Abstract
We examine the nonsymmetric Macdonald polynomials \(\mathrm {E}_\lambda \) at \(q=1\), as well as the more general permutedbasement Macdonald polynomials. When \(q=1\), we show that \(\mathrm {E}_\lambda (\mathbf {x};1,t)\) is symmetric and independent of t whenever \(\lambda \) is a partition. Furthermore, we show that, in general \(\lambda \), this expression factors into a symmetric and a nonsymmetric part, where the symmetric part is independent of t, and the nonsymmetric part only depends on \(\mathbf {x}\), t, and the relative order of the entries in \(\lambda \). We also examine the case \(q=0\), which gives rise to the socalled permutedbasement tatoms. We prove expansion properties of these tatoms, and, as a corollary, prove that Demazure characters (key polynomials) expand positively into permutedbasement atoms. This complements the result that permutedbasement atoms are atompositive. Finally, we show that the product of a permutedbasement atom and a Schur polynomial is again positive in the same permutedbasement atom basis. Haglund, Luoto, Mason, and van Willigenburg previously proved this property for the identity basement and the reverse identity basement, so our result can be seen as an interpolation (in the Bruhat order) between these two results. The common theme in this project is the application of basementpermuting operators as well as combinatorics on fillings, by applying results in a previous article by Per Alexandersson.
1 Introduction
The nonsymmetric Macdonald polynomials \(\mathrm {E}_\lambda (\mathbf {x};q,t)\) were introduced by Macdonald and Opdam in [19, 22]. They can be defined for arbitrary root systems. We only consider the type A for which there is a combinatorial rule, discovered by Haglund et al. [11]. These nonsymmetric Macdonald polynomials specialize to the Demazure characters, \(\mathcal {K}_\lambda \), (or key polynomials) at \(q=t=0\) (or at \(t=0\)), they are affine Demazure characters, see [14]. Furthermore, at \(q=t=\infty \), \(\mathrm {E}_\lambda (\mathbf {x};\infty ,\infty )\) reduces to the socalled Demazure atoms, \(\mathcal {A}_\lambda \) (also known as standard bases), see [16, 21]. The stable limit of \(\mathrm {E}_\lambda (\mathbf {x};q,t)\) gives the classical symmetric Macdonald polynomials (up to a rational function in q and t, depending on \(\lambda \)), denoted \(P_\lambda (\mathbf {x};q,t)\), see [18]. For a quick overview, see the diagram (1.1) below, where \(*\) denotes this stable limit:
The topic of this paper is a generalization that arises naturally from the Haglund–Haiman–Loehr (HHL) combinatorial formula, namely the permutedbasement Macdonald polynomials, see [1, 9]. Recently, an alcove walk model was given for these, as well, see [7, 8]. This generalizes the alcove walk model by Ram and Yip [24] for general type nonsymmetric Macdonald polynomials.
The permutedbasement Macdonald polynomials are indexed with an extra parameter, \(\sigma \), which is a permutation. For each fixed \(\sigma \in S_n\), the set \(\{ \mathrm {E}^\sigma _\lambda (\mathbf {x};q,t)\}_\lambda \) is a basis for the polynomial ring \(\mathbb {C}[x_1,\cdots ,x_n]\), as \(\lambda \) ranges over weak compositions of length n.
The current paper is the only one (to our knowledge) that studies this property in the permutedbasement setting. There has been previous research regarding various factorization properties of Macdonald polynomials; for example, [5, 6] concern symmetric Macdonald polynomials and the modified Macdonald polynomials when t is taken to be a root of unity. In [4], various factorization properties of nonsymmetric 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 \(\mathrm {E}^\sigma _\lambda (\mathbf {x};1,t)\). We show that, for any fixed basement \(\sigma \) and composition \(\lambda \):
where \(\mu \) is the weak standardization (defined below) of \(\lambda \). Observe that \(\mathrm {e}_{\lambda '}(\mathbf {x}) / \mathrm {e}_{\mu '}(\mathbf {x})\) is an elementary symmetric polynomial independent of t. We also show that, in the case, \(\lambda \) is a partition, we have the following:
which is independent of \(\sigma \) and t. This property is rather surprising and not evident from the HHL combinatorial formula [10]. Our proofs mainly use properties of Demazure–Lusztig operators, see (2.5) below for the definition.
