Poc-Sets and Dual Graphs in Combinatorial Algebra

• Poc-sets and dual graphs are finite structures where composition posets are organized via specific algebraic operators and partial orders.
• They employ four operator families, including box-removing and box-adding operators, derived from Pieri rules to build graded and filtered graph frameworks.
• Dual graded graphs emerge when up and down operators satisfy precise commutation relations, ensuring a robust duality that underpins combinatorial and algebraic symmetry.

A poc-set (partially ordered composition set) is a finite set of compositions of positive integers equipped with a partial order, typically constructed from algebraic or combinatorial operations such as those arising from the Pieri rules of noncommutative and quasisymmetric Schur functions. Dual graphs, in this context, refer to pairs of graded or filtered graphs on the same underlying set, endowed with up and down operators whose commutation relations encode deep combinatorial and algebraic structures. The theory of poc-sets and their dual graph structures provides an essential framework for understanding the interplay between combinatorics of compositions and algebraic structures such as (quasi)symmetric functions (Willigenburg, 2019).

1. Compositions and Fundamental Operators

A composition is defined as a finite sequence of positive integers $\alpha=(\alpha_1,\dots,\alpha_\ell)$, with its size given by $|\alpha| = \sum_{i=1}^\ell \alpha_i$. Allowing weak compositions (possibly containing zeroes) enables the natural extension of various operators critical for the construction of poc-sets and dual graphs.

Four principal families of linear operators act on (weak) compositions, indexed by $i\geq 0$:

• Box-removing operators $d_i$: $d_i(\alpha)$ subtracts 1 from the rightmost part equal to $i$, if it exists; $d_0$ is the identity.
• Appending operators $a_i$: $a_i(\alpha)$ appends $i$ to the end.
• Jeu-de-taquin (jdt) operators $u_i$: Defined as $u_i = a_i d_{[i-1]}$, where $d_{[i-1]}= d_1 d_2 \cdots d_{i-1}$.
• Box-adding operators $t_i$: For $i=1$, prepends 1; for $i\geq 2$, increments the leftmost part equal to $i-1$ by 1.

These operators satisfy a system of commutation relations, such as $u_i d_j = d_j u_i$ for $i\neq j$ and $u_i d_i = d_{i+1} u_{i+1}$, which are instrumental in establishing the dual graph structures.

2. Partial Orders and Composition Posets

Three principal partial orders—or poc-sets—are constructed on the set of compositions using the above operators, each graded by composition size:

• Right composition poset ($R$): $\beta$ covers $\alpha$ if $\alpha = u_i(\beta)$ for some $i\geq 1$.
• Left composition poset ($L$): $\beta$ covers $\alpha$ if $\alpha = t_i(\beta)$ for some $i\geq 1$.
• Quasisymmetric composition poset ($Q$): $\beta$ covers $\alpha$ if $d_i(\alpha) = \beta$ for some $i\geq 1$.

A deformation of $Q$, denoted $\widetilde Q$, is defined by allowing all possible box-removal sets: $\beta <^t \alpha$ iff $d_I(\alpha)=\beta$, for nonempty $I\subset\mathbb{N}_{>0}$. This deformation gives rise to strong filtered graph structures, where edges can join vertices across or within ranks.

3. Up and Down Operators

To analyze graded or filtered graphs on the set $P$ of all (weak) compositions, one introduces corresponding vector spaces $KP$ over a characteristic-zero field $K$. For a graded (or filtered) graph, linear up ($U$) and down ($D$) operators are defined as follows: $$U(\beta) = \sum_{\beta \to \alpha} m(\beta, \alpha)\, \alpha,\qquad D(\alpha) = \sum_{\beta \to \alpha} m(\beta, \alpha)\, \beta$$ where $m(\beta, \alpha)$ denotes the number of edges from $\beta$ to $\alpha$.

Specific instances include:

• On $R$: $U_R = \sum_{i\geq 1} u_i$
• On $Q$: $D_Q = \sum_{i\geq 1} d_i$
• On $L$: $U_L = \sum_{i\geq 1} t_i$, $D_L = \sum_{i\geq 1} d_i$
• On $\widetilde Q$: $D_{\widetilde Q} = \sum_{\emptyset\neq I} d_I$

The structure and interplay of these operators on poc-sets are foundational to the duality properties of the associated graphs.

4. Dual Graded and Filtered Graphs

Dual graded graphs arise from pairs of graded graphs $(G_1, G_2)$ on the same vertex set with operators $U$, $D$ satisfying the commutator identity: $D U - U D = \mathrm{Id}_{KP}$ Key dual graded graph pairs include $(R, Q)$ and $(L, Q)$. The proof relies on commutation relations between up and down operators, such as $u_i d_j = d_j u_i$ for $i\ne j$, and a case analysis for $i = j$. Specifically, the $(R, Q)$ pair satisfies

$$\sum_i d_i \sum_j u_j - \sum_j u_j \sum_i d_i = \mathrm{Id}$$

Filtered dual graphs generalize this structure. In this setting, weak and strong filtered graphs are defined based on the allowable edge directions in rank. Dual filtered graphs satisfy

$$D \widetilde{U} - \widetilde{U} D = D + \mathrm{Id}$$

The strong filtered graph $\widetilde Q$ (arising from deforming $Q$ to allow multiple simultaneous removals) admits $(R, \widetilde Q)$ and $(L, \widetilde Q)$ as dual filtered pairs.

5. Illustrative Small-Rank Examples

Explicit Hasse diagrams at low rank clarify the dual structures:

• Weight 0: Only the empty composition $()$.
• Weight 1: $(1)$.
• Weight 2: $(2)$ and $(1,1)$.

For $(R, Q)$ at rank 2:

• $R$: $(1) \xrightarrow{u_1} (1,1)$ and $(1)\xrightarrow{u_2} (2)$
• $Q$: $(2)\xrightarrow{d_1} (1,1)$, $(2)\xrightarrow{d_2} (1)$

The commutation relations ensure that the up-down commutator yields the identity on these elements.

6. General Framework and Significance

The emergence of dual graded or filtered graph structures is closely linked to the existence of commutation relations among raising and lowering operators derived from algebraic structures. Differential posets and dual graded graphs are characterized by collections of operators $\{U_i\}, \{D_j\}$ satisfying

$$D_j U_i = U_i D_j\;\;(i\neq j),\qquad D_i U_i - U_i D_i = \delta_{i,1}\,\mathrm{Id}$$

Within the context of symmetric or quasisymmetric functions, Pieri rules govern the addition of one-part functions, guiding the structure of the operators and enabling the deduction of dual graph frameworks. In particular:

• $R$ and $L$ correspond to the right and left Pieri rules of noncommutative Schur functions.
• $Q$ corresponds to the classical Pieri rule for quasisymmetric Schur functions.
• $\widetilde Q$ arises as the natural “all–subsets” deformation of $Q$.

The established commutation relations among $u_i$, $t_i$, and $d_i$ guarantee dual graded structure for $(R, Q)$ and $(L, Q)$, and dual filtered structure for $(R, \widetilde Q)$ and $(L, \widetilde Q)$. This framework answers the structural question: a poset with Pieri-type add-box and remove-box operators admits dual (filtered) graph structure precisely when those operators satisfy the classical commutation relations (Willigenburg, 2019).