#Macdonald E polynomials
Non-symmetric Macdonald polynomials were introduced in [Opd95], [Mac96], and [Che95]. They are closely related to affine root systems and the double affine Hecke algebra. Their definition is rather indirect, and does not give an efficient way of computing these non-symmetric polynomials.
In [HHL08], J. Haglund, M. Haiman, and N. Loehr found an explicit formula for the non-symmetric Macdonald polynomials in type \(A,\) as a sum over non-attacking fillings. This model is the basis for the permuted basement Macdonald polynomials, and we refer to that page for definitions. The specialization \(q=t=0\) gives key polynomials, while the limit \(q,t\to\infty\) gives Demazure atoms [Mas09].
Example (A tiny non-attacking filling).
The permuted-basement page uses English augmented diagrams. For shape \((2,0,1)\) and basement \((3,2,1),\) one filling is \[F= \begin{array}{c|cc} 3 & 2 & 1\\ 2 & {} & {}\\ 1 & 1 & {} \end{array}.\] Among equal entries, the only adjacent-column pairs have their right-hand entry in the same row as, or in a row above, the other entry. Hence no right-hand entry lies strictly below the other one. This is one of the non-attacking augmented fillings which contributes \(x_1^2x_2\) to \(\key_{(1,0,2)}(x_1,x_2,x_3).\)
Later, an alternative combinatorial formula using alcove walks was proved by A. Ram and M. Yip [RY11], which works for all Lie types. At \(q=t=0,\) their model reduces to the Littelmann path model.
In [BW19], an integrable vertex model is used to give an alternative formula for non-symmetric Macdonald polynomials.
Example
The local building block in [Eq. (3.7), BW19] is a face \[L_x(I,j;K,\ell) = \begin{array}{ccc} & K & \\ j & \boxed{x} & \ell \\ & I & \end{array},\] where the vertical labels \(j,\ell\) are colors in \(\{0,1,\dotsc,n\},\) and the horizontal labels \(I,K\) are occupation vectors in \(\setZ_{\geq 0}^n.\) For \(n=3,\) one of their sample faces is \[I=(1,1,1),\qquad j=1,\qquad K=(2,0,1),\qquad \ell=2.\] Here the conservation rule \(I+e_1=K+e_2\) holds, and the corresponding Boltzmann weight is \[L_x((1,1,1),1;(2,0,1),2)=x(1-t)t.\] The factor of \(x\) appears because the right edge has nonzero color. In the full model, these faces are concatenated into row operators and then read as colored lattice paths on a cylinder; the product of the local \(x\)-factors gives the monomial part of a path configuration.
See the book [Hag07] for more background on the type \(A\) non-symmetric Macdonald polynomials. Further related work on non-symmetric Macdonald polynomials and their generalizations can be found in [Ale19, AS17, AS19].
C. F. Dunkl evaluates families of nonsymmetric Macdonald superpolynomials at special points [Dun21]. These superpolynomials take values in Hecke-algebra modules built from anti-commuting variables, and the evaluations involve \((q,t)\)-hook products.
Milo Bechtloff Weising constructs stable-limit nonsymmetric Macdonald functions in type \(A\) [Wei23]. They form a simultaneous eigenbasis for the limit Cherednik operators in Ion–Wu’s stable-limit double affine Hecke algebra representation, connecting stable DAHA theory with the Carlsson–Mellit double Dyck path algebra. Milo Bechtloff Weising and A. E. Black prove that the supports of nonsymmetric Macdonald polynomials are \(M\)-convex [WB25]. This proves the saturated Newton polytope property conjectured by Monical–Tokcan–Yong, and also gives \(M\)-convex supports for affine Demazure characters of type \(\mathrm{GL}.\)
A model using multiline queues is introduced in [CMW18], and further explored in [CHMM+19].
Example
The following small example fixes the row and weight conventions at \(t=0.\) In the notation of [Defs. 3.1, 3.12 and Ex. 3.16, MV24], let \(n=6\) and consider the multiline queue \[M= \bigl( \{1,2,3,4\}, \{1,3,5,6\}, \{2,3\}, \{3,5\} \bigr),\] where the rows are listed from bottom to top. As a diagram, this is \[\begin{array}{c|cccccc} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 4 & & & \bullet & & \bullet & \\ 3 & & \bullet & \bullet & & & \\ 2 & \bullet & & \bullet & & \bullet & \bullet \\ 1 & \bullet & \bullet & \bullet & \bullet & & \end{array}\] The column word is \[21 \mid 31 \mid 4321 \mid 1 \mid 42 \mid 2.\] Running the Ferrari–Martin pairing algorithm gives the pairing data \[(4,4,0),(4,4,1),(3,4,0),(3,4,0), (2,4,0),(2,4,1),(2,2,0),(2,2,1),\] where the last coordinate records whether the pairing wraps around the ring. Thus the major index is \[\operatorname{maj}(M)=1+3+1=5,\] and the queue contributes \[\operatorname{wt}(M)=x_1^2x_2^2x_3^4x_4x_5^2x_6q^5.\] This is the Ferrari–Martin, or TASEP-side, part of the \(t=0\) theory; the enhanced and twisted versions of multiline queues add the extra data needed for the Macdonald-polynomial formulas.