In the second half of the paper, we study properties of the specialization \(\mathrm {E}^\sigma _\lambda (\mathbf {x};0,t)\). At \(\sigma = {{\,\mathrm{id}\,}}\), these are tdeformations of the socalled Demazure atoms, so it is natural to introduce the notation \(\mathcal {A}^{\sigma }_\lambda (\mathbf {x};t) {:}{=}\mathrm {E}^\sigma _\lambda (\mathbf {x};0,t)\), which are referred to as tatoms. The tatoms for \(\sigma ={{\,\mathrm{id}\,}}\) were initially considered in [12], where they proved a close connection with Hall–Littlewood polynomials. The Hall–Littlewood polynomials \(\mathrm {P}_\lambda (\mathbf {x};t)\) are obtained as the specialization \(q=0\) of the classical Macdonald polynomial \(\mathrm {P}_\lambda (\mathbf {x};q,t)\). In fact, it was proven in [12] that the ordinary Hall–Littlewood polynomials \(\mathrm {P}_\mu (\mathbf {x};t)\) can be expressed as follows:
whenever \(\mu \) is a partition, and \(\mathrm {par}(\gamma )\) denotes the unique partition with the parts of \(\gamma \) rearranged in decreasing order.
Our main result regarding the tatoms is as follows: If \(\tau \ge \sigma \) in Bruhat order, then \(\mathcal {A}^{\tau }_\gamma (\mathbf {x};t)\) admits the expansion:
with the sum being over all compositions \(\lambda \) whose parts rearrange to the parts of \(\gamma \) and the \(c^{\tau \sigma }_{\gamma \lambda }(t)\) are polynomials in t with the property that \(c^{\tau \sigma }_{\gamma \lambda }(t) \ge 0\) whenever \(0\le t \le 1\).
Equation (1.2) is a generalization of the fact that key polynomials and permutedbasement atoms expand positively into Demazure atoms, see e.g. [20, 23]. Letting \(t=0\), we obtain the general result that whenever \(\tau \ge \sigma \) in Bruhat order:
Figure 1 illustrates how various bases of polynomials are related under expansion. We prove the dashed relations (1.2) and (1.3) in this paper. In the figure, we have the permutedbasement atoms, \(\mathcal {A}^{\tau }_\gamma (\mathbf {x}) {:}{=}\mathcal {A}^{\tau }_\gamma (\mathbf {x};0)\), the key polynomials \(\mathcal {K}_\gamma (\mathbf {x}) {:}{=}\mathcal {A}^{\omega _0}_\gamma (\mathbf {x})\), and the Demazure atoms \(\mathcal {A}_\gamma (\mathbf {x}) {:}{=}\mathcal {A}^{{{\,\mathrm{id}\,}}}_\gamma (\mathbf {x})\). Finally, \(\omega _0\) denotes the longest permutation (in \(S_n\)).
As a final corollary, by taking \(\tau =\omega _0\), we see that key polynomials expand positively into permutedbasement Demazure atoms:
2 Preliminaries
We now give the necessary background on the combinatorial model for the permutedbasement Macdonald polynomials. The notation and some of the preliminaries is taken from [1].
Let \(\sigma = (\sigma _1,\dots ,\sigma _n)\) be a list of n different positive integers and let \(\lambda =(\lambda _1,\dots ,\lambda _n)\) be a weak integer composition, that is, a vector with nonnegative integer entries. An augmented filling of shape \(\lambda \) and basement\(\sigma \) is a filling of a Young diagram of shape \((\lambda _1,\cdots ,\lambda _n)\) with positive integers, augmented with a zeroth column filled from top to bottom with \(\sigma _1,\cdots ,\sigma _n\). Note that we use English notation rather than the skyline fillings used in [10, 21]. We motivate this choice with the fact that row operations are easier to work with compared with column operations.
Definition 2.1
Let F be an augmented filling. Two boxes a and b are attacking if \(F(a)=F(b)\) and the boxes are either in the same column, or they are in adjacent columns with the rightmost box in a row strictly below the other box:
A filling is nonattacking if there are no attacking pairs of boxes.
Definition 2.2
A triple of type A is an arrangement of boxes, a, b, c, located, such that a is immediately to the left of b, and c is somewhere below b, and the row containing a and b is at least as long as the row containing c.
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 counterclockwise 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. (2.1) is considered to be largest:
If \(u = (i,j)\) let d(u) denote \((i,j1)\), i.e., the box to the left of u. A descent in F is a nonbasement box u, such that \(F(d(u)) < F(u)\). The set of descents in F is denoted by \({{\,\mathrm{\mathrm {Des}}\,}}(F)\).
Example 2.3
Below is a nonattacking 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:
The leg of a box, denoted by \({{\,\mathrm{leg}\,}}(u)\), 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 \({{\,\mathrm{arm}\,}}(u)\), in an augmented diagram \(\lambda \) is defined as the cardinality of the union of the sets:
We illustrate the boxes x and y (in the first and second set in the union, respectively) contributing to \({{\,\mathrm{arm}\,}}(u)\) below. The boxes marked l contribute to \({{\,\mathrm{leg}\,}}(u)\). The \({{\,\mathrm{arm}\,}}\) values for all boxes in the diagram are shown in the diagram on the right.
