Stanley symmetric functions and Peterson algebras
These are (mostly) expository notes for lectures on affine Stanley symmetric functions given at the Fields Institute in 2010. We focus on the algebraic and combinatorial parts of the theory. The notes contain a number of exercises and open problems.
Stanley symmetric functions are a family of symmetric functions indexed by permutations. They were invented by Stanley [Sta] to enumerate the reduced words of elements of the symmetric group. The most important properties of the Stanley symmetric functions are their symmetry, established by Stanley, and their Schur positivity, first proven by Edelman and Greene [EG], and by Lascoux and Schützenberger [LSc82].
Recently, a generalization of Stanley symmetric functions to affine permutations was developed in [Lam06]. These affine Stanley symmetric functions turned out to have a natural geometric interpretation [Lam08]: they are pullbacks of the cohomology Schubert classes of the affine flag variety to the affine Grassmannian (or based loop space) under the natural map . The combinatorics of reduced words and the geometry of the affine homogeneous spaces are connected via the nilHecke ring of Kostant and Kumar [KK], together with a remarkable commutative subalgebra due to Peterson [Pet]. The symmetry of affine Stanley symmetric functions follows from the commutativity of Peterson’s subalgebra, and the positivity in terms of affine Schur functions is established via the relationship between affine Schubert calculus and quantum Schubert calculus [LS10, LL]. The affine-quantum connection was also discovered by Peterson.
The affine generalization also connects Stanley symmetric functions with the theory of Macdonald polynomials [Mac] – my own involvement in this subject began when I heard a conjecture of Mark Shimozono relating the Lapointe-Lascoux-Morse -Schur functions [LLM] to the affine Grassmannian.
While the definition of (affine) Stanley symmetric functions does not easily generalize to other (affine) Weyl groups (see [BH, BL, FK96, LSS10, Pon]), the algebraic and geometric constructions mentioned above do.
This article introduces Stanley symmetric functions and affine Stanley symmetric functions from the combinatorial and algebraic point of view. The goal is to develop the theory (with the exception of positivity) without appealing to geometric reasoning. The notes are aimed at an audience with some familiarity with symmetric functions, Young tableaux and Coxeter groups/root systems.
The first third (Sections 1 - 3) of the article centers on the combinatorics of reduced words. We discuss reduced words in the (affine) symmetric group, the definition of the (affine) Stanley symmetric functions, and introduce the Edelman-Greene correspondence. Section 4 reviews the basic notation of Weyl groups and affine Weyl groups. In Sections 5-9 we introduce and study four algebras: the nilCoxeter algebra, the Kostant-Kumar nilHecke ring, the Peterson centralizer subalgebra of the nilHecke ring, and the Fomin-Stanley subalgebra of the nilCoxeter algebra. The discussion in Section 9 is new, and is largely motivated by a conjecture (Conjecture 5.5) of the author and Postnikov. In Section 10, we give a list of geometric interpretations and references for the objects studied in the earlier sections.
We have not intended to be comprehensive, especially with regards to generalizations and variations. There are four such which we must mention:
We have included exercises and problems throughout which occasionally assume more prerequisites. The exercises vary vastly in terms of difficulty. Some exercises essentially follow from the definitions, but other problems are questions for which I do not know the answer to.
1. Stanley symmetric functions and reduced words
For an integer , let . For a partition (or composition) , we write . The dominance order on partitions is given by if for some we have for and . The descent set of a word is given by .
1.1. Young tableaux and Schur functions
We shall assume the reader has some familiarity with symmetric functions and Young tableaux [Mac] [EC2, Ch. 7]. We write for the ring of symmetric functions. We let , where is a partition, denote the monomial symmetric function, and let and , for integers , denote the homogeneous and elementary symmetric functions respectively. For a partition , we define , and similarly for . We let denote the Hall inner product of symmetric functions. Thus .
