Double Stanley Symmetric Functions

• Double Stanley symmetric functions are a two-parameter generalization that unifies type A and type C Stanley functions via a bivariate combinatorial framework.
• They employ a bicrystal structure with primed-tableau insertion methods to encode reduced signed increasing factorizations, extending classical Schubert calculus techniques.
• Their Schur function expansions reveal rich algebraic properties and motivate new enumeration problems and conjectures related to unknotted signed permutations.

Double Stanley symmetric functions are a two-parameter generalization that interpolates between the type $A$ and type $C$ Stanley symmetric functions via a bivariate combinatorial framework. Defined over two separate alphabets, these functions unify the classical Schubert calculus approaches for types $A_n$ and $C_{n+1}$ by encoding the combinatorics of reduced signed increasing factorizations in a single generating series. Specializing to one or the other alphabet recovers the respective Stanley symmetric functions, while the mixed setting introduces novel representation-theoretic and crystal structures with deep algebraic consequences (Hawkes, 2018).

1. Stanley Symmetric Functions of Types A and C

Let $k$ be a positive integer, and consider two alphabets $x=(x_1,\dotsc,x_k)$ and $y=(y_1,\dotsc,y_k)$, with the infinite-variable limits denoted by $\mathbf{x}$ and $\mathbf{y}$. Classical Stanley symmetric functions arise from enumerating certain reduced factorizations of permutations:

• Type A: For $\omega\in A_n$, a reduced increasing factorization into $k$ parts is a reduced word split into $k$ strictly increasing blocks. For $v\in RIF_k(\omega)$ with weights $\wt(v)=(w_1,\dotsc,w_k)$, the degree-$k$ Stanley polynomial is

$$F^A_\omega(x)=\sum_{v\in RIF_k(\omega)} x_1^{w_1}\dotsb x_k^{w_k}.$$

Taking $k\to\infty$ yields $F^A_\omega(\mathbf{x})$.

• Type C: For $\omega\in C_{n+1}$, unimodal reduced factorizations (blocks first decreasing, then increasing) are considered. The degree-$k$ polynomial is

$F^C_\omega(x) = \sum_{v\in RUF_k(\omega)} 2^{ne(v)} x^{\wt(v)},$

where $ne(v)$ is the count of nonempty factors. The infinite-variable version is $F^C_\omega(\mathbf{x})$ (Hawkes, 2018).

2. Definition and Combinatorics of Double Stanley Symmetric Functions

The double Stanley symmetric function arises by simultaneously allowing positive and negative indices in the underlying generators $s_{-n},\dotsc,s_{-1},s_0,s_1,\dotsc,s_n$ (with $s_{-i}=s_i$ as group elements). For $\omega\in C_{n+1}$, a reduced signed increasing factorization into $k$ parts is a reduced word with each block increasing under the total order $s_{-n}<\cdots<s_{-1}<s_0<s_1<\cdots<s_n$.

For $v\in RSIF_k(\omega)$, the double weight is a pair $(X,Y)$ where $X_i$ and $Y_i$ count the negative and nonnegative indices in block $i$, respectively. The double Stanley polynomial is

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

and in the infinite-variable limit, $$F^d_\omega(\mathbf{x},\mathbf{y})\in\Lambda_{\mathbf{x}}\otimes\Lambda_{\mathbf{y}}$$.

Specializations include:

Specialization	Output
$F^d_\omega(\mathbf{0}, \mathbf{x})$	$F^A_\omega(\mathbf{x})$
$F^d_\omega(\mathbf{x}, \mathbf{x})$	$F^C_\omega(\mathbf{x})$

Symmetry in the variables $(\mathbf{x},\mathbf{y})$ corresponds to the map $v\mapsto v^{-1}$ at the combinatorial level (Hawkes, 2018).

3. Bicrystal Structure and Tableaux Model

A principal insight is the identification of a bicrystal structure of type $A_{k-1}\oplus A_{k-1}$ on double Stanley symmetric functions, extending known constructions for the type $A$ and $C$ cases.

• Tableaux model: For $\omega\in A_n$, a “primed‐recording” variant of Edelman–Greene insertion constructs a bijection

$$RSIF_k(\omega)\longleftrightarrow\bigsqcup_{P\in E(\omega)}\{(P,Q)\},$$

where $P$ is an unsigned Edelman–Greene tableau and $Q$ is a primed tableau of the same shape, filled with marked and unmarked entries and satisfying weak monotonicity and column/row uniqueness conditions for markers.

The generating function over such tableau pairs recovers the double Stanley function:

$F^d_\omega(x, y) = \sum_{P\in E(\omega)}\sum_{Q\in PT_k(\sh P)} x^{\dw(Q,1)} y^{\dw(Q,2)}.$

• Crystal operators: The $A_{k-1}$ crystal and its dual are implemented via operators $f_i$, $e_i$, $f_{\bar{i}}$, $e_{\bar{i}}$, which act on specific subwords of a primed tableau according to local rewriting rules. These operators are mutual inverses on their respective sides and satisfy crystal axioms, such as

$\text{if }f_i(T)=T'\neq0\text{ then }\dw(T')=\dw(T)-(0,\alpha_i),$

and commutation relations $f_i f_{\bar{j}}=f_{\bar{j}} f_i$ for all $i,j$.