The major index, \({{\,\mathrm{maj}\,}}(F)\), of an augmented filling F is given by the following:
The number of inversions, denoted by \({{\,\mathrm{inv}\,}}(F)\), of a filling is the number of inversion triples of either type. The number of coinversions, \({{\,\mathrm{coinv}\,}}(F)\), is the number of type A and type B triples which are not inversion triples.
Let \(\mathrm {NAF}_\sigma (\lambda )\) denote all nonattacking fillings of shape \(\lambda \) with basement \(\sigma \in S_n\) and entries in \(\{1,\cdots ,n\}\).
Example 2.4
The set \(\mathrm {NAF}_{3124}(1,1,0,2)\) consists of the following augmented fillings:
Recall that the length of a permutation, \({{\,\mathrm{\ell }\,}}(\sigma )\), is the number of inversions in \(\sigma \). We let \(\omega _0\) denote the unique longest permutation in \(S_n\). Furthermore, given an augmented filling F, the weight of F is the composition \(\mu _1,\mu _2,\dots ,\) such that \(\mu _i\) is the number of nonbasement entries in F that are equal to i. We then let \(\mathbf {x}^F\) be a shorthand for the product \(\prod _i x_i^{\mu _i}\).
Definition 2.5
(Combinatorial formula) Let \(\sigma \in S_n\) and let \(\lambda \) be a weak composition with n parts. The nonsymmetric permutedbasement Macdonald polynomial\(\mathrm {E}^\sigma _\lambda (\mathbf {x};q,t)\) is defined as follows:
The product is over all boxes u in F, such that either u is in the basement or \(F(d(u))\ne F(u)\).
When \(\sigma = \omega _0\), we recover the nonsymmetric Macdonald polynomials \(\mathrm {E}_\lambda (\mathbf {x};q,t)\) defined in [10].
Note that the number of variables which we work over is always finite and implicit from the context. For example, if \(\sigma \in S_n\), then \(\mathbf {x}{:}{=}(x_1,\cdots ,x_n)\) in \(\mathrm {E}^\sigma _\lambda (\mathbf {x}; q,t)\), and it is understood that \(\lambda \) has n parts.
2.1 Bruhat Order, Compositions, and Operators
If \(\omega \in S_n\) is a permutation, we can decompose \(\omega \) as a product \(\omega = s_{i_1}s_{i_2}\cdots s_{i_k}\) of elementary transpositions, \(s_i = (i,i+1)\). When k is minimized, \(s_{i_1}s_{i_2}\cdots s_{i_k}\) is a reduced word of \(\omega \), and k is the length of \(\omega \), which we denote by \({{\,\mathrm{\ell }\,}}(\omega )\).
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 \({{\,\mathrm{\ell }\,}}(u) + 1 = {{\,\mathrm{\ell }\,}}(v)\) 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.
Given a composition \(\alpha \), let \(\mathrm {par}(\alpha )\) be the unique integer partition where the parts of \(\alpha \) have been rearranged in decreasing order. For example, \(\mathrm {par}(2,0,1,4,9)\) is equal to (9, 4, 2, 1, 0). We can act with permutations on compositions (and partitions) by permutation of the entries:
where \(\omega \) is given in oneline notation.
To prove the main result of this paper, we rely heavily on the Knop–Sahi recurrence, basementpermuting operators, and shapepermuting operators. The Knop–Sahi recurrence relation for Macdonald polynomials [15, 25] is given by the relation:
where \({\hat{\lambda }} = (\lambda _2,\cdots ,\lambda _n, \lambda _1 +1)\). Furthermore, note that the combinatorial formula implies that
We need some brief background on certain tdeformations of divided difference operators. Let \(s_i\) be a simple transposition on indices of variables and define
The operators \(\pi _i\) and \(\theta _i\) are used to define the key polynomials and Demazure atoms, respectively. Now, define the following tdeformations of the above operators:
The \({\tilde{\theta }}_i\) are called the Demazure–Lusztig operators. They generate the affine Hecke algebra, see e.g. [10] (where \({\tilde{\theta }}_i\) correspond to \(T_i\)). Note that these operators satisfy the braid relations, and that \({\tilde{\theta }}_i {\tilde{\pi }}_i = {\tilde{\pi }}_i {\tilde{\theta }}_i = t\).
Example 2.6
As an example, \({\tilde{\theta }}_1( x_1^2 x_2 ) = (1  t) x_1 x_2^2 + t x_1 x_2^2\).
With these definitions, we can now state the following two propositions which were proved in [1]:
Proposition 2.7
(Basementpermuting operators). Let \(\lambda \) be a composition and let \(\sigma \) be a permutation. Furthermore, let \(\gamma _i\) be the length of the row with basement label i, that is, \(\gamma _i = \lambda _{\sigma ^{1}_i}\).
