Bumpless Pipedreams in Algebraic Combinatorics

• Bumpless pipedreams are combinatorial models in symmetric function theory that generalize classical Stanley symmetric functions using double weight enumerations and bijective tableau correspondences.
• They employ reduced signed increasing factorizations and primed tableaux to connect type A and type C Stanley symmetric functions, yielding explicit Schur function expansions.
• The framework integrates Coxeter group theory, tableau conversions, and crystal operators to offer new insights and conjectural extensions in algebraic combinatorics.

Bumpless pipedreams, in the context of symmetric function theory and representation-theoretic combinatorics, refer to combinatorial models that generalize the classical Stanley symmetric functions and their connections to crystal structures, tableaux, and algebraic identities. In particular, bumpless pipedreams facilitate the combinatorial description of double Stanley symmetric functions, which interpolate between the type $A_n$ and type $C_n$ Stanley symmetric functions via specializations in two sets of variables. These constructions interface Coxeter group theory, tableau combinatorics, and the structure of algebraic symmetric functions in a unified framework (Hawkes, 2018).

1. Coxeter Group Framework and Factorizations

The foundation utilizes Coxeter groups: let $A_n$ denote the group generated by $s_1, \dots, s_n$ with braid and commutation relations; $C_{n+1}$ is generated by $s_0, s_1, ..., s_n$ with $(s_0s_1)^4=1$, $(s_is_{i+1})^3=1$ for $i\geq1$, and $(s_is_j)^2=1$ otherwise. A “reduced signed increasing factorization” ($\mathrm{RSIF}_k(\omega)$) of $\omega\in C_{n+1}$ into $k$ parts is a reduced word $u=u_1u_2\cdots u_\ell$ (with letters in $\{s_{-n},…,s_{-1},s_0,s_1,…,s_n\}$), subdivided into $k$ contiguous factors $\mathbf{v}=(v^{(1)})(v^{(2)})\cdots(v^{(k)})$ such that each $v^{(i)}$ is strictly increasing under $s_{-n}<…<s_{-1}<s_0<s_1<…<s_n$.

Define the double-weight of $v$ by

$$dw(v,1)=(\,\#\{\text{bars in }v^{(1)}\},\dots,\#\{\text{bars in }v^{(k)}\}),$$

$$dw(v,2)=(\,\#\{\text{non-negative-index letters in }v^{(1)}\},\dots,\#\{\text{non-negative-index letters in }v^{(k)}\})$$

where “bars” indicate generators $s_{-i}$ $(i>0)$. The double Stanley symmetric polynomial for $\omega$ in variables $x=(x_1,…,x_k)$ and $y=(y_1,…,y_k)$ is

$$F^d_\omega(x,y) = \sum_{v\in RSIF_k(\omega)} x^{dw(v,1)} y^{dw(v,2)}.$$

Letting $k\to\infty$ yields the formal power series $$F^d_\omega(\mathbf{x}, \mathbf{y}) \in \mathbb{Q}[[x_1, x_2, ...; y_1, y_2, ...]]$$.

2. Specialization to Stanley Symmetric Functions

The double Stanley symmetric function encapsulates established symmetric functions as special cases. If $\omega\in A_n$, restricting to reduced increasing factorizations with no barred letters retrieves the type $A$ Stanley symmetric function $F^A_\omega(\mathbf{x})$. If $\omega\in C_{n+1}$, removing bar-signs and weighting each nonempty factor by $2$ gives the type $C$ symmetric function $F^C_\omega(\mathbf{x})$. This specialization is formalized: $$F^d_\omega(\mathbf{0}, \mathbf{x}) = F^A_\omega(\mathbf{x}), \qquad F^d_\omega(\mathbf{x}, \mathbf{x}) = F^C_\omega(\mathbf{x})$$ for $\omega\in A_n\subset C_{n+1}$ (Hawkes, 2018).

3. Primed Tableaux and Expansion Formulas

To connect reduced signed increasing factorizations to tableau combinatorics, intermediate generating functions are introduced via primed and barred entries in skew tableaux. For fixed $\mu\subset\lambda$, $k\in\mathbb{N}$, vectors $X, Y \in \mathbb{Z}_{\geq 0}^k$, and $0\leq j\leq k$, a primed-signed tableau $T$ of shape $\lambda/\mu$ is filled from

$$\bar X_k' = \{ \bar k <\dots< \bar2 <\bar1 < 1' <1<2' <2< \cdots<k'<k \}$$

with rules enforcing weakly increasing order, limited markings per row and column, and prescribed barred/primed/unmarked counts. The double-weight $(X, Y)$ associates powers in the generating function $R_{\lambda/\mu}(x,y)$.

For $\omega\in A_n$, a mixed Edelman–Greene insertion yields a bijection between $\mathrm{RSIF}_k(\omega)$ and pairs $(P,Q)$ with $P$ an Edelman–Greene tableau of shape $\operatorname{sh}(P)$ and $Q$ a primed tableau of matching shape. This leads to the expansion: $$F^d_\omega(\mathbf{x}, \mathbf{y}) = \sum_{P \in E(\omega)} R_{\operatorname{sh}(P)}(\mathbf{x}, \mathbf{y})$$ where $E(\omega)$ is the set of Edelman–Greene increasing tableaux whose row-reading word is a reduced word for $\omega$.

