#Modified Macdonald polynomials
The modified (or transformed) Macdonald polynomials \(\macdonaldH_{\lambda}(\xvec;q,t)\) appear as a combinatorial version of the Macdonald \(P\) polynomials. This family is indexed by partitions \(\lambda\) and is symmetric in \(\xvec.\) It was proved by M. Haiman in [Hai01] that the modified Macdonald polynomials are bigraded Frobenius characteristics of certain \(\symS_n\)-modules that appear in diagonal harmonics. For details on this story, see J. P. Swanson’s notes from the \(n!\) conjecture seminar [Swa17].
A combinatorial formula for the modified Macdonald polynomials was proved in [HHL05], where the connection with LLT polynomials gives an LLT-positive expansion. The canonical reference on modified Macdonald polynomials is the book by J. Haglund [Hag07]. A. Garbali and M. Wheeler construct a two-alphabet family \(W_\lambda(x;q,t;z)\) using integrable lattice models [GW20]. Under specializations this family reduces to the integral form \(\macdonaldJ_\lambda,\) to modified Macdonald polynomials, and to dual counterparts; the same lattice construction gives manifestly positive monomial formulas for \(\macdonaldH_\lambda,\) with symmetry following from the Yang–Baxter equation. J. Blasiak, M. Haiman, J. Morse, A. Pun, and G. Seelinger give an explicit raising-operator formula for the modified Macdonald polynomials [BHMP+23]. Their formula follows from a formula for \(\nabla\) on LLT polynomials together with the Haglund–Haiman–Loehr expansion. The same method produces \(1,n\)-Macdonald polynomials, conjecturally Schur-positive with coefficients in \(\setN[q,t].\) A. M. Garsia and M. D. Haiman factor the Stanley–Macdonald Pieri rules by embedding Macdonald polynomials into polynomials indexed by lattice square diagrams [GH95]. The resulting recursions give partial results for the Macdonald–Kostka coefficients, including special cases for two-row shapes. It is a major open problem in algebraic combinatorics to give a combinatorial proof (by using RSK, dual equivalence or crystals) that the \(\macdonaldH_{\lambda}(\xvec;q,t)\) are Schur-positive. Kato’s categorification gives the stronger refinement that \(\macdonaldH_\lambda(\xvec;q,t)\) is \(k\)-Schur-positive when \(\lambda\) is \(k\)-bounded [Cor. 9.4, Kat25]. D. Kim, S. J. Lee, and J. Oh introduced a column-exchange rule for LLT equivalence, proved monomial positivity, and settled several cases of Butler’s conjecture [KLO26]. The full conjecture has now been proved by P. L. Guo, M. Kang, and R. Xiong [GKX26].
Theorem (Butler positivity).
Let \(\nu\vdash n+1,\) and let \(\lambda,\mu\subset\nu\) be distinct partitions of \(n,\) each obtained by deleting one cell. Put \[T_\rho=t^{n(\rho)}q^{n(\rho')}.\] Then the Macdonald intersection polynomial \[I_{\lambda,\mu}(X;q,t) \coloneqq \frac{ T_\lambda\macdonaldH_\mu(X;q,t) -T_\mu\macdonaldH_\lambda(X;q,t)} {T_\lambda-T_\mu}\] is Schur-positive with coefficients in \(\setN[q,t].\)
More strongly, \(I_{\lambda,\mu}\) is the bigraded Frobenius characteristic of an \(\symS_n\)-module. The construction uses the Procesi bundle on the Hilbert scheme of points in the plane.
#Definition
The modified Macdonald polynomials were originally defined via the Macdonald \(J\) polynomials, as \[\macdonaldH_{\mu}(\xvec;q,t) \coloneqq t^{n(\mu)} \macdonaldJ_\mu[X/(1-t^{-1});q,1/t].\] Note that we use plethystic notation here.
#Inner product characterization
The modified Macdonald polynomials \(\{\macdonaldH_{\mu} \}_{\mu \vdash n}\) are the unique family of symmetric functions with coefficients in \(\setQ(q,t),\) such that
\(\macdonaldH_{\mu}[X(1-q);q,t] \in \setQ(q,t)\{ \schurS_\lambda : \lambda \trianglerighteq \mu \}.\)
\(\macdonaldH_{\mu}[X(1-t);q,t] \in \setQ(q,t)\{ \schurS_\lambda : \lambda \trianglerighteq \mu' \}.\)
\(\langle \macdonaldH_{\mu} , \schurS_\mu \rangle = 1.\)
Alternatively, they are the unique family of polynomials that satisfies
\(\langle \macdonaldH_{\mu}[X;q,t], \schurS_\lambda[X/(t-1)] \rangle =0\) whenever \(\lambda \triangleright \mu,\)
\(\langle \macdonaldH_{\mu}[X;q,t], \schurS_\lambda[X/(1-q)] \rangle =0\) whenever \(\lambda \triangleleft \mu,\)
\(\langle \macdonaldH_{\mu} , \schurS_{(n)} \rangle = 1.\)
#Haglund’s combinatorial formula
A combinatorial formula for the modified Macdonald polynomials was obtained in [HHL05]. It is given by \[\macdonaldH_{\lambda}(\xvec;q,t) = \sum_{T:\lambda \to \setP} q^{\inv_\lambda(T)} t^{\maj_\lambda(T)} \xvec^T,\] where \(\inv_\lambda(T)\) and \(\maj_\lambda(T)\) are certain combinatorial statistics.
