The symmetric functions catalog

An overview of symmetric functions and related topics

2020-04-17

Stanley symmetric functions

The Stanley symmetric functions were introduced by R. Stanley in [Sta84]. They are used for studying the number of reduced words of permutations. It was later proved that these symmetric functions are Schur positive — and now there are many different proofs of this result.

Reduced word definition

Let $\omega \in \symS_n,$ and let $Red(\omega)$ be the set of reduced words of $\omega.$ Given a reduced word $a,$ let $I(a)$ be the set of integer sequences $1 \leq i_1 \leq i_2 \leq \dotsb \leq i_{\ell(\omega)}$ such that $a_j \lt a_{j+1}$ implies $i_j \lt i_{j+1}.$ Then the Stanley symmetric functions $\stanleySym_\omega(\xvec)$ are defined as

\begin{equation*} \stanleySym_\omega(\xvec) \coloneqq \sum_{a \in Red(\omega)} \sum_{i \in I(a)} x_{i_1}\dotsm x_{i_{\ell(\omega)}}. \end{equation*}

From this definition, it is fairly easy to obtain the expansion in the Gessel fundamental basis.

\begin{equation*} \stanleySym_\omega(\xvec) \coloneqq \sum_{a \in Red(\omega)} \gessel_{n,\DES(i_1i_2 \dotsc i_\ell))} \end{equation*}

where $a = s_{i_\ell} s_{i_{\ell-1}}\dotsb s_{i_1}.$ Using the slinky rule, this formula provides a way to compute the Schur expansion of these symmetric functions.

Decreasing factorization definition

A permutation is decreasing if it admits a reduced word $a_1 \dotsc a_\ell$ with $a_1 \gt \dotsb \gt a_{\ell}.$ If such a word exists, it is unique. A decreasing factorization of a permutation $\omega \in \symS_n$ is an expression of the form $\omega = v_1 \dotsm v_r$ where each $v_i$ is a decreasing permutation.

In [Sta84], the following formula was proved:

\begin{equation*} \stanleySym_\omega(x) = \sum_{\omega = v_1 \dotsm v_r } x^{\ell(v_1)} \dotsm x^{\ell(v_r)}. \end{equation*}

Stable limit definition

The Stanley symmetric functions can also be defined via the stable limit of the Schubert polynomials. Given a permutation $\omega \in \symS_n,$ we let

\begin{equation*} 1^k \times \omega \coloneqq 1,2,3,\dotsc,k,\omega_1+k,\omega_2+k,\dotsc,\omega_n+k. \end{equation*}

We have

\begin{equation*} \stanleySym_\omega(x) = \lim_{k\to \infty} \schubert_{1^k \times \omega}(x). \end{equation*}

The function $\stanleySym_\omega(x)$ is homogeneous with degree given by the number of inverions of $\omega.$ There is also a recursive definition.

Frobenius image of Specht modules

There is a generalization of Specht modules, indexed by diagrams $D,$ denoted $S^D,$ see [BP14]. The Frobenius image of $S^D$ is denoted $\schurS_D,$ where we obtain the irreducible Specht modules whenever $D$ is a Ferrers diagram. Given a permutation $\omega,$ $D(\omega)$ is the associated Rothe diagram:

\begin{equation*} D(\omega) \coloneqq \{ (i, \omega_j) : 1\leq i \lt j \leq n, \omega_i > \omega_j \} \end{equation*}

For a general diagram $D$ and a filling $T$ of $D,$ let

$y_T = \sum_{\substack{\sigma \in R(D) \\ \tau \in C(D)}} \sign(\tau) \tau \sigma \qquad \in \setC[\symS_n]$

where $R(D)$ is the set of permutations in $\symS_n$ permuting entries within rows of $D,$ and $C(D)$ is similar but for columns. The Specht module of $D$ is then $\setC[\symS_n]y_T,$ and we let $\schurS_D$ be its Frobenius image.