If \({{\,\mathrm{\ell }\,}}(\sigma s_i) < {{\,\mathrm{\ell }\,}}(\sigma )\), then
Similarly, if \({{\,\mathrm{\ell }\,}}(\sigma s_i) > {{\,\mathrm{\ell }\,}}(\sigma )\), then
Consequently, we see that \({\tilde{\pi }}_i\) and \({\tilde{\theta }}_i\) move the basement up and down, respectively, in the Bruhat order.
Proposition 2.8
(Shapepermuting operators). If \(\lambda _j < \lambda _{j+1}\), \(\sigma _j = i+1\) and \(\sigma _{j+1} =i\) for some i and j, then
where \(u = (j+1, \lambda _{j}+1)\) in the diagram of shape \(\lambda \).
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: \(\mathrm {E}_{(0,\ldots , 0)}(\mathbf {x};q,t)=1\).
Finally, we will need the following result from [1]:
Theorem 2.9
(Partial symmetry). Suppose \(\lambda _j = \lambda _{j+1}\) and \(\{\sigma _j, \sigma _{j+1} \}\) take the values \(\{i, i+1\}\) for some j, i, then \(\mathrm {E}^\sigma _\lambda (\mathbf {x};q,t)\) is symmetric in \(x_i, x_{i+1}\).
3 A Basement Invariance
Recall that the elementary symmetric function \(\mathrm {e}_\mu (\mathbf {x})\) with the partition \(\mu \) having \(\ell \) parts is defined as follows:
In this section, we use a bijective construction to prove that whenever \(\lambda \) is a partition, we have \(\mathrm {E}_\lambda ^\sigma (\mathbf {x};1,0) = \mathrm {e}_{\lambda '}(\mathbf {x})\). Note that this is independent of the basement \(\sigma \), which, at a first glance, might be surprising.
Lemma 3.1
Let D be a diagram of shape \(2^m 1^n\), where the first column has fixed distinct entries in \(\mathbb {N}\). If \(S\subseteq \mathbb {N}\) is 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 nonattacking 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) be the box to the left of C. 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 and 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 coinversionfree and that every choice for the element in the second column is forced.\(\square \)
Corollary 3.2
If \(\lambda \) is a partition with at most n parts and \(\sigma \in S_n\), then
Proof
Fix a basement \(\sigma \) 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 \(\mathrm {e}_{\lambda '}(\mathbf {x})\). By applying the previous lemma inductively column by column, it follows that there is a unique filling with the specified column sets. The combinatorial formula now implies that \(\mathrm {E}_\lambda ^\sigma (\mathbf {x};1,0) = \mathrm {e}_{\lambda '}(\mathbf {x})\) as desired.\(\square \)
We use a similar approach to give bijections among families of coinversionfree fillings of general composition shapes in [2].
Example 3.3
Here are the nine fillings associated with \(\mathrm {E}_{210}^{132}(\mathbf {x};1,0)\). In other words, it is the set of nonattacking, coinversionfree fillings of shape (2, 1, 0) and basement 132:
The sum of the weights is \(x_1^2x_2 + x_1^2x_3 + \cdots + x_2x_3^2 = \mathrm {e}_{210}(\mathbf {x})\).
4 The Factorization Property
Let \(\lambda \) be a composition. The weak standardization of \(\lambda \), denoted by \({\tilde{\lambda }}\), is the lexsmallest composition, such that, for all i, j, we have the following:
For example, \(\lambda = (6,2,5,2,3,3)\) gives \({\tilde{\lambda }} = (3,0,2,0,1,1)\).
Lemma 4.1
If \(\lambda = 1^m0^n\), then \(\mathrm {E}^{\sigma }_{\lambda }(\mathbf {x};1,t) = \mathrm {e}_m(\mathbf {x}).\)
Proof
We begin by showing this statement for \(\sigma = \text {id}\).
Using Theorem 2.9, we have that \(\mathrm {E}^{\text {id}}_{\lambda }(\mathbf {x};1,t)\) 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 nonattacking filling of weight \(\lambda \). This filling has major index 0. In other words:
It is, therefore, enough to show that the polynomial is symmetric in \(x_m\) and \(x_{m+1}\). A result in [10] implies that a polynomial f is symmetric in \(x_m,x_{m+1}\) if and only if \({\tilde{\pi }}_m(f)=f\). Hence, it suffices to show that
Proposition 2.7 gives that
Hence, it remains to show that \(\mathrm {E}^{\text {id}}_{\lambda }(\mathbf {x};1,t) = \mathrm {E}^{s_m}_{\lambda }(\mathbf {x};1,t)\). We do this by exhibiting a weightpreserving bijection between fillings of shape \(\lambda \) 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 nonattacking condition are preserved, so the result is a valid filling. Finally, since \({{\,\mathrm{arm}\,}}(u)=0\) for the box in position (m, 1), it is straightforward to verify that the weight is preserved under this map.