For background on the discovery of these statistics, see J. Haglund’s The genesis of the Macdonald polynomial statistics [Hag06].
We recall the statistics in the Haglund–Haiman–Loehr convention. Draw \(\lambda\) in French notation and let \(T\) be a positive-integer filling of its cells. A descent is a vertical adjacent pair with the upper entry larger than the lower entry; the upper cell is recorded in \(\mathrm{Des}(T).\) The major index is \[\maj_\lambda(T) = \sum_{u\in \mathrm{Des}(T)}(\leg(u)+1),\] where \(\leg(u)\) is the number of cells strictly above \(u\) in its column.
Two cells are attacking if they are in the same row, or if they are in adjacent rows and the cell in the upper row is strictly to the right of the cell in the lower row. Reading cells row by row from top to bottom and left to right, let \(\mathrm{Inv}(T)\) be the set of attacking pairs whose earlier entry is larger than the later entry. Then \[\inv_\lambda(T) = |\mathrm{Inv}(T)| - \sum_{u\in \mathrm{Des}(T)}\arm(u),\] where \(\arm(u)\) is the number of cells strictly to the right of \(u\) in its row. This subtraction is nonnegative; equivalently, it counts certain inversion triples together with the row inversions in the bottom row.
Example (Macdonald polynomials for \(|\lambda|=4.\)).
The modified Macdonald polynomials \(\macdonaldH_{\lambda}(\xvec;q,t)\) have the following Schur expansions:
\( \; \) \( \textbf{4} \) \( \textbf{31} \) \( \textbf{22} \) \( \textbf{211} \) \( \textbf{1111} \) \( \textbf{4} \) \( 1 \) \( t^3+t^2+t \) \( t^4+t^2 \) \( t^5+t^4+t^3 \) \( t^6 \) \( \textbf{31} \) \( 1 \) \( q+t^2+t \) \( q t+t^2 \) \( q t^2+q t+t^3 \) \( q t^3 \) \( \textbf{22} \) \( 1 \) \( q t+q+t \) \( q^2+t^2 \) \( q^2 t+q t^2+q t \) \( q^2 t^2 \) \( \textbf{211} \) \( 1 \) \( q^2+q+t \) \( q^2+q t \) \( q^3+q^2 t+q t \) \( q^3 t \) \( \textbf{1111} \) \( 1 \) \( q^3+q^2+q \) \( q^4+q^2 \) \( q^5+q^4+q^3 \) \( q^6 \)For example, \(\macdonaldH_{31}(\xvec;q,t)\) is given by \[\schurS_{4}(\xvec) + (q+t+t^2) \schurS_{31}(\xvec) +(q t+t^2) \schurS_{22}(\xvec) +(q t+q t^2+t^3) \schurS_{211}(\xvec) + q t^3 \schurS_{1111}(\xvec)\]
#Alternative combinatorial formulas
A second combinatorial formula is given by R. Kaliszewski and J. Morse in [KM19]. We have \[\macdonaldH_{\lambda}(\xvec;q,t) = \sum_{T} q^{betrayal(T)} t^{\cocharge(T)} \xvec^T,\] where the sum is over all tabloids with content \(\mu.\)
Another formula is given in [CHMM+22], where certain terms are grouped together: \[\macdonaldH_{\lambda'}(\xvec;q,t) = \sum_{\sigma \in \mathrm{ST}(\lambda)} \xvec^\sigma t^{\inv_\lambda(\sigma)} q^{\maj_\lambda(\sigma)} \mathrm{perm}_t(\sigma,\lambda)\] where \(\mathrm{perm}_t(\sigma,\lambda)\) is a certain \(t\)-multinomial coefficient, and \(\mathrm{ST}(\lambda)\) is the set of sorted tableaux of shape \(\lambda.\) In some sense, certain monomial terms in the original formula have been grouped together in order to produce multinomial coefficients.
N. A. Loehr gives bijective proofs of coinversion identities arising in the multiline-queue formula for modified Macdonald polynomials [Loe25]. These identities were first found by A. Ayyer, O. Mandelshtam, and J. Martin, and Loehr’s bijections answer their request for explicit combinatorial proofs.
