Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$

We examine the non-symmetric Macdonald polynomials $E_\lambda(x;q,t)$ at $q=1$, as well as the more general permuted-basement Macdonald polynomials. When $q=1$, we show that $E_\lambda(x;1,t)$ is symmetric and independent of $t$ whenever $\lambda$ is a partition. Furthermore, we show that for general $\lambda$, this expression factors into a symmetric and a non-symmetric part, where the symmetric part is independent of $t$, while the non-symmetric part only depends on the relative order of the entries in $\lambda$. We also examine the case $q=0$, which give rise to so called permuted-basement $t$-atoms. We prove expansion-properties of these, and as a corollary, prove that Demazure characters (key polynomials) expand positively into permuted-basement atoms. This complements the result that permuted-basement atoms are atom-positive. Finally, we show that a product of a permuted-basement atom and a Schur polynomial is again positive in the same permuted-basement atom basis, and thus interpolates between two results by Haglund, Luoto, Mason and van Willigenburg. The common theme in this project is the application of basement-permuting operators as well as combinatorics on fillings, by applying results in a previous article by the first author.


Introduction
The non-symmetric Macdonald polynomials, E λ (x; q, t) were introduced by Macdonald and Opdam in [Mac95a,Opd95]. They can be defined in other root systems. We only consider the type A for which there is a combinatorial rule, discovered by Haglund, Haiman and Loehr, [HHL + 05]. These non-symmetric Macdonald polynomials specialize to the Demazure characters, K λ , (or key polynomials) at q = t = 0, and at t = 0, they are affine Demazure characters, see [Bog03]. Furthermore, at q = t = ∞, E λ (x; ∞, ∞) reduces to the so called Demazure atoms, A λ , (also known as standard bases), see [Mas09,LS90]. The stable limit of E λ (x; q, t) gives the classical symmetric Macdonald polynomials (up to a rational function in q and t, depending on λ), denoted P λ (x; q, t), see [Mac95b]. For a quick overview, see the diagram (1) below, where * denotes this stable limit. (1) The topic of this paper is a generalization that arise naturally from Haglund's combinatorial formula, namely the permuted basement Macdonald polynomials, see [Ale15,Fer11]. Recently, an alcove walk model was given for these as well, see [FM15b,FM15a]. This generalize the alcove walk model by Ram and Yip, [RY11], for general type non-symmetric Macdonald polynomials.
The permuted basement Macdonald polynomials are indexed with an extra parameter, σ, which is a permutation. For each fixed σ ∈ S n , the set {E σ λ (x; q, t)} λ is a basis for the polynomial ring C[x 1 , . . . , x n ], as λ ranges over weak compositions of length n.
The current paper is the only one (to our knowledge) that studies this property in the permuted-basement setting. There has been previous research regarding various factorization properties of Macdonald polynomials, for example, [DM08,DMN12] concern symmetric Macdonald polynomials and the modified Madonald polynomials when t is taken to be a root of unity. In [CDL17], various factorization properties of non-symmetric Macdonald polynomials are observed experimentally (in particular, the specialization q = u −2 , t = u) in the last section of the article.
1.1. Main results. The first part of the paper concerns properties of the specialization E σ λ (x; 1, t). We show that for any fixed basement σ and composition λ, E σ λ (x; 1, t) = (e λ (x)/e µ (x)) E σ µ (x; 1, t) where µ is the weak standardization (defined below) of λ. Note that e λ (x)/e µ (x) is an elementary symmetric polynomial independent of t. We also show that in the case λ is a partition, we have E σ λ (x; 1, t) = e λ (x) which is independent of σ and t. This property is rather surprising and not evident from Haglund's combinatorial formula, [HHL08]. Our proofs mainly use properties of Demazure-Lusztig operators, see (11) below for the definition.
In the second half of the paper, we study properties of the specialization E σ λ (x; 0, t). At σ = id, these are t-deformations of so called Demazure atoms, so it is natural to introduce the notation A σ α (x; t) := E σ λ (x; 0, t), which are referred to as t-atoms. The t-atoms for σ = id were initially considered in [HLMvW11a], where they prove a close connection with Hall-Littlewood polynomials. The Hall-Littlewood polynomials P λ (x; t) are obtained as the specialization q = 0 in the classical Macdonald polynomial P λ (x; q, t). In fact, it was proven in [HLMvW11a] that the ordinary Hall-Littlewood polynomials P µ (x; t) can be expressed as whenever µ is a partition, and λ(γ) denotes the unique partition with the parts of γ rearranged in decreasing order.
Our main result regarding the t-atoms is as follows: If τ ≥ σ in Bruhat order, then A τ γ (x; t) admits the expansion where the c τ σ γα (t) are polynomials in t, with the property that c τ σ γα (t) ≥ 0 whenever 0 ≤ t ≤ 1.
As a final corollary, by taking τ = ω 0 , we see that key polynomials expand positively into permuted-basement Demazure atoms: pos. when 0≤t≤1 Figure 1. This graph shows various families of polynomials. The arrows indicate "expands positively in" which means that the coefficients are either non-negative numbers or polynomials in t with non-negative coefficients. Here, τ ≥ σ in Bruhat order, and Schur polynomials should be interpreted as polynomials in n variables or symmetric functions depending on context.