O. Mandelshtam and J. Valencia-Porras use twisted multiline queues to model Macdonald polynomials at \(t=0\) [MV24]. This adds another multiline-queue realization to the non-symmetric Macdonald polynomial circle.
A. Ayyer, J. Martin, and L. Williams connect ASEP polynomials and nonsymmetric Macdonald polynomials at \(q=1\) to the stationary distribution of an inhomogeneous multispecies \(t\)-PushTASEP on a ring [AMW24]. The normalizing partition function is the corresponding Macdonald \(P\) polynomial at \(q=1,\) and the proof uses multiline diagrams.
E. Feigin, A. Khoroshkin, and I. Makedonskyi introduce parasymmetric Macdonald polynomials, interpolating between symmetric and nonsymmetric Macdonald polynomials and depending on a parabolic subalgebra [FKM23]. They categorify specializations of these polynomials using cyclic modules for parahoric subalgebras of affine Kac–Moody Lie algebras.
#Monk’s rule
W. Baratta [Bar09, Bar11] gives Monk-type rules (terminology from Schubert calculus) for products of the form \[x_j \macdonaldE_\mu(x_1,\dotsc,x_n;q,t), \quad \elementaryE_1(\xvec) \cdot \macdonaldE_\mu(x_1,\dotsc,x_n;q,t), \quad \elementaryE_{r}(\xvec) \cdot\macdonaldE_\mu(x_1,\dotsc,x_n;q,t),\] expanded again in the \(\{\macdonaldE_\alpha\}\) basis. Baratta uses interpolation Macdonald polynomials in the proof.
In [HR22], the authors prove Monk-type formulas for \[x_j \macdonaldE_\mu, \qquad (x_1+\dotsb+x_j) \macdonaldE_\mu, \qquad x_j^{-1} \macdonaldE_\mu, \qquad (x_j^{-1}+\dotsb+x^{-1}_n) \macdonaldE_\mu.\] Halverson and Ram use a different method based on intertwiners.
#Recurrence relations
The HHL filling formula is often proved by checking the Knop–Sahi recurrence. There are two common indexing orientations. Haglund–Haiman–Loehr, and the skyline convention used by Moura–Mandelshtam, index a composition by column heights. This website follows Alexandersson’s English-diagram convention; when \(\sigma=\omega_0,\) our index \(\alpha\) corresponds to the HHL index \(\rev(\alpha).\) Thus the cyclic recurrence is reversed compared with the HHL display. In the website convention, let \(\alpha=(\alpha_1,\dotsc,\alpha_n)\) and \[\widehat{\alpha}\coloneqq(\alpha_2,\dotsc,\alpha_n,\alpha_1+1),\qquad \Psi f\coloneqq x_1f(x_2,\dotsc,x_n,q^{-1}x_1).\] Then \[\macdonaldE_{\widehat{\alpha}}(\xvec;q,t) = q^{\alpha_1}\Psi\macdonaldE_\alpha(\xvec;q,t)\] in Alexandersson’s convention [Sec. 3, Ale19]. Equivalently, after setting \(\mu=\rev(\alpha),\) this is the HHL recurrence [Sec. 2.1, HHL08].
In HHL skyline indexing, if \(\mu_i\gt{}\mu_{i+1}\) and \(u=(i,\mu_{i+1}+1)\) is the corresponding cell in the column diagram, then \[\macdonaldE_{s_i\mu}(\xvec;q,t) = \left( T_i+ \frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)}} \right) \macdonaldE_\mu(\xvec;q,t),\] where \(T_i\) is the affine Hecke operator in the HHL convention. Translating this adjacent-part recurrence to the website convention reverses the shape positions. Together with \(\macdonaldE_{(0,\dotsc,0)}=1,\) the cyclic recurrence and the special cases of the adjacent recurrence uniquely determine all non-symmetric Macdonald polynomials [Sec. 2.1, HHL08].