We shall draw Young diagrams in English notation. A tableau of shape is a filling of the Young diagram of with integers. A tableau is column-strict (resp. row-strict) if it is increasing along columns (resp. rows). A tableau is standard if it is column-strict and row-strict, and uses each number exactly once. A tableau is semi-standard if it is column-strict, and weakly increasing along rows. Thus the tableaux
are standard and semistandard respectively. The weight of a tableau is the composition where is equal to the number of ’s in . The Schur function is given by
where the summation is over semistandard tableaux of shape , and for a composition , we define . For a standard Young tableau we define . We also write for the number of standard Young tableaux of shape . Similar definitions hold for skew shapes .
Let be a partition. Then
1.2. Permutations and reduced words
Let denote the symmetric group of permutations on the letters . We think of permutations as bijections , so that the product is the composition as functions. The simple transposition , swaps the letters and , keeping the other letters fixed. The symmetric group is generated by the , with the relations
The length of a permutation is the length of the shortest expression for as a product of simple generators. Such a shortest expression is called a reduced expression for , and the word is a reduced word for . Let denote the set of reduced words of . We usually write permutations in one-line notation, or alternatively give reduced words. For example has reduced word .
There is a natural embedding and we will sometimes not distinguish between and its image in under this embedding.
1.3. Reduced words for the longest permutation
The longest permutation is in one-line notation. Stanley [Sta] conjectured the following formula for the number of reduced words of , which he then later proved using the theory of Stanley symmetric functions. Let denote the staircase of size .
Theorem 1.2 ([Sta]).
The number of reduced words for is equal to the number of staircase shaped standard Young tableaux.
1.4. The Stanley symmetric function
Definition 1 (Original definition).
Let . Define the Stanley symmetric function
Theorem 1.3 ([Sta]).
The generating function is a symmetric function.
A word is decreasing if . A permutation is decreasing if it has a (necessarily unique) decreasing reduced word. The identity is considered decreasing. A decreasing factorization of is an expression such that are decreasing, and .
Definition 2 (Decreasing factorizations).
Let . Then
Consider . Then . Thus
The decreasing factorizations which give are .
1.5. The code of a permutation
Let . The code is the sequence of nonnegative integers given by for , and for . Note that the code of is the same regardless of which symmetric group it is considered an element of.
Let be the partition conjugate to the partition obtained from rearranging the parts of in decreasing order.
Let . Then , and . Thus .
For a symmetric function , let denote the coefficient of in .
Proposition 1.4 ([Sta]).
Suppose . Then .
Left multiplication of by acts on by
whenever . Thus factorizing a decreasing permutation out of from the left will decrease different entries of each by . (1) follows easily from this observation.
To obtain (2), one notes that there is a unique decreasing permutation of length such that . ∎
1.6. Fundamental Quasi-symmetric functions
Let . Define the (Gessel) fundamental quasi-symmetric function by
Note that depends not just on the set but also on .
A basic fact relating Schur functions and fundamental quasi-symmetric functions is:
Let be a partition. Then
Definition 3 (Using quasi symmetric functions).
Let . Then
Continuing Example 1, we have , where all subsets are considered subsets of . Note that these are exactly the descent sets of the tableaux
Prove that is equal to .
Let , where permutations are identified under the embeddings . Prove that is a bijection between and nonnegative integer sequences with finitely many non-zero entries.
What happens if we replace decreasing factorizations by increasing factorizations in Definition 2?
What is the relationship between and ?
(Grassmannian permutations) A permutation is Grassmannian if it has at most one descent.
Characterize the codes of Grassmannian permutations.
Show that if is Grassmannian then is a Schur function.
Which Schur functions are equal to for some Grassmannian permutation ?
(321-avoiding permutations [BJS]) A permutation is 321-avoiding if there does not exist such that . Show that is 321-avoiding if and only if no reduced word contains a consecutive subsequence of the form . If is 321-avoiding, show directly from the definition that is a skew Schur function.
2. Edelman-Greene insertion
2.1. Insertion for reduced words
We now describe an insertion algorithm for reduced words, due to Edelman and Greene [EG], which establishes Theorem 1.3, and in addition stronger positivity properties. Related bijections were studied by Lascoux-Schützenberger [LSc85] and by Haiman [Hai].
