Partially Symmetric Macdonald Polynomials

• Partially symmetric Macdonald polynomials are Macdonald-theoretic objects obtained by enforcing symmetry on only a subset of variables, bridging symmetric and nonsymmetric cases.
• They are constructed via partial symmetrization of nonsymmetric Macdonald polynomials using partition-composition indexing and special Hecke symmetrizers.
• Their study involves combinatorial models, inversion identities, and geometric connections to parabolic flag Hilbert schemes, advancing positivity and algebraic formulations.

Partially symmetric Macdonald polynomials are type $A$ Macdonald-theoretic objects obtained by imposing symmetry on only one distinguished block of variables. In one standard convention, the polynomial $P_{(\lambda\mid\gamma)}(x;q,t)$ is symmetric in the first $n-k$ variables and not necessarily symmetric in the final $k$ variables, where $\lambda\in \mathbb Z_{\ge 0}^{\,n-k}$ is a partition and $\gamma\in \mathbb Z_{\ge 0}^{\,k}$ is a composition. In the equivalent $m$-symmetric convention, one instead works with polynomials symmetric in the tail variables $x_{m+1},x_{m+2},\dots$ and unrestricted in $x_1,\dots,x_m$. These conventions differ by a change of variables, and the family interpolates between ordinary symmetric Macdonald polynomials and nonsymmetric Macdonald polynomials (Goodberry, 2023, Concha et al., 2023).

1. Definitions and indexing conventions

The type $A$ partially symmetric Macdonald polynomials are indexed by a partition-composition pair. In the convention of Goodberry and Orr, one writes a sequence as $P_{(\lambda\mid\gamma)}(x;q,t)$0, with $P_{(\lambda\mid\gamma)}(x;q,t)$1 the “symmetric” part and $P_{(\lambda\mid\gamma)}(x;q,t)$2 the “nonsymmetric” part, and defines $P_{(\lambda\mid\gamma)}(x;q,t)$3 so that it is symmetric in $P_{(\lambda\mid\gamma)}(x;q,t)$4 but not necessarily in $P_{(\lambda\mid\gamma)}(x;q,t)$5. In the stable $P_{(\lambda\mid\gamma)}(x;q,t)$6-symmetric framework, the ambient space is

$P_{(\lambda\mid\gamma)}(x;q,t)$7

and its bases are indexed by $P_{(\lambda\mid\gamma)}(x;q,t)$8-partitions $P_{(\lambda\mid\gamma)}(x;q,t)$9, where $n-k$0 is a composition and $n-k$1 is an ordinary partition. The associated monomials are $n-k$2 (Concha et al., 2023, Lapointe, 2022).

The interpolating character of the theory is fundamental. If the symmetric block occupies all variables, one recovers the usual symmetric Macdonald polynomials. If the symmetric block is empty, or equivalently if $n-k$3 is large enough compared with the length of the indexing composition in the $n-k$4-symmetric formalism, one recovers nonsymmetric Macdonald polynomials. The papers therefore treat partially symmetric Macdonald polynomials not as an auxiliary deformation, but as a genuine intermediate regime linking the symmetric and nonsymmetric theories (Goodberry, 2023, Lapointe, 2022).

A second notational point is that the “parabolic variables” in geometric papers are often denoted $n-k$5, so that

$n-k$6

This separation of symmetric and nonsymmetric variables is built into both the algebraic and geometric formulations (Orr et al., 2024).

2. Construction from nonsymmetric Macdonald polynomials

The basic construction starts from nonsymmetric Macdonald polynomials $n-k$7 and applies a partial Hecke symmetrizer. In the finite-variable convention with symmetry in the first $n-k$8 variables, one sets

$n-k$9

where

$k$0

is the partial symmetrizer in the first $k$1 variables, and $k$2 is the Poincaré polynomial of the stabilizer of $k$3. The result is symmetric in the prescribed block and reduces to ordinary or nonsymmetric Macdonald theory in the extreme cases (Goodberry, 2023).

In the $k$4-symmetric convention, if $k$5 and $k$6, one forms

$k$7

and defines

$k$8

with $k$9 the $\lambda\in \mathbb Z_{\ge 0}^{\,n-k}$0-symmetrizer acting only on $\lambda\in \mathbb Z_{\ge 0}^{\,n-k}$1. This yields a basis of $\lambda\in \mathbb Z_{\ge 0}^{\,n-k}$2, or equivalently of $\lambda\in \mathbb Z_{\ge 0}^{\,n-k}$3 in the notation of Lapointe-style $\lambda\in \mathbb Z_{\ge 0}^{\,n-k}$4-symmetric functions (Concha et al., 2023, Lapointe, 2022).