The statement for general \(\sigma \) now follows by applying the basementpermuting operators \({\tilde{\pi }}_i\) repeatedly on both sides of the identity \(\mathrm {E}^{\sigma }_{\lambda }(\mathbf {x};1,t) = \mathrm {e}_m(\mathbf {x})\). The righthand side is unchanged, since these operators preserve symmetric functions. \(\square \)
We say that \(\lambda \le \mu \) in the Bruhat order if there is a sequence of transpositions, \(s_{i_1}\cdots s_{i_k}\), such that \(s_{i_1}\cdots s_{i_k} \lambda = \mu \) and where each application of a transposition increases the number of inversions.
Lemma 4.2
If \(\lambda \) and \(\mu \) are compositions, such that \(\lambda \le \mu \) in the Bruhat order, then the following implication holds:
where \(F_\lambda (\mathbf {x})\) is any function symmetric in \(x_1,\cdots ,x_n\).
Proof
It suffices to show the implication for any simple transposition, \(s_i \lambda = \mu \) that increases the number of inversions. Suppose that
for some composition \(\lambda \). By Proposition 2.8, we note that the shapepermuting operator is the same on both sides for \(q=1\). That is, for any composition \(\lambda \) with \(\lambda _i < \lambda _{i+1}\), we have the following:
and
where \({{\,\mathrm{arm}\,}}(u)\ge 1\) has the same value in both diagrams \(\lambda \) and \({\tilde{\lambda }}\). \(\square \)
To simplify typesetting of the upcoming proofs, we will sometimes use the notation:
where \(\lambda \) is the composition:
Lemma 4.3
We have the identity:
Proof
We prove this lemma by induction on k, where the base case \(k=1\) is given by Lemma 4.1. For \(k >1\), by Proposition 2.8 and a similar reasoning as in Lemma 4.2, it is enough to prove that
Furthermore, through repeated application of the Knop–Sahi recurrence [Eq. (2.3)], it suffices to prove that
Again using Proposition 2.8, we reduce the above to the \(k1\) case:
which is true by induction. \(\square \)
Proposition 4.4
If \(\lambda \) is a composition, then
where \(F_\lambda (\mathbf {x})\) is an elementary symmetric polynomial.
Proof
We prove the proposition by induction on \(\lambda \) and the number of inversions of \(\lambda \). Note that the result is trivial if \(\lambda  \le 1\).
Given \(\lambda \), there are several cases to consider:
Case 1: \(\min _i\lambda _i \ge 1\). The result follows by inductive hypothesis on the size of the composition using Eq. (2.4) in the numerator.
Case 2: \(\lambda \) is not weakly increasing. We can reduce this case to a composition with fewer inversions using Lemma 4.2.
Case 3: \(\lambda \) is weakly increasing. It is enough to prove that
is an elementary symmetric polynomial where \(0=a_0<a_1<a_2<\cdots \). Using the cyclic shift relation (2.3) in the numerator and denominator, it suffices to show that
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. (4.1), 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 4.2, the second fraction is also an elementary symmetric polynomial. \(\square \)
Theorem 4.5
If \(\lambda \) is a composition and \(\sigma \in S_n\), then
where \(F_\lambda (\mathbf {x})\) is an elementary symmetric polynomial independent of t.
Proof
From Proposition 4.4, we have that
where \(F_\lambda \) is an elementary symmetric polynomial. Applying basementpermuting operators from Proposition 2.7 on both sides, then gives
Note that applying a basementpermuting operator might give an extra factor of t, but, since \(\lambda _{i} \le \lambda _{j}\) if and only if \(\tilde{\lambda _{i}}\le \tilde{\lambda _{j}}\), these extra factors cancel. \(\square \)
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 4.6
If \(\lambda \) is a partition and \(\sigma \in S_n\), then
Proof
It is enough to prove that \(\mathrm {E}^{w_0}_\lambda (\mathbf {x};1,t)=\mathrm {e}_{\lambda '}(\mathbf {x})\) as the more general statement follows from using Proposition 2.7.
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 2.8 implies that it is enough to prove that
Using the Knop–Sahi recurrence (2.3), it suffices to show that
which now follows from induction.\(\square \)
Corollary 4.7
The previous proof can be extended to show that
for partition \(\lambda \).
Note that the parts of \(\lambda '\) are a superset of the parts of \(({\tilde{\lambda }})'\), so the above expression is, indeed, some elementary symmetric polynomials.
Our results are in some sense optimal: for general compositions \(\lambda \), it happens that \(\mathrm {E}^\sigma _{{\tilde{\lambda }}}(\mathbf {x};1,t)\) cannot be factorized further. For example, Mathematica computations suggest that
are irreducible.