Preliminaries
We now give the necessary background on the combinatorial model for the permuted basement Macdonald polynomials. The notation and some of the preliminaries is taken from [Ale15].
Let σ = (σ 1 , . . . , σ n ) be a list of n different positive integers and let λ = (λ 1 , . . . , λ n ) be a weak integer composition, that is, a vector with non-negative integer entries. An augmented filling of shape λ and basement σ is a filling of a Young diagram of shape (λ 1 , . . . , λ n ) with positive integers, augmented with a zeroth column filled from top to bottom with σ 1 , . . . , σ n . Note that we use English notation rather than the skyline fillings used in [HHL08,Mas09].  Similarly, a triple of type B is an arrangement of boxes, a, b, c, located such that a is immediately to the left of b, and c is somewhere above a, and the row containing a and b is strictly longer than the row containing c.
A type A triple is an inversion triple if the entries ordered increasingly form a counter-clockwise orientation. Similarly, a type B triple is an inversion triple if the entries ordered increasingly form a clockwise orientation. If two entries are equal, the one with the largest subscript in Eq. (7) is considered to be largest.
Type A: . The set of descents in F is denoted by Des(F ).

Example 3.
Below is a non-attacking filling of shape (4, 1, 3, 0, 1) with basement (4, 5, 3, 2, 1). The bold entries are descents and the underlined entries form a type A inversion triple. There are 7 inversion triples (of type A and B) in total. 4 2 1 2 4 5 5 3 3 4 3 2 1 1 The leg of a box, denoted by leg(u), in an augmented diagram is the number of boxes to the right of u in the diagram. The arm of a box u = (r, c), denoted by arm(u), in an augmented diagram λ is defined as the cardinality of the sets {(r , c) ∈ λ : r < r and λ r ≤ λ r } and {(r , c − 1) ∈ λ : r < r and λ r < λ r }.
We illustrate the boxes x and y (in the first and second set in the union, respectively) contributing to arm(u) below. The boxes marked l contribute to leg(u). The arm values for all boxes in the diagram are shown in the diagram on the right. Recall, the length of a permutation, (σ), is the number of inversions in σ. We let ω 0 denote the unique longest permutation in S n . Furthermore, given an augmented filling F , the weight of F is the composition µ 1 , µ 2 , . . . , such that µ i is the number of non-basement entries in F that are equal to i. We then let x F be a shorthand for the product i x µi i .
Definition 5 (Combinatorial formula). Let σ ∈ S n and let λ be a weak composition with n parts. The non-symmetric permuted basement Macdonald polynomial When σ = ω 0 , we recover the non-symmetric Macdonald polynomials defined in [HHL08], E λ (x; q, t).
Note that the number of variables we work over is always finite and implicit from the context. For example, if σ ∈ S n , then x := (x 1 , . . . , x n ) in E σ λ (x; q, t), and it is understood that λ has n parts.

