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 [MS15cLM08], 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.

## References

- [BP14] Sara C. Billey and Brendan Pawlowski. Permutation patterns, Stanley symmetric functions, and generalized Specht modules. Journal of Combinatorial Theory, Series A, 127:85–120, September 2014.
- [EG87] Paul Edelman and Curtis Greene. Balanced tableaux. Advances in Mathematics, 63(1):42–99, January 1987.
- [FK96a] Sergey V. Fomin and Anatol N. Kirillov. Combinatorial $B_N$-analogues of Schubert polynomials. Transactions of the American Mathematical Society, 348(09):3591–3621, September 1996.
- [Haw18] Graham Hawkes. Symmetric function theory at the border of An and Cn. arXiv e-prints, 2018.
- [HMP17] Zachary Hamaker, Eric Marberg and Brendan Pawlowski. Schur P-positivity and involution Stanley symmetric functions. International Mathematics Research Notices, November 2017.
- [HPS17] Graham Hawkes, Kirill Paramonov and Anne Schilling. Crystal analysis of type C Stanley symmetric functions. The Electronic Journal of Combinatorics, 24(3), September 2017.
- [Lam06a] Thomas Lam. Affine Stanley symmetric functions. American Journal of Mathematics, 128(6):1553–1586, 2006.
- [Liu10] Ricky Ini Liu. Specht modules and Schubert varieties for general diagrams. Massachusetts Institute of Technology. 2010.
- [LM08] Luc Lapointe and Jennifer Morse. Quantum cohomology and the k-Schur basis. Transactions of the American Mathematical Society, 360(4):2021–2040, 2008.
- [MS15c] Jennifer Morse and Anne Schilling. Crystal approach to affine Schubert calculus. International Mathematics Research Notices, 2016(8):2239–2294, July 2015.
- [MZ18] Eric Marberg and Yifeng Zhang. Affine transitions for involution Stanley symmetric functions. arXiv e-prints, 2018.
- [Sta84] Richard P. Stanley. On the number of reduced decompositions of elements of Coxeter groups. European Journal of Combinatorics, 5(4):359–372, December 1984.
- [Zha19] Yifeng Zhang. Quasiparabolic sets and Stanley symmetric functions for affine fixed-point-free involutions. arXiv e-prints, 2019.