4.1 Discussion
It is natural to ask whether or not there are bijective proofs of the identities which we consider.
Question 4.8
Is there a bijective proof of the case \(\sigma =\omega _0\) of Theorem 4.5 that establishes
Since a priori \(\mathrm {E}^\sigma _{{\lambda }}(\mathbf {x};1,t)\) is only a rational function in t, this seems like a difficult challenge. We, therefore, pose a more conservative question:
Question 4.9
Is there a combinatorial explanation of the identity \(\mathrm {E}^\sigma _\lambda (\mathbf {x};1,t) = \mathrm {e}_{\lambda '}(\mathbf {x})\) whenever \(\lambda \) is a partition?
We finish this section by discussing properties of the family \(\{\mathrm {E}_\lambda (\mathbf {x};1,0)\}\) as \(\lambda \) ranges over compositions with n parts. It is a basis for \(\mathbb {C}[x_1,\cdots ,x_n]\) and is a natural generalization of the elementary symmetric functions in the same manner the key polynomials extend the family of Schur polynomials. For example, in a recent paper [3], it is proved that \(\mathrm {E}_\lambda (\mathbf {x};1,0)\) expands positively into key polynomials, where the coefficients are given by the classical Kostka coefficients. This generalizes the classical result that elementary symmetric functions expand positively into Schur polynomials. Furthermore, \(\{\mathrm {E}_\lambda (\mathbf {x};q,0)\}\) 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 [2, 3] and references therein for details.
It is known that the product of a Schur polynomial and a key polynomial is keypositive (see e.g. Proposition 5.8 below), and thus, the product of an elementary symmetric polynomial and a key polynomial is keypositive. It is, therefore, natural to ask if this extends to the nonsymmetric elementary polynomials. However, a quick computer search reveals that
does not expand positively into key polynomials.
5 Positive Expansions at \(q=0\)
By specializing the combinatorial formula [Eq. (2.2)] with \(q=0\), we obtain a combinatorial formula for the permutedbasement Demazure tatoms.
Example 5.1
As an example, \(\mathcal {A}^{1423}_{2301}(x_1,x_2,x_3,x_4;t)\) is equal to
where the corresponding fillings are as follows:
In this section, we show how to construct permutedbasement Demazure tatoms via Demazure–Lusztig operators. First, consider Proposition 2.7 and Proposition 2.8 at \(q=0\). Note that Proposition 2.8 simplifies, where we use the fact that \({\tilde{\theta }}_i +(1t) = {\tilde{\pi }}_i\). Hence, the shapepermuting operator reduces to a basementpermuting operator. This “duality” between shape and basement was first observed at \(t=0\) in [21], where S. Mason gave an alternative combinatorial description of key polynomials which is not immediate from the combinatorial formula for the nonsymmetric Macdonald polynomials. A similar duality holds for general values of t, see [1].
To get a better overview of Propositions 2.7 and 2.8, we present the statements as actions on the basement and shape as follows:
Example 5.2
The operators \({\tilde{\pi }}_i\) and \({\tilde{\theta }}_i\) 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 follows:
which are easily obtained from Proposition 2.8 at \(q=0\), together with the fact that \({\tilde{\theta }}_i {\tilde{\pi }}_i = t\).
The following proposition also appeared in [1]; however, the proof that we present here is different and more constructive.
Proposition 5.3
Given a composition \(\lambda \) with n parts and a permutation \(\sigma \in S_n\), there is a sequence of operators \({\tilde{\rho }}_{i_1}, \cdots ,{\tilde{\rho }}_{i_\ell }\), such that
where \(\mathrm {par}(\lambda )\) is the partition with the parts of \(\lambda \) in decreasing order and each operator \({\tilde{\rho }}_{i_j}\) is in the set \(\big \{ {\tilde{\theta }}_1,\cdots , {\tilde{\theta }}_{n1}, {\tilde{\pi }}_1,\cdots , {\tilde{\pi }}_{n1} \big \}\).
Proof
Given \((\sigma ,\lambda )\), let the number of monotone pairs be the number of pairs (i, j), such that
We do induction over the number of monotone pairs. First note that if there are no monotone pairs in \((\sigma ,\lambda )\), 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 \(\sigma \) and shape \(\lambda \) has to be constant, implying that \(\mathcal {A}_\lambda ^\sigma (\mathbf {x};t) = \mathbf {x}^\lambda \).