Bruhat order, compositions and operators.
If ω ∈ S n is a permutation, we can decompose ω as a product ω = s 1 s 2 · · · s k of elementary transpositions, s i = (i, i + 1). When k is minimized, s 1 s 2 · · · s k is a reduced word of ω, and k is the length of ω, which we denote by (ω).
The strong order on permutations in S n is a partial order defined via the cover relations that u covers v if (a, b)u = v and (u) + 1 = (v) for some transposition (a, b). The Bruhat order is defined in a similar fashion, where only elementary transpositions are allowed in the covering relations. We illustrate these partial orders in Fig. 2.
In order to prove the main result of this paper we rely heavily on the Knop-Sahi recurrence, basement permuting operators, and shape permuting operators. The Knop-Sahi recurrence relations for Macdonald polynomials [Kno97,Sah96] is given by the relation whereλ = (λ 2 , . . . , λ n , λ 1 + 1). Furthermore, note that the combinatorial formula implies that We need some brief background on certain t-deformations of divided difference operators. Let s i be a simple transposition on indices of variables and define The operators π i and θ i are used to define the key polynomials and Demazure atoms, respectively. Now define the following t-deformations of the above operators: Theθ i are called the Demazure-Lusztig operators generates the affine Hecke algebra, see e.g. [HHL08] (whereθ i correspond to T i ). Note that these operators satisfy the braid relations, and thatθ iπi =π iθi = t.
Example 6. As an example, With these definitions, we can now state the following two propositions which were proved in [Ale15]: Proposition 7 (Basement permuting operators). Let λ be a composition and let σ be a permutation. Furthermore, let γ i be the length of the row with basement label i, that is, Consequently, we see thatπ i andθ i move the basement up and down, respectively, in the Bruhat order.
Note that these formulas together with the Knop-Sahi recurrence uniquely define the Macdonald polynomials recursively, with the initial condition that for the empty composition, E 0...0 (x; q, t) = 1.
Finally, we will need the following result from [Ale15]:

A basement invariance
In this section, we prove bijectively that whenever λ is a partition, we have E σ λ (x; 1, 0) = e λ (x). Note that this is independent of the basement σ, which at a first glance might be surprising.
Lemma 10. Let D be a diagram of shape 2 m 1 n , where the first column has fixed distinct entries in N. Furthermore, if S ⊆ N be a set of m integers then there is a unique way of placing the entries in S into the second column of D such that the resulting filling has no coinversions.
Proof. We provide an algorithm for filling in the second column of the diagram. Begin by letting C be the topmost box in the second column and let L(C) to be the box to the left of C. In order to pick an entry for C, we do the following: If there is an element in S which is less than or equal to L(C), remove it from S let it be the value of C.
Otherwise, remove the maximal element in S and let this be the value of C.
Iterate this procedure for the remaining entries in the second column while moving C downwards. It is straightforward to verify that the result is coinversion-free and that every choice for the element in second column is forced.
Corollary 11. If λ is a partition with at most n parts and σ ∈ S n , then Proof. Fix a basement σ and choose sets of elements for each of the remaining columns. Note that all such choices are in natural correspondence with the monomials whose sum is e λ . By applying the previous lemma inductively column by column, it follows that there is a unique filling with the the specified column sets. The combinatorial formula now implies that E σ λ (x; 1, 0) = e λ (x) as desired.
We use a similar approach to give bijections among families of coinversion-free fillings of general composition shapes in [AS17].
Proof. We begin by showing this statement for σ = id.
Using Theorem 9, we have that E id λ (x; 1, t) is symmetric in x 1 , . . . , x m and symmetric in x m+1 , . . . , x m+n . Furthermore, using the combinatorial formula, we can easily see that there is exactly one non-attacking filling of weight λ. This filling has major index 0. In other words, λ (x; 1, t) = 1. It is therefore enough show that the polynomial is symmetric in x m and x m+1 . A result in [HHL08] implies that a polynomial f is symmetric in x m , x m+1 if and only ifπ m (f ) = f . Hence, it suffices to show Proposition 7 gives thatπ We do this by exhibiting a weight-preserving bijection between fillings of shape λ with identity basement, and those with s m as basement. The bijection is given by simply permuting the basement labels in row m and m + 1, since both coinversions and the non-attacking condition are preserved, so the result is a valid filling. Finally, since arm(u) = 0 for the box in position (m, 1), it is straightforward to verify that the weight is preserved under this map.
The statement for general σ now follows by applying the basement permuting operatorsπ i repeatedly on both sides of the identity E σ λ (x; 1, t) = e m (x). The right hand side is unchanged since these operators preserve symmetric functions.
We say that λ ≤ µ in the Bruhat order if there is a sequence of transpositions, s i1 · · · s i k such that s i1 · · · s i k λ = µ and where each application of a transposition increases the number of inversions.
Lemma 14. If λ and µ are compositions such that λ ≤ µ in the Bruhat order, then the following implication holds: where F λ (x) is any function symmetric in x 1 , . . . , x n .
Proof. It suffices to show the implication for any simple transposition, s i λ = µ that increases the number of inversions. Suppose that E w0 λ (x; 1, t) = F λ (x)E w0 λ (x; 1, t) for some composition λ. By Proposition 8, we note that the shape permuting operator is the same on both sides for q = 1. That is, for any composition λ with λ i < λ i+1 we have where arm(u) ≥ 1 has the same value in both diagrams λ andλ.
To simplify typesetting of the upcoming proofs, we will sometimes use the notation where λ is the composition ).