Then $\stanleySym_{\omega}(x) = \schurS_{D(\omega)}(x),$ which implies that $\stanleySym_{\omega}$ is Schur positive.

Problem (See [Liu10]).

Find a combinatorial description of the decomposition of $\schurS_D$ into Schur polynomials.

Liu's conjecture, [Liu10] states that the coefficients of the Schur expansion of $\schurS_D$ are the same as certain coefficients appearing when studying cohomology classes of Schubert varieties defined by $D.$

Schur expansion

Let $EG(\omega)$ be the set of semi-standard tableaux whose column reading word (reading columns left to right, bottom to top) is a reduced word for $\omega.$ Then

\begin{equation*} \stanleySym_\omega = \sum_{T \in EG(\omega)} \schurS_{sh(T)^t}. \end{equation*}

This is proved using the Edelman–Greene correspondence, see [EG87].

There is also a crystal structure, see [MS15c] proving Schur positivity.

Type B/C Stanley symmetric functions

The type $B$ and $C$ Stanley symmetric functions were introduced in [FK96a], where the authors also introduce type $B$ and $C$ analogs of Schubert polynomials.

Let $W_C$ be the type $B_n/C_n$ Coxeter group, consisting of signed permutations. The group $W_C$ is generated by $s_0,\dotsc,s_{n-1}$ subject to

$s_i s_j =s_j s_i \text{ if } |i-j|>1, \quad s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1} \text{ if } i>1, \text{ and } s_0 s_{1} s_0 s_1 = s_1 s_0 s_{1} s_0.$

We have the notion of reduced words of generators. A reduced word $w_1,w_2,\dotsc,w_k$ is unimodal if $w_1 \lt w_2 \lt \dotsb \lt w_j \gt \dotsb \gt w_k$ for some $j.$ A unimodal factorization of a reduced word $w$ is a factorization

$\omega = (w_1,\dotsc w_{\ell_1})(w_{\ell_1+1},\dotsc w_{\ell_2}) \dotsm (w_{\ell_{m-1}+1},\dotsc w_{\ell_m})$

where each factor is unimodal. Factors can be empty. Let $UF(\omega)$ be the set of unimodal factorizations of $\omega.$ Given such a factorization $F,$ let $w(F)$ be the vector of the number of elements in each factor, and let $nz(F)$ be the number of non-zero factors.

Given $\omega \in W_C,$ the type $C$ Stanley symmetric function is defined as

$\stanleySym^C_\omega(\xvec) = \sum_{F \in UF(\omega)} 2^{nz(F)}\xvec^{w(F)}.$

and the type $B$ Stanley symmetric function is defined as

$\stanleySym^B_\omega(\xvec) = 2^{-zero(\omega)} \stanleySym^C_\omega(\xvec)$

where $zero(\omega)$ count the number of zeros in a reduced word for $\omega.$

Schur expansion

The type $B$ and $C$ Stanley symmetric functions are Schur positive, a crystal proof is given in [HPS17], where a combinatorial interpretation of the coefficients are given.

In this paper by T. Lam, it is shown that the type $B$ Stanley symmetric functions are Schur's $P$-function positive, which is a stronger statement.

Double Stanley symmetric functions

In [Haw18], an interpolaton of the type $A$ and type $C$ Stanley symmetric functions is introduced. Hawkes show that his symmetric functions, $\stanleySym_\omega(\xvec;\yvec)$ are Schur-positive for all $\omega \in A_n,$ by a version of the Edelman–Greene insertion algorithm.

Affine Stanley symmetric functions

The affine Stanley symmetric functions were introduced by Thomas Lam in [Lam06a]. The family $\stanleySymAffine_\omega(\xvec)$ are indexed by affine permutations $\omega \in \asymS_n.$ Whenevere $\omega \in \symS_n,$ they agree with the classical Stanley symmetric function $\stanleySym_\omega(\xvec).$

There is a "skew" version of affine Stanley symmetric functions, where

$\stanleySymAffine_{w/v}(\xvec) = \stanleySymAffine_{wv^{-1}}(\xvec).$

In other words, the family of affine Stanley symmetric functions contain these skew versions.

