Stanley symmetric functions

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#Stanley symmetric functions

The Stanley symmetric functions were introduced by R. Stanley [Sta84]. They are used for studying the number of reduced words of permutations. P. Edelman and C. Greene later proved Schur positivity by giving a tableau model for the Schur expansion [EG87]. E. Marberg constructs bialgebras for Stanley symmetric functions and several analogues in other types [Mar20].

#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 \[\stanleySym_\omega(\xvec) \coloneqq \sum_{a \in Red(\omega)} \sum_{i \in I(a)} x_{i_1}\dotsm x_{i_{\ell(\omega)}}.\]

From this definition, it is fairly easy to obtain the expansion in the Gessel fundamental basis. \[\stanleySym_\omega(\xvec) \coloneqq \sum_{a \in Red(\omega)} \gessel_{n,\DES(i_1i_2 \dotsc i_\ell)}\] 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. S. Assaf and D. Searles relate Schubert polynomials, slide polynomials, Stanley symmetric functions, and quasi-Yamanouchi pipe dreams [AS17].

#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.

R. Stanley proved the following formula [Sta84]: \[\stanleySym_\omega(x) = \sum_{\omega = v_1 \dotsm v_r } x^{\ell(v_1)} \dotsm x^{\ell(v_r)}.\]

#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 \[1^k \times \omega \coloneqq 1,2,3,\dotsc,k,\omega_1+k,\omega_2+k,\dotsc,\omega_n+k.\]

We have \[\stanleySym_\omega(x) = \lim_{k\to \infty} \schubert_{1^k \times \omega}(x).\]

The function $\stanleySym_\omega(x)$ is homogeneous of degree equal to the number of inversions of $\omega.$ There is also a recursive definition.

#Frobenius image of Specht modules

There is a generalization of Specht modules, indexed by diagrams $D$ and denoted $S^D$; see [BP14]. The Frobenius image of $S^D$ is denoted $\schurS_D.$ For Ferrers diagrams $D,$ this recovers the irreducible Specht modules. Given a permutation $\omega,$ $D(\omega)$ is the associated Rothe diagram: \[D(\omega) \coloneqq \{ (i, \omega_j) : 1\leq i \lt j \leq n, \omega_i \gt{} \omega_j \}\] 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 $D.$

P. Diaconis and J. Fulman study Foulkes characters, Eulerian idempotents, and related transition matrices [DF12].

#Schur expansion

Let $EG(\omega)$ be the set of semistandard tableaux whose column reading word (reading columns left to right, bottom to top) is a reduced word for $\omega.$ Then \[\stanleySym_\omega = \sum_{T \in EG(\omega)} \schurS_{sh(T)^t}.\] This is proved using the Edelman–Greene correspondence, see [EG87]. Billey and Pawlowski study pattern classes controlling the number of Schur terms in Stanley symmetric functions [BP14].

There is also a crystal structure; see [MS15] for a proof of Schur positivity.

#Type B/C Stanley symmetric functions

S. Fomin and A. N. Kirillov introduced the type $B$ and $C$ Stanley symmetric functions together with type $B$ and $C$ analogues of Schubert polynomials [FK96].

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 \[s_i s_j =s_j s_i \text{ if } |i-j|\gt{}1, \quad s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1} \text{ if } i\gt{}1, \text{ and } s_0 s_{1} s_0 s_1 = s_1 s_0 s_{1} s_0.\] 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 nonzero 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 $zero(\omega)$ counts 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, together with a combinatorial interpretation of the coefficients, is given in [HPS17].

T. Lam shows that the type $B$ Stanley symmetric functions are Schur $P$-function positive [Lam96], which is a stronger statement.

#Double Stanley symmetric functions

G. Hawkes introduces an interpolation of the type $A$ and type $C$ Stanley symmetric functions [Haw20]. He shows that his double Stanley 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 T. Lam in [Lam06]. The family $\stanleySymAffine_\omega(\xvec)$ is indexed by affine permutations $\omega \in \asymS_n.$ Whenever $\omega \in \symS_n,$ they agree with the classical Stanley symmetric function $\stanleySym_\omega(\xvec).$ S. Pon extends affine Stanley symmetric functions to the other classical affine types [Pon12].

#Skew affine Stanley symmetric functions

There is a skew version of affine Stanley symmetric functions, where \[\stanleySymAffine_{w/v}(\xvec) = \stanleySymAffine_{wv^{-1}}(\xvec).\] Thus the skew affine Stanley symmetric functions contain the ordinary affine Stanley symmetric functions as the special case $v=e.$

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

#Coproduct

T. Lam proves that \[\stanleySymAffine_w(x_1,y_1,x_2,y_2,\dotsc) = \sum_{uv=w} \stanleySymAffine_u(\xvec)\stanleySymAffine_v(\yvec).\]

#Schur expansion

Affine Stanley symmetric functions are not Schur-positive in general. However, J. Morse and A. Schilling prove that the coefficient of $\schurS_\mu$ in an affine Stanley symmetric function is a nonnegative affine Littlewood–Richardson coefficient whenever $\mu \subseteq (n-r)^r,$ for $1 \leq r \lt n.$ Their crystal gives highest-weight formulas for these coefficients in several cases [Prop. 5.9 and Thm. 5.10, MS15].

#Affine Schur expansion

Conjecture (See [Lam06]).

For $w\in \asymS_n,$ the expansion in the affine Schur functions \[\stanleySymAffine_w(\xvec) = \sum_{\lambda} a_{w\lambda} \stanleySymAffine_\lambda(\xvec)\] is nonnegative.