Let be a column and row strict Young tableau. The reading word is the word obtained by reading the rows of from left to right, starting with the bottom row.
Let . We say that a tableau is a EG-tableau for if is a reduced word for . For example,
has reading word , and is an EG-tableau for .
Theorem 2.1 ([Eg]).
Let . There is a bijection between and the set of pairs , where is an EG-tableau for , and is a standard Young tableau with the same shape as . Furthermore, under the bijection we have .
Let . Then , where is equal to the number of EG-tableau for . In particular, is Schur positive.
As a consequence we obtain Theorem 1.3.
Suppose is an EG-tableau for . Then the shape of is contained in the staircase .
Since is row-strict and column-strict, the entry in the -th row and -th column is greater than or equal to . But EG-tableaux can only be filled with the numbers , so the shape of is contained inside . ∎
Proof of Theorem 1.2.
The proof of Theorem 2.1 is via an explicit insertion algorithm. Suppose is an EG-tableau. We describe the insertion of a letter into . If the largest letter in the first row of is less than , then we add to the end of the first row, and the insertion is complete. Otherwise, we find the smallest letter in greater than , and bump to the second row, where the insertion algorithm is recursively performed. The first row of changes as follows: if both and were present in (and thus ) then the row remains unchanged; otherwise, we replace by in .
For a reduced word , we obtain by inserting , then , and so on, into the empty tableau. The tableau is the standard Young tableau which records the changes in shape of the EG-tableau as this insertion is performed.
Let . Then the successive EG-tableau are
2.2. Coxeter-Knuth relations
Let be a reduced word. A Coxeter-Knuth relation on is one of the following transformations on three consecutive letters of :
Since Coxeter-Knuth relations are in particular Coxeter relations for the symmetric group, it follows that if two words are related by Coxeter-Knuth relations then they represent the same permutation in . The following result of Edelman and Greene states that Coxeter-Knuth equivalence is an analogue of Knuth-equivalence for reduced words.
Theorem 2.4 ([Eg]).
Suppose . Then if and only if and are Coxeter-Knuth equivalent.
2.3. Exercises and Problems
For let denote the permutation obtained from by adding to every letter in the one-line notation, and putting a in front. Thus if , we have . Show that .
Suppose is 321-avoiding (see Section 1.7). Show that Edelman-Greene insertion of is the usual Robinson-Schensted insertion of .
(Vexillary permutations [BJS]) A permutation is vexillary if it avoids the pattern 2143. That is, there do not exist such that . In particular, is vexillary.
The Stanley symmetric function is equal to a Schur function if and only if is vexillary [BJS, p.367]. Is there a direct proof using Edelman-Greene insertion?
(Shape of a reduced word) The shape of a reduced word is the shape of the tableau or under Edelman-Greene insertion. Is there a direct way to read off the shape of a reduced word? (See [TY] for a description of .)
For example, Greene’s invariants (see for example [EC2, Ch. 7]) describe the shape of a word under Robinson-Schensted insertion.
(Coxeter-Knuth relations and dual equivalence (graphs)) Show that Coxeter-Knuth relations on reduced words correspond exactly to elementary dual equivalences on the recording tableau (see [Hai]). They thus give a structure of a dual equivalence graph [Ass] on .
An independent proof of this (in particular not using EG-insertion), together with the technology of [Ass], would give a new proof of the Schur positivity of Stanley symmetric functions.
(Lascoux-Schützenberger transition) Let denote the transposition which swaps and . Fix and . The Stanley symmetric functions satisfy [LSc85] the equality
where the last term with is only present if .
One obtains another proof of the Schur positivity of as follows. Let be the last descent of , and let be the largest index such that . Set . Then the left hand side of (2.1) has only one term . Recursively repeating this procedure for the terms on the right hand side one obtains a positive expression for in terms of Schur functions.
(Little’s bijection) Little [Lit] described an algorithm to establish (2.1), which we formulate in the manner of [LS06]. A -marked nearly reduced word is a pair where is a word with letters in and is an index such that is a reduced word for , where denotes omission. We say that is a marked nearly reduced word if it is a -marked nearly reduced word for some . A marked nearly reduced word is a marked reduced word if is reduced.