4. A Type A Bicrystal Structure and Schur Product Formulas

Primed tableaux possess an $A_{k−1}\times A_{k−1}$ “bicrystal” structure: crystal operators $f_i, e_i$ handle unprimed letters, while $f_{\bar i}, e_{\bar i}$ handle primed letters. These operators satisfy commuting relations and define a bicrystal. By bicrystal theory,

$$F^d_\omega(\mathbf{x}, \mathbf{y}) = \sum_{P\in E(\omega)} \sum_{Q\in PT_k(\operatorname{sh}(P)), Q \text{ highest-weight in both crystals}} s_{dw(Q,1)}(\mathbf{x})\ s_{dw(Q,2)}(\mathbf{y})$$

where $s_\lambda(\mathbf{x})$ and $s_\lambda(\mathbf{y})$ are Schur polynomials.

For $\omega = 121 \in A_2$, the corresponding expansion involves highest-weight primed tableaux of shape $(2,1)$, explicitly enumerating the case and yielding a summation over products of Schur functions in both alphabets.

5. Tableaux Conversions and Algebraic Relationships

Inward and outward conversion algorithms allow the translation between primed and signed tableaux. A local swap repeatedly replaces primed $j$'s with barred $j$'s or vice versa, establishing a bijection $$PST(\lambda/\mu; X,Y; j) \leftrightarrow PST(\lambda/\mu; X,Y; j-1)$$, weight-preserving. Specifically, primed tableaux $PT_k(\lambda/\mu)$ correspond to signed tableaux $ST_k(\lambda/\mu)$, and

$$R_{\lambda/\mu}(x,y) = \sum_{T\in ST_k(\lambda/\mu)} x^{dw(T,1)} y^{dw(T,2)} = s_{\lambda/\mu}(\mathbf{x}/\mathbf{y})$$

i.e., the skew Schur function in the difference $\mathbf{x}/\mathbf{y}$.

The algebraic relationships derived include

$$F^d_\omega(\mathbf{x}, \mathbf{y}) = \sum_{P\in E(\omega)} s_{\operatorname{sh}(P)}(\mathbf{x}/\mathbf{y}) = F^A_\omega(\mathbf{x}/\mathbf{y}),$$

$$F^d_\omega(\mathbf{0}, \mathbf{x}) = F^A_\omega(\mathbf{x}), \quad F^d_\omega(\mathbf{x}, \mathbf{x}) = F^C_\omega(\mathbf{x}), \quad F^d_\omega(\mathbf{x}, \mathbf{y}) = F^d_{\omega^{-1}}(\mathbf{y}, \mathbf{x})$$

for $\omega\in A_n$.

6. Conjectural Extensions in Type $C$

For general signed permutations in $C_{n+1}$, “unknotted” elements are defined as those whose reduced words avoid certain forbidden patterns ($s_0s_1s_0s_1\dots s_2$ and $s_is_{i+1}s_i\dots s_{i+2}$). For unknotted $\omega$, analogs of EG-tableaux (signed Edelman–Greene tableaux) can be defined, and the following conjectures are proposed:

• For unknotted $\omega$,

$$F^d_\omega(\mathbf{x}, \mathbf{x}) = \sum_\lambda \bar E^\lambda_\omega\, s_\lambda(\mathbf{x}).$$

• If in every reduced word for $\omega$, at most one $s_0$ occurs,

$$F^d_\omega(\mathbf{x}, -\mathbf{x}) = \sum_{r\text{ even}} \sum_\lambda \bar E^{\lambda,r}_\omega s_\lambda(\mathbf{x}) - \sum_{r\text{ odd}} \sum_\lambda \bar E^{\lambda,r}_\omega s_\lambda(\mathbf{x})$$

where $\bar E^{\lambda,r}_\omega$ counts signed EG-tableaux of shape $\lambda$ with $r$ barred entries.

• For $\omega\in A_n$ (so no $s_0$ at all),

$$F^d_\omega(\mathbf{x}, t\mathbf{x}) = \sum_{r,\lambda} \bar E^{\lambda,r}_\omega s_\lambda(\mathbf{x}) t^r$$

with verification for small unknotted $\omega$.

7. Context and Future Directions

The bumpless pipedreams constructions for double Stanley symmetric functions unify combinatorial models for type $A$ and type $C$ symmetric functions, crystal-theoretic perspectives, and Schur function expansions. The bicrystal structure and tableau expansions enable new algebraic relationships and conjectural ties to broader symmetric function families. Further research directions include establishing the conjectures for general type $C$ permutations, explicit realization of crystal operators on signed tableaux, and deeper connections to Schubert calculus and other representation-theoretic domains (Hawkes, 2018).