---
title: Double Stanley Symmetric Functions
url: https://www.emergentmind.com/topics/double-stanley-symmetric-functions
type: topic
---

# Double Stanley Symmetric Functions

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 [1809.04433].

## 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 $F^C_\omega(\mathbf{x})$ [1809.04433].

## 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 [1809.04433].

## 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 $RSIF_k(\omega)$ [1809.04433].

## 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) [1809.04433].

## 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 [1809.04433].

## 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 [1809.04433].

## 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 [1809.04433].

Source: https://www.emergentmind.com/topics/double-stanley-symmetric-functions