Assume that there are some monotone pairs determined by \((\sigma ,\lambda )\). 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 righthand sides of (5.2) and (5.1). By induction, \(\mathcal {A}_\lambda ^\sigma (\mathbf {x};t)\) can, therefore, be obtained from some \(\mathcal {A}_\lambda ^{\sigma s_i}(\mathbf {x};t)\) by applying either \({\tilde{\pi }}_i\) or \({\tilde{\theta }}_i\). \(\square \)
Example 5.4
We illustrate the above proposition by expressing \(\mathcal {A}^{3142}_{3102}(\mathbf {x};t)\) in terms of operators. The shape and basement associated with this atom is given in the first augmented diagram in (5.5):
The rows with labels 2 and 3 constitute a monotone pair and can be obtained using (5.2), which explains the \({\tilde{\pi }}_2\)arrow. Continuing on with \({\tilde{\pi }}_1\) followed by \({\tilde{\theta }}_2\) leads to an augmented diagram without any monotone pairs, so \(\mathcal {A}^{1342}_{3102}(\mathbf {x};t) = \mathbf {x}^{(3,2,1,0)}\). Finally, following the arrows yields the operator expression:
Proposition 5.5
If \(\sigma = s_i \tau \) with \({{\,\mathrm{\ell }\,}}(\sigma )>{{\,\mathrm{\ell }\,}}(\tau )\), then
where \({{\,\mathrm{stat}\,}}(\lambda ,\sigma ,i)\) is a nonnegative integer depending on \(\lambda \), \(\sigma \), and i.
Proof
We prove this statement via induction over \({{\,\mathrm{\ell }\,}}(\tau )\).
Case \(\tau = {{\,\mathrm{id}\,}}\) and \(\lambda _{i} \le \lambda _{i+1}\): We need to show that \(\mathcal {A}^{s_i}_\lambda (\mathbf {x};t) = \mathcal {A}^{{{\,\mathrm{id}\,}}}_{s_i\lambda }(\mathbf {x};t)\). Since \({\tilde{\pi }}_i\) is invertible, it suffices to show that
This equality now follows from using (5.3) on the lefthand side and (5.2) on the righthand side.
Case \(\tau = {{\,\mathrm{id}\,}}\) and \(\lambda _{i} > \lambda _{i+1}\): It suffices to prove that
Note that the lefthand side is equal to \({\tilde{\pi }}_i \mathcal {A}^{{{\,\mathrm{id}\,}}}_\lambda (\mathbf {x};t)\) using (5.2), while the lefthand side is equal to \([{\tilde{\theta }}_i + (1t)] \mathcal {A}^{{{\,\mathrm{id}\,}}}_\lambda (\mathbf {x};t)\) where we use (5.3). Since \({\tilde{\pi }}_i = [{\tilde{\theta }}_i + (1t)]\), this proves the identity.
This proves the base case. The general case now follows from applying \({\tilde{\pi }}_j\) on both sides, thus, increasing the lengths of the basements. We examine the details in the following two cases.
Case \(\tau \in S_n\) and \(\lambda _{i} \le \lambda _{i+1}\): suppose \(\mathcal {A}^{\sigma }_\lambda (\mathbf {x};t) = \mathcal {A}^{\tau }_{s_i\lambda }(\mathbf {x};t)\). As diagrams, we have the equality:
for rows i and \(i+1\), \(b>a\), while the remaining rows are identical. If \({{\,\mathrm{\ell }\,}}(\sigma s_j) > {{\,\mathrm{\ell }\,}}(\sigma )\), we can conclude that if \(a=j\), then \(b \ne j+1\). We now compare the row lengths of the rows with basement label j and \(j+1\) and apply the basementpermuting \({\tilde{\pi }}_j\) from (5.2) on both sides. Note that the row lengths that are compared are the same on both sides, meaning that if we need (5.2) to increase the basement on the lefthand side, the same relation acts the same way on the righthand side. In other words, we have the implication:
whenever \({{\,\mathrm{\ell }\,}}(\sigma s_j) > {{\,\mathrm{\ell }\,}}(\sigma )\) and \(\lambda _{i} \le \lambda _{i+1}\).