This result would 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 Grassmann permutation, corresponding to $\lambda = \lambda(\omega),$ and define $\stanleySymAffine_\lambda(\xvec) \coloneqq \stanleySymAffine_\omega(\xvec).$ These are dual to the $k$-Schur functions, see [MS15, LM08], and are thus sometimes referred to as dual $k$-Schur functions. C. Berg, F. Saliola, and L. Serrano study Pieri operators on the affine nilCoxeter algebra and use them to prove Lam–Lapointe–Morse–Shimozono conjectures about strong Schur functions [BSS13]. In particular, the strong Schur functions are symmetric and lie in $\spaceSym^{(k)}$; see [Thms. 5.2 and 5.4, BSS13]. For the purposes of this page we therefore treat them as part of the same affine Schur and $k$-Schur picture, rather than as a separate family node. T. Ikeda, S. Iwao, and M. Shimozono introduce dual affine Schur $P$-functions in their study of the equivariant homology of the symplectic affine Grassmannian [IIS25]. These functions play the type $C$ analogue of the dual affine Schur functions in the affine Grassmannian Schur-basis story.

#Pieri operators

We recall one useful operator model. Set $k=n-1$ and let $\mathcal{A}_k$ be the affine nilCoxeter algebra with generators $u_0,u_1,\dotsc,u_k.$ For a proper subset $D\subset \{0,1,\dotsc,k\},$ let $u_D$ denote the nilCoxeter element corresponding to the cyclically decreasing word with letter set $D,$ and define \[h_r \coloneqq \sum_{\substack{D\subsetneq \{0,1,\dotsc,k\}\\ |D|=r}} u_D.\] The elements $h_0,h_1,\dotsc,h_k$ commute and generate the affine Fomin–Stanley subalgebra. Under the isomorphism $\completeH_r\mapsto h_r,$ the image of the $k$-Schur function $\kSchur^{(k)}_\lambda$ is the noncommutative $k$-Schur element $\mathbf{s}^{(k)}_\lambda.$

Let $D_i$ be the down operator obtained by summing over strong strips of length $i$ in the marked strong-order graph. Berg–Saliola–Serrano prove [Thm. 4.7, BSS13] that \[D_i\bigl(\mathbf{s}^{(k)}_\lambda\bigr) = \sum_{\substack{\mu\\ \operatorname{size}(w_\lambda \rightsquigarrow w_\mu)=i}} \mathbf{s}^{(k)}_\mu,\] where $w_\lambda$ is the $0$-Grassmannian affine permutation corresponding to $\lambda,$ and the sum is over strong strips from $w_\lambda$ to $w_\mu.$ More generally, for a composition $J,$ the restriction of the operator $D_J$ to the affine Fomin–Stanley subalgebra is the adjoint $\schurS_J^\perp$ of multiplication by the ribbon Schur function $\schurS_J$ [Thm. 4.9, BSS13].

Example

For $k=2,$ the affine nilCoxeter algebra has generators $u_0,u_1,u_2,$ with indices read modulo $3.$ The first two affine homogeneous elements are \[h_1=u_0+u_1+u_2, \qquad h_2=u_1u_0+u_2u_1+u_0u_2.\] Thus left multiplication by $h_r$ gives \[U_1(u_0)=h_1u_0=u_2u_0+u_1u_0\] and \[U_2(u_0)=h_2u_0=u_0u_2u_0+u_2u_1u_0,\] using $u_i^2=0.$ This is the small computation in [Ex. 3.1, BSS13].

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

Following Lam’s definition [Sec. 5, Lam06], let $h_r(u)$ be the affine homogeneous operator acting on the set of $n$-cores by the nilCoxeter action. For two $n$-cores $\mu\subseteq \lambda,$ the skew affine Schur function is \[\stanleySymAffine_{\lambda/\mu}(\xvec) = \sum_{a=(a_1,\dotsc,a_t)} \bigl\langle h_{a_t}(u)\dotsm h_{a_1}(u)\cdot\mu,\lambda\bigr\rangle x_1^{a_1}\dotsm x_t^{a_t},\] where the sum is over weak compositions with finite support, written without trailing zero parts, and with $0\leq a_i\lt{}n.$ Here the inner product is the standard one on the free vector space spanned by $n$-cores, so the coefficient counts the ways to obtain $\lambda$ from $\mu$ by applying the indicated affine Pieri operators.

Let $\mu \subseteq \lambda$ be two $n$-cores such that some $w \in \asymS_n$ satisfies $u_w \cdot \mu = \lambda.$ This means that $\lambda$ can be obtained from $\mu$ under the $\asymS_n$-action given by $u_w.$ Lam gives a formula of the form \[\stanleySymAffine_{\lambda/\mu}(\xvec) = \sum_{T} \xvec^{w(T)}\] where the sum is over certain $k$-tableaux of shape $\lambda/\mu$ [Lam06]. He also shows 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

L. Lapointe and J. Morse expand the skew affine Schur functions in terms of affine Schur functions [Thm. 6.9, LM08].

#Involution Stanley symmetric functions

Z. Hamaker, E. Marberg, and B. Pawlowski introduced the involution Stanley symmetric functions as the stable limit of the involution Schubert polynomials [HMP17].

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 [Cor. 4.37, HMP17]).

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

#Affine involution Stanley symmetric functions

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

#Affine fixed-point free Stanley symmetric functions

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

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}

[Zha19]Yifeng Zhang. Quasiparabolic sets and Stanley symmetric functions for affine fixed-point-free involutions. arXiv:1911.05961, 2019.

.bib

@article{Zhang2019,
Author = {Yifeng Zhang},
Title = {Quasiparabolic sets and {S}tanley symmetric functions for affine fixed-point-free involutions},
Year = {2019},
Eprint = {1911.05961},
  url = {https://arxiv.org/abs/1911.05961},
journal = {arXiv e-prints}
}