Lemma 15. We have the identity
Proof. We prove this lemma by induction on k, where the base case k = 1 is given by Lemma 13. For k > 1, by Proposition 8 and a similar reasoning as in Lemma 14, it is enough to prove that Furthermore, through repeated application of the Knop-Sahi recurrence Eq. (9) it suffices to prove Again using Proposition 8, we reduce the above to the k − 1 case which is true by induction.

Proposition 16. If λ is a composition, then
where F λ (x) is an elementary symmetric polynomial.
Proof. We prove the proposition by induction on |λ| and the number of inversions of λ. Note that the result is trivial if |λ| ≤ 1.
Given λ, there are several cases to consider: Case 1: min i λ i ≥ 1. The result follows by inductive hypothesis on the size of the composition by using Eq. (10) in the numerator.
Case 2: λ is not weakly increasing. We can reduce this case to a composition with fewer inversions using Lemma 14.
Case 3: λ is weakly increasing. It is enough to prove that is an elementary symmetric polynomial where 0 = a 0 < a 1 < a 2 < · · · . Using the cyclic shift relation (9) in the numerator and denominator, it suffices to show that E[ (a 1 − 1) b1 , (a 2 − 1) b2 , . . . , (a k − 1) is an elementary symmetric polynomial. If a 1 = 1, the result follows by the inductive hypothesis on the size of the composition. Otherwise, by rewriting Eq. (19), it is enough to prove that is an elementary symmetric polynomial. The first fraction is an elementary symmetric polynomial by induction, since it is of the right form with a smaller size. According to Lemma 14, the second fraction is also an elementary symmetric polynomial.

Theorem 17. If λ is a composition and
where F λ (x) is an elementary symmetric polynomial independent of t.
Proof. From Proposition 16, we have that where F λ is an elementary symmetric polynomial. Applying basement-permuting operators from Proposition 7 on both sides then gives ; 1, t). Note that applying a basement-permuting operator might give an extra factor of t, but since λ i ≤ λ j if and only ifλ i ≤λ j , these extra factors cancel.
We are now ready to prove the following surprising identity, which was first observed through computational evidence by J. Haglund and the first author.
Theorem 18. If λ is a partition and σ ∈ S n , then Proof. It is enough to prove that E w0 λ (x; 1, t) = e λ (x) as the more general statement follows from using Proposition 7.
By using the previous theorem, it is enough to prove that We show this via induction on k. The base case k = 1 is trivial, so assume k > 1 and note that repeated use of Proposition 8 implies that it is enough to prove By using the Knop-Sahi recurrence (9), it suffices to show that which now follows from induction.