The family of affine Stanley symmetric functions contains the cylindrical Schur functions.

Coproduct

It is proved in [Lam06a] that

$\stanleySymAffine_w(x_1,y_1,x_2,y_2,\dotsc) = \sum_{uv=w} \stanleySymAffine_u(\xvec)\stanleySymAffine_v(\yvec).$

Schur expansion

In [MS15c], the authors give a crystal proof that the (some?) affine Stanley symmetric functions are Schur positive.

Affine Schur expansion

Conjecture (See [Lam06a]).

For $w\in \asymS_n,$ the expansion in the affine Schur functions

$\stanleySymAffine_w(\xvec) = \sum_{\lambda} a_{w\lambda} \stanleySymAffine_\lambda(\xvec)$

is non-negative.

This result be analogous to the Schur expansion of Stanley symmetric functions.

Affine Schur functions

The affine Schur functions are obtained as a special case of the affine Stanley symmetric functions.

Let $\omega$ be an affine Grassman permutation, corresponding to $\lambda = \lambda(\omega),$ we let $\stanleySymAffine_\lambda(\xvec) \coloneqq \stanleySymAffine_\omega(\xvec).$ These are called the affine Schur functions. These are dual to the $k$-Schur functions, see , and are thus sometimes referred to as dual $k$-Schur functions.

Let $Par^n$ denote the set of partitions with $\lambda_1 \leq n-1.$

$\{ \stanleySymAffine_{\lambda}(\xvec) : \lambda \in Par^n \}$

form a basis for the space $\spaceSym^{(n)}$ where

$\spaceSym^{(n)} \coloneqq\{ \monomial_{\lambda}(\xvec) : \lambda \in Par^n \}, \qquad \spaceSym_{(n)} \coloneqq\{ \completeH_{\lambda}(\xvec) : \lambda \in Par^n \}.$

Skew affine Schur functions

Let $\mu \subseteq \lambda$ be two $n$-cores such that there is some $w \in \asymS_n$ such that $u_w \cdot \mu = \lambda.$ This means that $\lambda$ can be obtained from $\mu$ under a certain $\asymS_n$-action given by $u_w.$ Then [Lam06a], a formula of the form

$\stanleySymAffine_{\lambda/\mu}(\xvec) = \sum_{T} \xvec^{w(T)}$

is presented, where the sum is over certain $k$-tableaux of shape $\lambda/\mu,$ and it is shown that whenever $\lambda \subseteq ((n-m)^m)$ for some $1\leq m\leq n-1,$ then $\stanleySymAffine_{\lambda}(\xvec) = \schurS_\lambda(\xvec).$

The family of skew affine Schur functions contains the cylindrical Schur functions.

Remark.

Note that some skew affine Schur functions are not obtained as (skew) affine Stanley symmetric functions.

Expansion in affine Schur functions

See [Theorem 6.9, LM08], which expands the skew affine Schur functions in terms of affine Schur functions.

Involution Stanley symmetric functions

The involution Stanley symmetric functions were introduced in [HMP17], as the stable limit of the involution Schubert polynomials.

The involution Stanley symmetric functions are by definition a positive linear combination of the usual Stanley symmetric functions, and are therefore Schur positive.

Theorem (See [Corollary 4.37, HMP17]).

The involution Stanley symmetric functions are $P$-Schur positive.

Affine involution Stanley symmetric functions

There is a unification of the involution Stanley symmetric functions and the affine Stanley symmetric functions, introduced by E. Marberg and Y. Zhang 2018 in [MZ18]. It is expected that these are related to geometry of affine analogues of certain symmetric varieties.

Affine fixed-point free Stanley symmetric functions

In [MZ18], the authors also introduced the fixed-point free Stanley symmetric functions which are indexed by fixed-point-free permutations. An affine extension is introduced by Zhang [Zha19]. These are indexed by self-inverse permutations without fixed-points.