The relation to the nonsymmetric basis remains explicit. Goodberry and Orr derive recursion and closed formulas for the expansion of $\lambda\in \mathbb Z_{\ge 0}^{\,n-k}$5 into nonsymmetric Macdonald polynomials, while later combinatorial work refines such expansions to individual permuted-basement terms. This is significant because it preserves access to the fine Hecke-algebraic structure of $\lambda\in \mathbb Z_{\ge 0}^{\,n-k}$6 inside the partially symmetrized theory (Goodberry, 2023, Orr et al., 28 Aug 2025).

3. Stability, integral forms, orthogonality, and degree-one structure

Several foundational properties now parallel the classical symmetric theory. Stability is established in both finite and stable formulations: the partially symmetric polynomials behave compatibly with adding a zero part on the symmetric side, and in the $\lambda\in \mathbb Z_{\ge 0}^{\,n-k}$7-symmetric setting they stabilize as elements of $\lambda\in \mathbb Z_{\ge 0}^{\,n-k}$8. Integral forms are defined using Young-diagram statistics. One writes

$\lambda\in \mathbb Z_{\ge 0}^{\,n-k}$9

where $\gamma\in \mathbb Z_{\ge 0}^{\,k}$0 is an explicit product of arm and leg terms, and one obtains

$\gamma\in \mathbb Z_{\ge 0}^{\,k}$1

The same circle of ideas leads, in the modified theory, to the partially symmetric analogues $\gamma\in \mathbb Z_{\ge 0}^{\,k}$2 of Haiman’s modified Macdonald polynomials (Goodberry, 2023, Orr et al., 2024).

Orthogonality is developed in the $\gamma\in \mathbb Z_{\ge 0}^{\,k}$3-symmetric framework via a natural scalar product on $\gamma\in \mathbb Z_{\ge 0}^{\,k}$4-symmetric power sums. If $\gamma\in \mathbb Z_{\ge 0}^{\,k}$5, then

$\gamma\in \mathbb Z_{\ge 0}^{\,k}$6

With respect to this scalar product, the $\gamma\in \mathbb Z_{\ge 0}^{\,k}$7-symmetric Macdonald polynomials are characterized as the unique basis satisfying orthogonality together with unitriangularity in the monomial basis. Explicit formulas are given for squared norms, principal specialization, and inclusion, and a Cauchy-type identity is proved that specializes to the usual symmetric Macdonald Cauchy identity for $\gamma\in \mathbb Z_{\ge 0}^{\,k}$8 and to the nonsymmetric reproducing kernel when the symmetric part is absent (Concha et al., 2023).

The degree-one multiplication theory is also substantially developed. Goodberry and Orr construct Pieri-type rules for $\gamma\in \mathbb Z_{\ge 0}^{\,k}$9 with $m$0 in the nonsymmetric block and for $m$1. They further show substantial combinatorial simplification of the $m$2-multiplication formula, and the final coefficients can be expressed through arm-leg data read directly from the Young diagram (Goodberry, 2023).

4. Combinatorial models and inversion identities

The modern combinatorics of partially symmetric Macdonald polynomials is closely tied to the Haglund–Haiman–Loehr filling formula for nonsymmetric Macdonald polynomials and to Alexandersson’s permuted-basement polynomials. For a permutation $m$3, the permuted-basement Macdonald polynomial is

$m$4

These objects refine the terms appearing in partially symmetrized sums and make it possible to formulate identities at the level of individual basement contributions rather than only after summation (Orr et al., 28 Aug 2025).

A central identity is the Concha–Lapointe inversion formula for partially symmetric Macdonald polynomials: $m$5 When $m$6, this reduces to the classical invariance of ordinary Macdonald polynomials under $m$7. When $m$8, it becomes the established nonsymmetric inversion symmetry (Orr et al., 28 Aug 2025).

The 2025 combinatorial proof refines this to a permuted-basement identity. If $m$9, $x_{m+1},x_{m+2},\dots$0, $x_{m+1},x_{m+2},\dots$1, $x_{m+1},x_{m+2},\dots$2, and $x_{m+1},x_{m+2},\dots$3, then

$x_{m+1},x_{m+2},\dots$4

where $x_{m+1},x_{m+2},\dots$5 and $x_{m+1},x_{m+2},\dots$6 are the reverses in their respective blocks. The proof constructs an explicit bijection between the relevant non-attacking fillings and tracks the statistics $x_{m+1},x_{m+2},\dots$7, $x_{m+1},x_{m+2},\dots$8, and $x_{m+1},x_{m+2},\dots$9 so that the weights transform correctly under reversal of variables and inversion of parameters. This refinement shows that the global partially symmetric identity is already visible at the level of the combinatorial summands (Orr et al., 28 Aug 2025).