Corollary 19. The previous proof can be extended to show that
for partition λ.
Note that the parts of λ is a super-set of the parts of (λ) , so the above expression is indeed some elementary symmetric polynomial.
Our results are in some sense optimal: for general compositions λ, it happens that E σ λ (x; 1, t) cannot be factorized further. For example, Mathematica computations suggest that E Since a priori E σ λ (x; 1, t) is only a rational function in t, this seems like a difficult challenge. We therefore pose a more conservative question: Question 21. Is there a combinatorial explanation of the identity E σ λ (x; 1, t) = e λ (x) whenever λ is a partition?
We finish this section by discussing properties of the family {E λ (x; 1, 0)} as λ ranges over compositions with n parts. It is a basis for C[x 1 , . . . , x n ] and naturally extends the elementary symmetric functions. Furthermore, it is shown in [Ass17] that E λ (x; 1, 0) expands positively into key polynomials, where the coefficients are given by the classical Kostka coefficients. Furthermore, {E λ (x; q, 0)} exhibit properties very similar to those of modified Hall-Littlewood polynomials. In particular, these expand positively into key polynomials with Kostka-Foulkes polynomials (in q) as coefficients. There are representation-theoretical explanations for these expansions as well, see [Ass17,AS17] and references therein for details.
It is known that a product of a Schur polynomial and a key polynomial is keypositive (see e.g. Proposition 29 below), and thus a product of an elementary symmetric polynomial and a key polynomial is key positive. It is therefore natural to ask if this extends to the non-symmetric elementary polynomials. However, a quick computer search reveals that E 030 (x; 1, 0)K 201 (x) does not expand positively into key polynomials.

Positive expansions at t = 0
By specializing the combinatorial formula Eq. (8) with q = 0, we obtain a combinatorial formula for the permuted-basement Demazure t-atoms.

