height $m$ $P$-partitions of a poset $P.$ Hopkins (see [Hop20a]) conjectured a CSP for the action of piecewise-linear rowmotion on these $P$-partitions when $P$ is root poset 'of coincidental type', where the sieving polynomial is the so-called '$q$-$\Phi$-multi-Catalan number'. This conjecture extend the aforementioned result of Armstong–Stump–Thomas, which is the case $m=1.$

Minuscule posets

In [RS12], the authors consider rowmotion acting on the antichains of minuscule posets. The corresponding CSP-polynomial is the rank-generating function of the poset, which for minuscule posets enjoys some particularly nice properties, see [Exercise 170, Sta11].

For the product of two chains, this action is a special case of the plane partitions CSP.

Hopkins (see [Hop20a]) conjectured a CSP for the action of piecewise-linear rowmotion on height $m$ $P$-partitions when $P$ is a minuscule poset, where the sieving polynomial is the rank-generating of $P\times[m],$ which again has nice properties (in particular, a product formula). This conjecture extends the aforementioned result of Rush–Shi, which is the case $m=1.$

BiCSP on 01-matrices and Hall–Littlewood

In [BRS08], a bi-cyclic sieving phenomenon on permutation matrices is described. Let $X_n$ be the set of $n\times n$ permutation matrices, and let $\grpc_n \times \grpc_n $ act on $X_n$ by cyclic shift on rows, and columns. Then

\[ \left(X_n, \grpc_n \times \grpc_n, \epsilon(q,t)\sum_{\lambda \vdash n} f^\lambda(q) f^\lambda(t) \right) \]

exhibit the bi-cyclic sieving phenomenon. Here, $\epsilon(q,t)$ is $(qt)^{n/2}$ if $n$ is even, and$1$ if $n$ is odd. In fact, they prove a much stronger statement regarding complex reflection groups.

This is further generalized in [Rho10b] to $01$-matrices, where the column sums and row sums in the matrices are given by fixed integer partitions.

B. Rhoades further generalizes this to matrices with non-negative integer entries, and prove a bi-cyclic phenomenon using

\[ \epsilon(q,t)\sum_{\lambda \vdash n} K_{\lambda,\mu}(q) K_{\lambda,\nu}(t) \]

as biCSP polynomial. Note that this is closely related to the \hyperref[rsk}{Robinson–Schensted–Knuth algorithm}.

Tensor products

In [Wes16], cyclic sieving phenomena are created via representation theory and crystals, generalizing the promotion CSP by Rhoades. This is related to the principal specialization of Frobenius image of characters of the symmetric group.

Lattice polytopes

J. Propp asks for a solution to the following problem. Let $P \subset \setR^d$ be a lattice polytope and $g$ be a linear map $\setR^d \to \setR^d,$ such that $\langle g \rangle$ is a cyclic group of order $n$ where $g(P) = P.$

Can we find a linear function $\sigma:\setZ^d \to \setZ$ such that for the $q$-analogue of the Ehrhart function of $P,$ defined as

\[ p_m(q) \coloneqq \sum_{x \in mP \cap \setZ^d} q^{\sigma(x)}, \]

we have that $(mP \cap \setZ^d, \langle g \rangle, p_m(q))$ is a CSP-triple, for all $m \in \setN$? That is, we get a family of cyclic sieving phenomena on the lattice points of dilations of $P.$

Hamming codes

A. Mason, V. Reiner and S. Sridhar have some results on cyclic codes and (dual) Hamming codes in [MRS20].

Dominant maximal weights

A cyclic sieving phenomenon on dominant maximal weights in affine Kac–Moody algebras is described in