5. Kazhdan–Lusztig involution and parabolic flag Hilbert schemes

The inversion identity has an equivalent reformulation in terms of the Kazhdan–Lusztig involution. In the partially symmetric setting, the involution acts by

$x_1,\dots,x_m$0

and one obtains

$x_1,\dots,x_m$1

where

$x_1,\dots,x_m$2

The 2025 paper proves that the Concha–Lapointe identity is equivalent to the statement that normalized partially symmetric Macdonald polynomials are fixed under the Kazhdan–Lusztig involution (Orr et al., 28 Aug 2025).

A geometric realization was first formulated conjecturally through the parabolic flag Hilbert schemes $x_1,\dots,x_m$3, which parametrize chains of ideals

$x_1,\dots,x_m$4

in $x_1,\dots,x_m$5 with $x_1,\dots,x_m$6. The conjectural correspondence uses the Carlsson–Gorsky–Mellit module isomorphism

$x_1,\dots,x_m$7

and identifies normalized fixed-point classes with modified partially symmetric Macdonald polynomials. As evidence, compatibility was proved with the Hecke generators $x_1,\dots,x_m$8 and with the degree-one Pieri element corresponding to multiplication by $x_1,\dots,x_m$9 on the polynomial side (Goodberry et al., 2023).

A later stable-limit result computes the images of the normalized fixed-point classes explicitly. If

$A$0

and $A$1, then

$A$2

This identifies the basis of normalized $A$3-fixed point classes in equivariant $A$4-theory with the basis of modified partially symmetric Macdonald polynomials, thereby extending Haiman’s $A$5 correspondence from Hilbert schemes to the parabolic flag setting. The same work also determines the involution $A$6 on the polynomial representation: $A$7 so each $A$8 is fixed by $A$9 (Orr et al., 2024).

6. Specializations, prescribed symmetry, and positivity conjectures

The partially symmetric theory sits inside a larger landscape of prescribed-symmetry constructions. Earlier work on Macdonald polynomials with prescribed symmetry develops the general mechanism of $P_{(\lambda\mid\gamma)}(x;q,t)$00-symmetrization, $P_{(\lambda\mid\gamma)}(x;q,t)$01-antisymmetrization, and normalization applied to nonsymmetric Macdonald polynomials. In that framework, partially symmetric Macdonald polynomials may be viewed as the purely symmetrized sector, whereas more general prescribed-symmetry families mix symmetric and antisymmetric blocks and admit explicit expansion and normalization formulas together with constant-term applications (Baratta, 2010).

Specializations of permuted-basement nonsymmetric Macdonald polynomials illuminate how partial symmetry emerges combinatorially. At $P_{(\lambda\mid\gamma)}(x;q,t)$02, $P_{(\lambda\mid\gamma)}(x;q,t)$03 is symmetric and independent of $P_{(\lambda\mid\gamma)}(x;q,t)$04 whenever $P_{(\lambda\mid\gamma)}(x;q,t)$05 is a partition, and for general compositions it factors into a symmetric part, independent of $P_{(\lambda\mid\gamma)}(x;q,t)$06, and a nonsymmetric part depending only on the relative order of the entries of $P_{(\lambda\mid\gamma)}(x;q,t)$07. The same work proves a local symmetry statement: if $P_{(\lambda\mid\gamma)}(x;q,t)$08 and $P_{(\lambda\mid\gamma)}(x;q,t)$09, then $P_{(\lambda\mid\gamma)}(x;q,t)$10 is symmetric in $P_{(\lambda\mid\gamma)}(x;q,t)$11. These results do not define partially symmetric Macdonald polynomials in the modern sense, but they exhibit the same phenomenon of symmetry restricted to selected variables (Alexandersson et al., 2018).

A major current direction is positivity. In the $P_{(\lambda\mid\gamma)}(x;q,t)$12-symmetric setting, $P_{(\lambda\mid\gamma)}(x;q,t)$13-symmetric Schur functions are defined through a dual-basis construction involving tableaux combinatorics and Hecke algebra generators. The central conjecture states that suitably normalized and plethystically modified $P_{(\lambda\mid\gamma)}(x;q,t)$14-symmetric Macdonald polynomials expand positively in the $P_{(\lambda\mid\gamma)}(x;q,t)$15-symmetric Schur basis,

$P_{(\lambda\mid\gamma)}(x;q,t)$16

The resulting $P_{(\lambda\mid\gamma)}(x;q,t)$17-Koska coefficients generalize the classical ones, and the usual $P_{(\lambda\mid\gamma)}(x;q,t)$18-Koska coefficients occur as special cases. When $P_{(\lambda\mid\gamma)}(x;q,t)$19 is large, modulo a certain subspace, the conjecture becomes a positivity statement for the expansion of nonsymmetric Macdonald polynomials in terms of nonsymmetric Hall–Littlewood polynomials. This suggests that the partially symmetric regime is not merely interpolatory but may also provide the correct ambient setting for relating the symmetric and nonsymmetric positivity problems (Lapointe, 2022).