Define the directed Little graph on marked nearly reduced words, where each vertex has a unique outgoing edge as follows: is obtained from by changing to . If , then we also increase every letter in by . If is reduced then . If is not reduced then is the unique index not equal to such that is reduced. (Check that this is well-defined.)
For a marked reduced word such that is reduced, the forward Little move sends to where is the first marked reduced word encountered by traversing the Little graph.
Beginning with and one has
Note that is a reduced word for which covers . The word is a reduced word for .
(Dual Edelman-Greene equivalence) Let denote the set of all reduced words of permutations. We say that are dual EG-equivalent if the recording tableaux under EG-insertion are the same: .
Two reduced words are dual -equivalent if and only if they are connected by forward and backwards Little moves.
For example, both and of Example 6 Edelman-Greene insert to give recording tableau
Combine this with Edelman-Greene insertion to obtain an explicit weight-changing bijection on the monomials of a Stanley symmetric function, which exhibits the symmetry of a Stanley symmetric function. Compare with Stanley’s original bijection [Sta].
3. Affine Stanley symmetric functions
3.1. Affine symmetric group
For basic facts concerning the affine symmetric group, we refer the reader to [BB].
Let be a positive integer. Let denote the affine symmetric group with simple generators satisfying the relations
Here and elsewhere, the indices will be taken modulo without further mention. The length and reduced words for affine permutations are defined in an analogous manner to Section 1.2. The symmetric group embeds in as the subgroup generated by .
One may realize as the set of all bijections such that for all and . In this realization, to specify an element it suffices to give the “window” . The product of two affine permutations is then the composed bijection . Thus is obtained from by swapping the values of and for every . An affine permutation is Grassmannian if . For example, the affine Grassmannian permutation has reduced words and .
A word with letters in is called cyclically decreasing if (1) each letter occurs at most once, and (2) whenever and both occur in the word, precedes .
An affine permutation is called cyclically decreasing if it has a cyclically decreasing reduced word. Note that such a reduced word may not be unique.
There is a bijection between strict subsets of and cyclically decreasing affine permutations , sending a subset to the unique cyclically decreasing affine permutation which has reduced word using exactly the simple generators .
We define cyclically decreasing factorizations of in the same way as decreasing factorizations in .
Let . The affine Stanley symmetric function is given by
where the summation is over cyclically decreasing factorizations of .
Theorem 3.2 ([Lam06]).
The generating function is a symmetric function.
is a homogeneous of degree .
If , then a cyclically decreasing factorization of is just a decreasing factorization of , so .
The coefficient of in is equal to .
Consider the affine permutation . The reduced words are . The other cyclically decreasing factorizations are
Let . The code is a vector of non-negative entries with at least one 0. The entries are given by .
Let denote the set of partitions satisfying , called the set of -bounded partitions.
Lemma 3.3 ([Bb]).
The map is a bijection between and .
The analogue of Proposition 1.4 has a similar proof.
Proposition 3.4 ([Lam06]).
Suppose . Then .
Let be the subalgebra generated by , and let where is the ideal generated by for . A basis for is given by . A basis for is given by .
The ring of symmetric functions is a Hopf algebra, with coproduct given by . Equivalently, the coproduct of can be obtained by writing in the form where and are symmetric in ’s and ’s respectively. Then .
The ring is self Hopf-dual under the Hall inner product. That is, one has for . Here the Hall inner product is extended to in the obvious way. The rings and are in fact Hopf algebras, which are dual to each other under the same inner product. We refer the reader to [Mac] for further details.
3.5. Affine Schur functions
Stanley symmetric functions expand positively in terms of the basis of Schur functions (Corollary 2.2). We now describe the analogue of Schur functions for the affine setting.
For , we let where is the unique affine Grassmannian permutation with . These functions are called affine Schur functions (or dual -Schur functions, or weak Schur functions).
