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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.

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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