Properties of Nonsymmetric Macdonald Polynomials at \(q=1\) and \(q=0\)
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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.
Keywords
Macdonald polynomials Elementary symmetric functions Key polynomials Hall–Littlewood Demazure characters FactorizationMathematics Subject Classification
05E10 05E051 Introduction
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
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
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.
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
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
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
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 \({{\,\mathrm{\ell }\,}}(\omega )\).
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, \(\gamma _i = \lambda _{\sigma ^{1}_i}\).
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
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
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
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
4 The Factorization Property
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 \(\sigma = \text {id}\).
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
Proof
Lemma 4.3
Proof
Proposition 4.4
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.
Theorem 4.5
Proof
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
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.
Corollary 4.7
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.
4.1 Discussion
It is natural to ask whether or not there are bijective proofs of the identities which we consider.
Question 4.8
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.
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
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 following proposition also appeared in [1]; however, the proof that we present here is different and more constructive.
Proposition 5.3
Proof
Example 5.4
Proposition 5.5
Proof
We prove this statement via induction over \({{\,\mathrm{\ell }\,}}(\tau )\).
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.

\(\epsilon = 1\) 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 \(\gamma _a \ge \gamma _{b+1} > \gamma _b\),
Corollary 5.6
Corollary 5.7
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
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^{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.
Footnotes
 1.
What they call the typeB condition.
Notes
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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