• Haiman insertion and isomorphism: Mixed (unshifted) Haiman insertion on words in the primed alphabet provides a bicrystal isomorphism from the tensor-product word crystal to the primed-tableau crystal, intertwining with the Edelman–Greene insertion and thus lifting to $RSIF_k(\omega)$ (Hawkes, 2018).

4. Algebraic Properties and Schur Function Expansions

The algebraic structure of double Stanley symmetric functions is characterized by Schur expansions over two alphabets and admits a plethystic interpretation.

• Schur expansion:

$F^d_\omega(\mathbf{x},\mathbf{y}) = \sum_{P\in E(\omega)} \sum_{S\in\mathcal{H}(\sh P)} s_{\dw(S,1)}(\mathbf{x}) s_{\dw(S,2)}(\mathbf{y}),$

where $\mathcal{H}(\lambda)$ is the set of highest-weight primed tableaux of shape $\lambda$ with both Yamanouchi property and its transpose+shift version.

• Plethystic Schur function relation: For $\lambda \vdash n$, the classical involution $\omega_{\mathbf{x}}$ on $\Lambda_{\mathbf{x}}$ yields

$$s_\lambda(\mathbf{x}/\mathbf{y}) := \omega_{\mathbf{x}}(s_\lambda(\mathbf{x}))\big|_{\mathbf{x}\mapsto (\mathbf{x},\mathbf{y})}.$$

There is a bijection between primed tableaux and signed tableaux, giving

$$R_\lambda(\mathbf{x},\mathbf{y}) = s_\lambda(\mathbf{x}/\mathbf{y}),$$

so for $\omega\in A_n$

$$F^d_\omega(\mathbf{x},\mathbf{y}) = F^A_\omega(\mathbf{x}/\mathbf{y}).$$

In particular,

$$F^C_\omega(\mathbf{x}) = F^d_\omega(\mathbf{x},\mathbf{x}) = F^A_\omega(\mathbf{x}/\mathbf{x}).$$

This recovers the relationships known in the literature (Lam ’95) (Hawkes, 2018).

5. Conjectures and Type C Generalizations

For general $\omega\in C_{n+1}$, $F^d_\omega(\mathbf{x},\mathbf{y})$ need not be symmetric. Several conjectures address expansions and special cases for the so-called unknotted signed permutations:

• Conjecture 4.1: For unknotted $\omega$, the specialization

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

holds, where $\bar E^\lambda_\omega$ counts signed Edelman–Greene tableaux of shape $\lambda$.

• Conjecture 4.2: If $\omega$ is unknotted and every reduced word contains at most one $s_0$,

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

with $\bar E^{\lambda,r}_\omega$ denoting the count of tableaux with exactly $r$ barred entries.

• Conjecture 4.3: For $\omega\in A_n\subset C_{n+1}$, for any $t$,

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

These conjectures suggest a rich interplay between the double Stanley functions and Schur expansions depending on subtle properties of signed permutations (Hawkes, 2018).

6. Worked Examples

Double Stanley symmetric functions encode nontrivial combinatorics even for small permutations. For instance:

• Example 1 ($\omega=s_1s_2s_1\in S_3$, one-line $132$):

$$F^d_{121}(\mathbf{x},\mathbf{y}) = s_{21}(\mathbf{x}) + s_2(\mathbf{x})s_1(\mathbf{y}) + s_{11}(\mathbf{x})s_1(\mathbf{y}) + s_1(\mathbf{x})s_2(\mathbf{y}) + s_1(\mathbf{x})s_{11}(\mathbf{y}) + s_{21}(\mathbf{y}).$$

The expansion simultaneously recovers the type $A$ and $C$ cases in the specializations $\mathbf{y}=0$ and $\mathbf{y}=\mathbf{x}$.

• Example 2 ($\omega=s_2s_1s_2\in S_3$, one-line $231$):

$$F^d_{212}(\mathbf{x},\mathbf{y}) = s_2(\mathbf{x}) + s_1(\mathbf{x})s_1(\mathbf{y}) + s_2(\mathbf{y}).$$

These explicit expansions demonstrate the full interplay of the two alphabets and generalize classical Schur function formulas (Hawkes, 2018).

7. Context and Significance

Double Stanley symmetric functions provide a unifying framework for the algebraic and combinatorial structures underlying types $A$ and $C$ Schubert calculus. Their bicrystal structure exposes new symmetry and representation-theoretic phenomena, and the Schur-in-two-alphabets expansions bridge plethystic and classical symmetric function theory. The conjectured relationships for unknotted signed permutations motivate new enumeration problems for signed Edelman–Greene tableaux and further generalizations in type $C$. This suggests ongoing connections to crystal theory, symmetric functions in noncommutative variables, and generalized Schubert calculus (Hawkes, 2018).