A. Ayyer, O. Mandelshtam, and J. Martin connect the same queue-inversion tableau model to a multispecies totally asymmetric zero-range process [AMM23]. Their sequel treats site-dependent jump rates [AMM24]. The stationary distribution is expressed using quinv-weighted tableaux, and the partition function is \(\macdonaldH_\lambda(\xvec;1,t).\)
#Properties
We have the following specializations: \[\macdonaldH_{\lambda}(\xvec;0,0) = \completeH_{n}(\xvec), \qquad \macdonaldH_{\lambda}(\xvec;1,0) = \prod_i \completeH_{\lambda_i}(\xvec), \qquad \macdonaldH_{\lambda}(\xvec;1,1) = (\elementaryE_1(\xvec))^{|\lambda|},\]
We have the following symmetries: \[\macdonaldH_{\lambda}(\xvec;q,t) = \macdonaldH_{\lambda'}(\xvec;t,q) = q^{n(\lambda)} t^{n(\lambda')}\omega \macdonaldH_{\lambda}(\xvec;q^{-1},t^{-1})\] where \(n(\lambda) = \sum_i (n-i)\lambda(i)\) when \(\lambda\) has size \(n.\) The first identity follows immediately from the fact that the modified Macdonald polynomials are bigraded Frobenius series with this symmetry. However, a bijective proof is not known. Some partial results have been found by M. Gillespie [Gil16].
The modified Macdonald polynomials specialize to the modified Hall–Littlewood polynomials at \(q=0,\) and these are in turn closely related to the transformed Hall–Littlewood polynomials.
#Schur expansion
M. Haiman defines a family of bigraded \(\symS_n\)-modules, which has the property that the Frobenius image of this family gives the modified Macdonald polynomials, see [Hai01]. It follows that the modified Macdonald polynomials are Schur-positive, although no combinatorial formula is known.
Problem (Macdonald, [Mac95]).
Find pairs of statistics \(\maj_\mu(\cdot)\) and \(\inv_\mu(\cdot)\) on standard Young tableaux, such that \[\macdonaldH_{\mu}(\xvec;q,t) = \sum_{T \in \SYT(n)} q^{\inv_\mu(T)} t^{\maj_\mu(T)} \schurS_{sh(T)}(\xvec).\]
This would give a combinatorial proof that the modified Macdonald–Kostka polynomials \(\tilde{K}_{\lambda\mu}(q,t),\) also called \(qt\)-Kostka polynomials, are elements in \(\setN[q,t].\) These polynomials are defined via the relation \[\macdonaldH_{\mu}(\xvec;q,t) = \sum_{\lambda} \tilde{K}_{\lambda\mu}(q,t) \schurS_{\lambda}(\xvec).\] Note that \(\tilde{K}_{\lambda\mu}(1,1) = f^\lambda,\) the number of standard Young tableaux.
#\(qt\)-Kostka polynomials
The following properties can be found in [p.32, Hag07].
The modified Kostka polynomials are \(\tilde{K}_{\lambda\mu}(q,t) = t^{n(\mu)}K_{\lambda \mu}(q,1/t),\) where the \(K_{\lambda \mu}\) are the \(qt\)-Kostka polynomials.
Recall that the \(qt\)-Kostka polynomials \(K_{\lambda \mu}(q,t)\) have the following properties: \[K_{\lambda \mu}(0,t) = K_{\lambda \mu}(t) = \sum_{T \in \SSYT(\lambda,\mu)} t^{\charge(T)}.\] and thus \(K_{\lambda \mu}(0,1) = K_{\lambda \mu}.\) We also define the cocharge polynomial, \[\tilde{K}_{\lambda \mu}(t) = t^{n(\mu)}K_{\lambda \mu}(1/t) = \sum_{T \in \SSYT(\lambda,\mu)} t^{\cocharge(T)}.\] Also, \(\tilde{K}_{\lambda\mu}(1,1) = K_{\lambda\mu}(1,1) = f^{\lambda},\) the number of standard Young tableaux of shape \(\lambda.\)
Then there are the following symmetries: \[\begin{aligned} K_{\lambda,\mu}(q,t) &= t^{n(\mu)} q^{n(\mu')} K_{\lambda',\mu}(1/q,1/t) \\ K_{\lambda,\mu}(q,t) &= K_{\lambda',\mu'}(t,q)\\ % \tilde{K}_{\lambda',\mu}(q,t) &= t^{n(\mu)} q^{n(\mu')} \tilde{K}_{\lambda',\mu}(1/q,1/t) \\ \tilde{K}_{\lambda,\mu}(q,t) &= \tilde{K}_{\lambda,\mu'}(t,q)\\ \end{aligned}\]
#Stretching symmetry property
The following property is stated as a conjecture in [LOR22], but it was later revealed that this is a known property, following from a result by A. M. Garsia and G. Tesler [Thm. I.1, GT96].
Proposition
Let \(k\) be a positive integer. Then \[\macdonaldH_{k \mu}(\xvec;q,q^k) = \macdonaldH_{k \mu'}(\xvec;q,q^k).\] That is, we have a type of symmetry under conjugation and stretching.
#General diagrams
In [Ban07], J. Bandlow studies more general shapes, in particular skew shapes.
#Macdonald cumulants
A generalization of the \(qt\)-Kostka coefficients is defined in [Do19]. These coefficients are related to a Schur-positivity problem generalizing Schur positivity of the modified Macdonald polynomials.
#Non-commutative Macdonald polynomials
In [BZ05], the authors introduce a non-commutative family of symmetric functions with properties similar to those of the modified Macdonald polynomials. Their construction gives \(q\)- and \((q,t)\)-analogues of noncommutative symmetric functions, with triangularity and specialization properties parallel to the commutative Hall–Littlewood and Macdonald settings.
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