Example 22. As an example,
where the corresponding fillings are .
In this section, we show how to construct permuted-basement Demazure t-atoms via Demazure-Luzstig operators. First consider Proposition 7 and Proposition 8 at q = 0. Note that Proposition 8 simplifies, where we use the fact thatθ i + (1 − t) =π i . Hence, the shape-permuting operator reduce to a basement-permuting operator. This "duality" between shape and basement was first observed at t = 0 in [Mas09], where S. Mason gave an alternative combinatorial description of key polynomials which is not immediate from the combinatorial formula for the non-symmetric Macdonald polynomials. A similar duality holds for general values of t, see [Ale15].
To get a better overview of Proposition 7 and Proposition 8, we present the statements as actions on the basement and shape as follows: Example 23. The operatorsπ i andθ i act as follows on diagram shapes and basements. Note that we only care about the relative order of row lengths. A box with a dot might either be present or not, indicating weak or strict difference between row lengths.
The operators acting on the shape can be described pictorially as which are easily obtained from Proposition 8 at q = 0, together with the fact that θ iπi = t.
The following proposition also appeared in [Ale15], however the proof we present here is different and more constructive.
Proposition 24. Given λ and σ, there is a sequenceρ i1 · · ·ρ i such that where λ is the partition with the parts of λ in decreasing order and eachρ ij is one ofθ i orπ i .
Proof. Given (σ, λ), let the number of monotone pairs be the number of pairs (i, j) such that We do induction over number of monotone pairs. First note that if there are no monotone pairs in (σ, λ), then the longest row has basement label 1, the second longest row has basement label 2 and so on. It then follows that every row in a filling with basement σ and shape λ has to be constant, implying that A σ λ (x; t) = x λ . Assume that there is some monotone pairs determined by (σ, λ). A permutation with at least one inversion must have a descent, and for a similar reason, there is at least one monotone pair of the form These match the right hand sides of (21) and (20). By induction, A σ λ (x; t) can therefore be obtained from some A σsi λ (x; t) by applying eitherπ i orθ i .
Example 25. We illustrate the above proposition by expressing A 3142 3102 (x; t) in terms of operators. The shape and basement associated with this atom is given in the first augmented diagram in (24). The rows with labels 2 and 3 constitute a monotone pair and can be obtained using (21), which explains theπ 2 -arrow. Continuing on withπ 1 followed byθ 2 leads to an augmented diagram without any monotone pairs, so A 1342 3102 (x; t) = x (3,2,1,0) . Finally, following the arrows yields the operator expression 2,1,0) .
where stat(λ, σ, i) is a non-negative integer depending on λ, σ and i.
Proof. We prove this statement via induction over (τ ).
Case τ = id and λ i ≤ λ i+1 : We need to show that A si λ (x; t) = A id siλ (x; t). Sinceπ i is invertible, it suffices to show that This equality now follows from using (22) on the left hand side and (21) on the right hand side.
Case τ = id and λ i > λ i+1 : It suffices to prove that Note that the left hand side is equal toπ i A id λ (x; t) using (21), while the left hand side is equal to , this proves the identity.
This proves the base case. The general case now follows from applyingπ j on both sides, thus increasing the lengths of the basements. We examine the details in the following two cases.
Case τ ∈ S n and λ i ≤ λ i+1 : Suppose A σ λ (x; t) = A τ siλ (x; t). As diagrams, we have the equality for rows i and i + 1, b > a, while the remaining rows are identical. If (σs j ) > (σ), we can conclude that if a = j, then b = j + 1. We now compare the row lengths of the rows with basement label j and j + 1 and apply the basement-permutingπ j from (21) on both sides. Note that the row lengths that are compared are the same on both sides, meaning that if we need (21) to increase the basement on the left hand side, the same relation acts the same way on the right hand side. In other words, we have the implication whenever (σs j ) > (σ) and λ i ≤ λ i+1 .
Case τ ∈ S n and λ i > λ i+1 : Again, suppose we have the diagram identity for some λ, σ and that (σs j ) > (σ). As in the previous case, if a = j then b = j + 1. If j / ∈ {a − 1, a, b − 1, b}, applyingπ j on both sides yield the implication because -depending on the relative row lengths of the rows with basement labels j, j + 1 -we either multiply each of the three terms by t or not at all.
It remains to verify the cases j ∈ {a − 1, a, b − 1, b}. Case by case study after applyingπ j on both sides shows that where (using the same notation as in Proposition 7, γ i being the length of the row with basement label i) and γ a ≥ γ b+1 > γ b and = 0 otherwise. Thus, we have that where the left hand side is a polynomial. Furthermore, A τ sj λ (x; t) is not a multiple of t -this follows from the combinatorial formula (8). Hence, + stat(λ, σ, i) must be non-negative.
In [HLMvW11b], the cases σ = id and σ = ω 0 of the following proposition were proved. We give an interpolation between these results: Proposition 29. The coefficients d µσ λγ in the expansion are non-negative integers.
Remember that x = (x 1 , . . . , x n ), so we evaluate s µ (x) in a finite alphabet.
Proof. With the case σ = id as a starting point (proved in [HLMvW11b]), we can apply π i on both sides, (π i commutes with any symmetric function, in particular s λ (x)), and thus we may walk upwards in the Bruhat order and obtain the statement for any basement σ. Note that Proposition 7 implies that π i applied to A σ γ (x) either increase σ in Bruhat order, or kills that term.
Note that the above result implies that the products e µ × A σ λ (x) and h µ × A σ λ (x) also expand non-negatively into σ-atoms. It would be interesting to give a precise rule for this expansion, as well as a Murnaghan-Nakayama rule for the permutedbasement Demazure atoms.
Remark 30. We need to mention the paper [LR13], which also concerns a different type of general Demazure atoms. These objects are also studied in [HLMvW11b], but are in general different from ours when σ = id. In particular, the polynomial families they study are not bases for C[x 1 , . . . , x n ], and they are not compatible with the Demazure operators. The authors of [LR13,HLMvW11b] construct these families by imposing an additional restriction 1 on Haglund's combinatorial model, which enables them to perform a type of RSK.
The introductions in the two papers mention the permuted-basement Macdonald polynomials, E σ µ (x; q, t), but the additional restriction breaks this connection whenever σ = id. This fact is unfortunately hidden since they use the same notationÊ γ is used for two different families of polynomials.