By Proposition 3.4, the leading monomial term of is . Thus is triangular with resepect to the basis , so that it is also a basis. ∎
We let denote the dual basis to . These are the (ungraded) -Schur functions, where . It turns out that the -Schur functions are Schur positive. However, affine Stanley symmetric functions are not. Instead, one has:
Theorem 3.6 ([Lam08]).
The affine Stanley symmetric functions expand positively in terms of the affine Schur functions .
3.6. Example: The case of
To illustrate Theorem 3.5, we completely describe the affine Schur functions for .
Let be the affine Grassmannian permutation corresponding to the partition . Then .
The affine Schur function is given by
The -Schur function is given by
For , we have and . Thus has cardinality , and .
3.7. Exercises and problems
(Coproduct formula [Lam06]) Show that .
( limit) Show that for a fixed partition , we have , where denotes the affine Schur function for .
Extend Proposition 3.8 to all affine Stanley symmetric functions in , and thus give a formula for the affine Stanley coefficients.
Is there an affine analogue of the fundamental quasi-symmetric functions? For example, one might ask that affine Stanley symmetric functions expand positively in terms of such a family of quasi-symmetric functions. Affine Stanley symmetric functions do not in general expand positively in terms of fundamental quasi-symmetric functions (see [McN, Theorem 5.7]).
Find closed formulae for numbers of reduced words in the affine symmetric groups , , extending Proposition 3.7. Are there formulae similar to the determinantal formula, or hook-length formula for the number of standard Young tableaux?
(-cores) A skew shape is a -ribbon if it is connected, contains squares, and does not contain a square. An -core is a partition such that there does not exist so that is a -ribbon. There is a bijection between the set of -cores and the affine Grassmannian permutaitons . Affine Schur functions can be described in terms of tableaux on -cores, called -tableau [LM05] (or weak tableau in [LLMS]).
(Cylindric Schur functions [Pos, McN]) Let denote the set of lattice paths in where every step either goes up or to the right, and which is invariant under the translation . Such lattice paths can be thought of as the boundary of an infinite periodic Young diagram, or equivalently of a Young diagram on a cylinder. We write if lies completely to the left of . A cylindric skew semistandard tableau is a sequence of where the region between and does not contain two squares in the same column. One obtains [Pos] a natural notion of a cylindric (skew) Schur function. Show that every cylindric Schur function is an affine Stanley symmetric function, and every affine Stanley symmetric function of a 321-avoiding permutation is a cylindric Schur function ([Lam06]).
(Affine Little bijection [LS06]) There is an affine analogue of Little’s bijection (Section 2.3) developed in [LS06]. It gives a combinatorial proof of the affine analogue of the transition formula (2.1). Can the affine Little bijection, or the affine transition formula lead to a proof of Theorem 3.6? Can one define a notion of dual EG-equivalence using the affine Little bijection?
4. Root systems and Weyl groups
4.1. Notation for root systems and Weyl groups
Let denote an affine Cartan matrix, where , so that is the corresponding finite Cartan matrix. For example, for type (corresponding to ) and we have and
The affine Weyl group is generated by involutions satisfying the relations , where for , one defines to be according as is . The finite Weyl group is generated by . For the symmetric group , we have , , and for .
Let be the root system for . Let denote the positive roots, denote the negative roots and denote the simple roots. Let denote the highest root of . Let denote the half sum of positive roots. Also let denote the simple coroots.
We write and for the affine root system, and positive affine roots. The positive simple affine roots (resp. coroots) are (resp. ). The null root is given by . A root is real if it is in the -orbit of the simple affine roots, and imaginary otherwise. The imaginary roots are exactly . Every real affine root is of the form , where . The root is positive if , or if and .
Let denote the root lattice and let denote the co-root lattice. Let and denote the weight lattice and co-weight lattice respectively. Thus and . We also have a map given by sending to (or equivalently, by sending to ). Let denote the pairing between and . In particular, one requires that .
The Weyl group acts on weights via (and via the same formula on or ), and on coweights via (and via the same formula on ). For a real root (resp. coroot