Case \(\tau \in S_n\) and \(\lambda _{i} > \lambda _{i+1}\): Again, suppose that we have the diagram identity:
for some \(\lambda \), \(\sigma \), and that \({{\,\mathrm{\ell }\,}}(\sigma s_j) > {{\,\mathrm{\ell }\,}}(\sigma )\). As in the previous case, if \(a=j\), then \(b \ne j+1\). If \(j \notin \{a1,a,b1,b\}\), applying \({\tilde{\pi }}_j\) on both sides yields 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 \(j \in \{a1,a,b1,b\}\). Casebycase study after applying \({\tilde{\pi }}_j\) on both sides shows that
where (using the same notation as in Proposition 2.7, \(\gamma _i\) being the length of the row with basement label i):

\(\epsilon = 1\) if \(j=a1\) and \(\gamma _a > \gamma _{a1} \ge \gamma _b\),

\(\epsilon = 1\) if \(j=a\) and \(\gamma _a \ge \gamma _{a+1} > \gamma _b\),

\(\epsilon = 1\) if \(j=b1\) and \(\gamma _a > \gamma _{b1} \ge \gamma _b\),

\(\epsilon = 1\) if \(j=b\) and \(\gamma _a \ge \gamma _{b+1} > \gamma _b\),
and \(\epsilon =0\) otherwise. Thus, we have that
where the lefthand side is a polynomial. Furthermore, \(\mathcal {A}^{\tau s_j}_\lambda (\mathbf {x};t)\) is not a multiple of t—this follows from the combinatorial formula (2.2). Hence, \(\epsilon +{{\,\mathrm{stat}\,}}(\lambda ,\sigma ,i)\) must be nonnegative.\(\square \)
Corollary 5.6
If \(\tau \ge \sigma \) in Bruhat order, then \(\mathcal {A}^{\tau }_\gamma (\mathbf {x};t)\) admits the expansion:
where the \(c^{\tau \sigma }_{\gamma \lambda }(t)\) are polynomials in t, with the property that \(c^{\tau \sigma }_{\gamma \lambda }(t) \ge 0\) whenever \(0\le t \le 1\).
Corollary 5.7
If \(\tau \ge \sigma \) in Bruhat order, then \(\mathcal {A}^{\tau }_\gamma (\mathbf {x})\) admits the expansion:
Proof
Let \(t=0\) in (5.6). It is then clear that all coefficients are nonnegative integers. Furthermore, since key polynomials \((\tau = \omega _0)\) expand into Demazure atoms \((\sigma = {{\,\mathrm{id}\,}})\) with coefficients in \(\{0,1\}\) (see e.g., [16, 21]), the statement follows. \(\square \)
In [13], the cases \(\sigma = {{\,\mathrm{id}\,}}\) and \(\sigma =\omega _0\) of the following proposition were proved. We give an interpolation (in the Bruhat order) between these results. Recall that \(\mathbf {x}= (x_1,\cdots ,x_n)\), so we evaluate \(\mathrm {s}_\mu (\mathbf {x})\) in a finite alphabet.
Proposition 5.8
The coefficients \(d^{\mu \sigma }_{\lambda \gamma }\) in the expansion
are nonnegative integers.
Proof
With the case \(\sigma ={{\,\mathrm{id}\,}}\) as a starting point (proved in [13]), we can apply \(\pi _i\) on both sides, (\(\pi _i\) commutes with any symmetric function, in particular \(\mathrm {s}_\lambda (\mathbf {x})\)), and thus, we may walk upwards in the Bruhat order and obtain the statement for any basement \(\sigma \). Note that Proposition 2.7 implies that \(\pi _i\) applied to \(\mathcal {A}^\sigma _\gamma (\mathbf {x})\) either increases \(\sigma \) in Bruhat order, or kills that term. \(\square \)
Note that the above result implies that the products \(\mathrm {e}_\mu \times \mathcal {A}^\sigma _\lambda (\mathbf {x})\) and \(h_\mu \times \mathcal {A}^\sigma _\lambda (\mathbf {x})\) also expand nonnegatively into \(\sigma \)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 5.9
We need to mention the paper [17], which also concerns a different type of general Demazure atoms. These objects are also studied in [13], but are, in general, different from ours when \(\sigma \ne {{\,\mathrm{id}\,}}\). In particular, the polynomial families which they study are not bases for \(\mathbb {C}[x_1,\cdots ,x_n]\), and they are not compatible with the Demazure operators. The authors of [13, 17] construct these families by imposing an additional restriction^{Footnote 1} on Haglund’s combinatorial model, which enables them to perform a type of RSK.
The introductions of both the papers [13, 17] mention the permutedbasement Macdonald polynomials \(\mathrm {E}^\sigma _\mu (\mathbf {x};q,t)\). However, the additional restriction imposed further on breaks this connection whenever \(\sigma \ne {{\,\mathrm{id}\,}}\). This fact is unfortunately hidden, since the same notation, \({\hat{E}}_\gamma \), is used for two different families of polynomials.
Notes
What they call the typeB condition.
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Acknowledgements
The authors would like to thank Jim Haglund for insightful discussions, and the anonymous referee for valuable corrections. The first author is funded by the Knut and Alice Wallenberg Foundation (2013.03.07).
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Alexandersson, P., Sawhney, M. Properties of Nonsymmetric Macdonald Polynomials at \(q=1\) and \(q=0\). Ann. Comb. 23, 219–239 (2019). https://doi.org/10.1007/s0002601900432z
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DOI: https://doi.org/10.1007/s0002601900432z
Keywords
 Macdonald polynomials
 Elementary symmetric functions
 Key polynomials
 Hall–Littlewood
 Demazure characters
 Factorization
Mathematics Subject Classification
 05E10
 05E05