For permuted-basement Macdonald polynomials, Demazure–Lusztig operators give a second recursive viewpoint. Let \[\widetilde{\pi}_i\coloneqq (1-t)\pi_i+t s_i,\qquad \widetilde{\theta}_i\coloneqq (1-t)\theta_i+t s_i,\] where \(\pi_i\) and \(\theta_i\) are the usual Demazure character and atom operators. If \(\gamma_i\) is the length of the row whose basement label is \(i,\) then Alexandersson’s basement-permuting formulas state [Prop. 15, Ale19] that \[\widetilde{\theta}_i\macdonaldE^\sigma_\alpha = \begin{cases} t\macdonaldE^{\sigma s_i}_\alpha, & \text{if } \ell(\sigma s_i)\lt{}\ell(\sigma) \text{ and } \gamma_i\leq\gamma_{i+1},\\ \macdonaldE^{\sigma s_i}_\alpha, & \text{if } \ell(\sigma s_i)\lt{}\ell(\sigma) \text{ and } \gamma_i\gt{}\gamma_{i+1}, \end{cases}\] and \[\widetilde{\pi}_i\macdonaldE^\sigma_\alpha = \begin{cases} t\macdonaldE^{\sigma s_i}_\alpha, & \text{if } \ell(\sigma s_i)\gt{}\ell(\sigma) \text{ and } \gamma_i\lt{}\gamma_{i+1},\\ \macdonaldE^{\sigma s_i}_\alpha, & \text{if } \ell(\sigma s_i)\gt{}\ell(\sigma) \text{ and } \gamma_i\geq\gamma_{i+1}. \end{cases}\]
There are also shape-permuting versions. For example, if \(\alpha_j\lt{}\alpha_{j+1}\) and the adjacent basement labels are \(\sigma_j=i+1,\) \(\sigma_{j+1}=i,\) then \[\macdonaldE^\sigma_{s_j\cdot\alpha} = \left( \widetilde{\theta}_i+ \frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)}} \right)\macdonaldE^\sigma_\alpha,\] where \(u=(j+1,\alpha_j+1)\) is computed in the diagram of \(\alpha\) [Prop. 17, Ale19]. The inverse case \(\alpha_j\gt{}\alpha_{j+1}\) is given in [Prop. 18, Ale19]. The more recent shape-changing identities of Moura–Mandelshtam remove the adjacent-basement restriction and give a straightening rule in the full permuted-basement family.
#Non-symmetric q-Whittaker polynomials
The non-symmetric \(q\)-Whittaker polynomials are obtained by letting \(t=0\) in \(\macdonaldE_\mu(\xvec;q,t).\) The result can be thought of as a non-symmetric analogue of the transformed Hall–Littlewood polynomials, and generalizes the \(q\)-Whittaker functions. We keep this family distinct from the modified Hall–Littlewood polynomials: the former comes from the \(t=0\) Macdonald specialization, while the latter is the \(q=0\) specialization of modified Macdonald polynomials. There are nevertheless direct comparisons between their filling models; see [Rat24, Bha25].
The polynomials \(\macdonaldE_\mu(\xvec;q,0)\) expand positively in key polynomials, and the coefficients are Kostka–Foulkes polynomials, see [AS17, Ass18, AG18]. S. Assaf and N. González construct an affine Demazure crystal on semistandard key tabloids whose characters realize these specialized nonsymmetric Macdonald polynomials [AG21]. Removing the affine edges recovers their finite Demazure crystals.
Models using quantum alcove walks and quantum Lakshmibai–Seshadri paths are considered in [LNSS+17].
#Non-symmetric elementary polynomials
The non-symmetric elementary polynomials are the specialization \(\macdonaldE_\lambda(\xvec;1,0),\) where \(\lambda\) ranges over weak compositions. For a fixed number of variables, these polynomials form a basis of the polynomial ring and naturally extend the ordinary elementary symmetric functions [Sec. 4.1, AS19]. More precisely, if \(\lambda\) is a partition, then the corresponding permuted-basement polynomial \(\macdonaldE^\sigma_\lambda(\xvec;1,t)\) is independent of \(\sigma\) and \(t,\) and is equal to \(\elementaryE_{\lambda'}(\xvec)\) [Thm. 18, AS19].
They expand positively in key polynomials, with coefficients given by ordinary Kostka numbers [Ass18]. This is the \(q=1\) specialization of the key-positive expansion of \(\macdonaldE_\lambda(\xvec;q,0)\) by Kostka–Foulkes polynomials.
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@article{WeisingBlack2025x, author = {Milo Bechtloff Weising and Alexander E. Black}, title = {Saturation for {N}on-{S}ymmetric {M}acdonald {P}olynomials}, year = {2025}, eprint = {2508.00336}, url = {https://arxiv.org/abs/2508.00336}, journal = {arXiv e-prints} } - [Hag07]James Haglund. The $q,t$-Catalan numbers and the space of diagonal harmonics (University lecture series). American Mathematical Society, 2007.
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@book{qtCatalanBook, Author = {James Haglund}, Title = {The $q,t$-{C}atalan numbers and the space of diagonal harmonics ({U}niversity lecture series)}, Publisher = {American Mathematical Society}, Year = {2007}, ISBN = {0821844113}, url = {https://www.math.upenn.edu/~jhaglund/books/qtcat.pdf} }