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Nonsymmetric Macdonald polynomials and a refinement of Kostka-Foulkes polynomials. (English) Zbl 1404.33017
An in-depth discussion on the analysis of the function $${E_a }({X;q,t})$$ for the special case $$t=0$$ is made in this paper, wherein the author shows that the specialized function ‘$${E_a}({X;q,0})$$ stabilizes to $$\omega {P_\mu }({X;0,t})$$’, where $${P_\mu }({X;0,t})$$ denotes the Hall-Littlewood polynomials and $$\omega$$ is the ‘involution on symmetric functions’. The author also ‘relates $${E_a}({X;q,0})$$ to the (finite) type A Demazure characters $$E_a ({X; 0,0})$$’.
In his another paper [“Weak dual equivalence for polynomials”, Preprint, arXiv:1702.04051], the author has developed the theory of weak dual equivalence and introduced the ‘standard key tableaux to develop a theory of type A Demazure characters’ which is invoked by him in this paper to give a combinatorial proof of the fact that on grouping ‘together the terms in the fundamental slide expansion of $${E_a}({X;q,0})$$, the coefficients of $${E_a}({X;q,0})$$, when expanded into Demazure characters, are polynomials in $$q$$ with nonnegative integer coefficients.’ This treatment here parallels the earlier treatment of ‘the use of dual equivalence’ by the author in his work [Forum Math. Sigma 3, Article ID e12, 33 p. (2015; Zbl 1319.05135)] regarding the fundamental quasisymmetric expansion of $${H_\mu}({X;q,t})$$ (the transformed Macdonald symmetric functions in type A).
The first significant result of the paper is:
Theorem 3.6. The specialized nonsymmetric Macdonald polynomial $${E_a}({X;q,0})$$ is given by ${E_a}({X;q,0}) = \sum_{T \in {\text{SKD}}(a)} {{q^{{\text{maj}}(T)}}{{\mathcal{F}}_{{\text{des}}(T)}}(X)},$ where $${{\text{SKD}}(a)}$$ denotes the standard key tabloids of shape $$a$$, $$\mathcal{F}_a$$ denotes the fundamental slide polynomial (see [S. Assaf and D. Searles, Adv. Math. 306, 89–122 (2017; Zbl 1356.14039)]) defined on the finite set $$X$$ of variables $$x_1, \ldots, x_n$$ by the relation $${{\mathcal{F}}_a}(X) = \sum_{b \geqslant a; {\text{flat}}(b){\text{ refines flat}}(a)}{{X^b}}$$ in which ‘$${\text{flat}}(a)$$ is the composition obtained by removing zero parts from $$a$$’, $${{\text{des}}(T)}$$ denotes the weak descent composition of $$T$$ for a standard filling $$T$$ of a key diagram and $${{\text{maj}}(T)}$$ represents ‘the sum of the legs of all cells $$c$$ (of a key diagram) such that the entry in $$c$$ is strictly greater than the entry immediately to its left.’ Another important result is the following theorem:
Theorem 4.7. For a weak composition $$a$$ such that $$\text{SKD}(a)$$ has no virtual elements, the maps $$\left\{ \psi_i \right\}$$ on $$\text{SKD}(a)$$ give a weak dual equivalence for $$(\text{SKD}(a),\text{des})$$.
The Demazure character $${\kappa _a}(X)$$ is given by the author in [loc. cit., arXiv:1702.04051] and in this paper he beautifully develops the relation between the functions $${E_a}({X;q,0})$$ and Demazure characters in the following result:
Theorem 4.9. The specialized nonsymmetric Macdonald polynomial $${E_a}({X;q,0})$$ given by $${E_a}({X;q,0}) = \sum_{T \in {\text{YKD}}(a)} {{q^{{\text{maj}}(T)}}{\kappa _{{\text{des}}(T)}}}.$$ In particular, $${E_a}({X;q,0})$$ is a positive graded sum of Demazure characters.
The following theorem is a landmark result of this paper:
Theorem 5.6. For a weak composition $$a$$, we have $\lim_{m \to \infty } {E_{{0^m} \times a}}({X;q,0}) = \omega {H_{{\text{sort}}(a)'}}({X;0,q}) = \omega {H_{{\text{sort}}(a)}}({X;q,0})$ By defining the nonsymmetric Kostka-Foulkes polynomial $${K_{a,b}}(q)$$ by the relation $${E_b}({X;q,0}) = \sum_a {{K_{a,b}}(q){\kappa _{\text{a}}}(X)}$$ the author redevelops the Theorem 5.6 in terms of Kostka-Foulkes polynomials as follows:
Corollary 5.7. Given a weak composition $$b$$ with column lengths $$\mu$$ such that $${{\text{SKT}}(b)}$$ has no virtual Yamanouchi elements, we have ${K_{\lambda ,\mu }}(t) = \sum_{{\text{sort}}({{\text{flat}}(a)}) = \lambda '} {{K_{a,b}}(t)}.$
The reviewer finds the paper an important and valuable contribution to the theory of nonsymmetric Macdonald polynomials and their interconnection with Demazure characters.

##### MSC:
 33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) 05E05 Symmetric functions and generalizations 14N15 Classical problems, Schubert calculus
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