Macdonald–Torrilhon Interpolants

• The Macdonald–Torrilhon interpolants are inhomogeneous generalizations of Macdonald polynomials defined by unique normalization and vanishing conditions at explicit interpolation points.
• They bridge algebraic combinatorics with interacting particle systems by linking symmetric functions to probabilistic models like the t-Push TASEP Markov chain at q=1.
• The framework, incorporating nonsymmetric ASEP polynomials, paves the way for further research into q-deformed dynamics and integrable probability models.

The Macdonald–Torrilhon interpolants, also known as interpolation Macdonald polynomials in the literature of Knop and Sahi, arise as inhomogeneous generalizations of Macdonald polynomials. They are uniquely characterized by normalization and vanishing properties at explicit interpolation points indexed by compositions, and at $q=1$ admit a probabilistic interpretation as partition functions of an inhomogeneous multispecies $t$-Push TASEP Markov chain. These polynomials, together with their nonsymmetric companions (the interpolation ASEP polynomials), link algebraic combinatorics with interacting particle systems and generalize earlier work connecting Macdonald polynomials to integrable Markov chains (Dali et al., 13 Feb 2026).

1. Definition and Characterization

The Knop–Sahi interpolation Macdonald polynomials $P^*_\lambda(x_1,\dots,x_n;q,t)$ are defined for a partition $\lambda=(\lambda_1\ge\cdots\ge\lambda_n\ge0)$, with $m_\lambda(x_1,\dots,x_n)$ the associated monomial symmetric function. For any composition $\nu=(\nu_1,\dots,\nu_n)$, the evaluation point is

$$\overline\nu = (q^{\nu_1} t^{-k_1(\nu)},\, \dots,\, q^{\nu_n} t^{-k_n(\nu)}),$$

where

$$k_i(\nu) = \#\{\,j<i:\nu_j>\nu_i\} + \#\{\,j>i:\nu_j\ge\nu_i\}.$$

The defining conditions are:

• Normalization: The coefficient of $m_\lambda$ in $P^*_\lambda$ is $t$0.
• Vanishing: For any partition $t$1 with $t$2, the interpolant vanishes at the evaluation point: $t$3.
• Homogeneous Limit: The top homogeneous component is the usual Macdonald polynomial $t$4.

At $t$5, these polynomials yield interpolation Schur functions via a determinantal formula; in general, tableaux-sum formulas of Okounkov type apply.

2. Interpolation ASEP Polynomials and Symmetrization

Given $t$6 as above, for each composition $t$7 in the $t$8-orbit of $t$9 (that is, each composition of $P^*_\lambda(x_1,\dots,x_n;q,t)$0 obtained by permuting parts), there exists a uniquely defined interpolation ASEP polynomial

$P^*_\lambda(x_1,\dots,x_n;q,t)$1

of total degree $P^*_\lambda(x_1,\dots,x_n;q,t)$2. The defining conditions are:

• Monomial Normalization: $P^*_\lambda(x_1,\dots,x_n;q,t)$3.
• Vanishing: For every $P^*_\lambda(x_1,\dots,x_n;q,t)$4 with $P^*_\lambda(x_1,\dots,x_n;q,t)$5, $P^*_\lambda(x_1,\dots,x_n;q,t)$6.

These $P^*_\lambda(x_1,\dots,x_n;q,t)$7 interpolate the homogeneous ASEP/Macdonald–basement polynomials $P^*_\lambda(x_1,\dots,x_n;q,t)$8 at leading degree. The symmetrization relation holds: $P^*_\lambda(x_1,\dots,x_n;q,t)$9

3. Probabilistic Interpretation at \texorpdfstring{$\lambda=(\lambda_1\ge\cdots\ge\lambda_n\ge0)$0}{q=1} via Markov Chains

Specializing to $\lambda=(\lambda_1\ge\cdots\ge\lambda_n\ge0)$1, the polynomials $\lambda=(\lambda_1\ge\cdots\ge\lambda_n\ge0)$2 and $\lambda=(\lambda_1\ge\cdots\ge\lambda_n\ge0)$3 acquire a probabilistic interpretation via the interpolation $\lambda=(\lambda_1\ge\cdots\ge\lambda_n\ge0)$4-Push TASEP Markov chain defined on the finite state space $\lambda=(\lambda_1\ge\cdots\ge\lambda_n\ge0)$5: $\lambda=(\lambda_1\ge\cdots\ge\lambda_n\ge0)$6 The transition mechanism, parameterized by $\lambda=(\lambda_1\ge\cdots\ge\lambda_n\ge0)$7 and $\lambda=(\lambda_1\ge\cdots\ge\lambda_n\ge0)$8, consists of:

• Step 0: Randomly select site $\lambda=(\lambda_1\ge\cdots\ge\lambda_n\ge0)$9 with explicit probability $m_\lambda(x_1,\dots,x_n)$0 as a function of inhomogeneities.
• Step 1: The particle at $m_\lambda(x_1,\dots,x_n)$1 initiates the $m_\lambda(x_1,\dots,x_n)$2-Push TASEP dynamic, displacing weaker particles or vacancies cyclically with probability depending on $m_\lambda(x_1,\dots,x_n)$3.
• Step 2: The vacancy attempts to displace other particles as it traverses the ring, with displacement rates depending on the $m_\lambda(x_1,\dots,x_n)$4 and $m_\lambda(x_1,\dots,x_n)$5.

It is established that the Markov chain is irreducible and admits a unique stationary measure. The stationary probability for $m_\lambda(x_1,\dots,x_n)$6 is

$m_\lambda(x_1,\dots,x_n)$7

rendering $m_\lambda(x_1,\dots,x_n)$8 as the partition function. This generalizes prior interpretations of symmetric Macdonald polynomials $m_\lambda(x_1,\dots,x_n)$9 via homogeneous multispecies $\nu=(\nu_1,\dots,\nu_n)$0-Push TASEP (Dali et al., 13 Feb 2026).

4. Illustrative Special Cases

Certain concrete partitions yield closed forms for the polynomials and the Markov chain rates:

• One-Column Partition $\nu=(\nu_1,\dots,\nu_n)$1: Here,

$\nu=(\nu_1,\dots,\nu_n)$2

and

$\nu=(\nu_1,\dots,\nu_n)$3

where $\nu=(\nu_1,\dots,\nu_n)$4 is a deformed elementary symmetric function. The corresponding chain is the inhomogeneous Push-TASEP for a single species.

• One-Row Partition $\nu=(\nu_1,\dots,\nu_n)$5: The interpolation polynomial admits

$\nu=(\nu_1,\dots,\nu_n)$6

and the corresponding $\nu=(\nu_1,\dots,\nu_n)$7 share this structure across $\nu=(\nu_1,\dots,\nu_n)$8. This case gives explicit closed forms for stationary distributions in the single-type limit of the Markov chain.

5. Structural and Theoretical Implications

The Macdonald–Torrilhon interpolants establish an explicit Markov-chain framework for interpolated symmetric functions and polynomials. The sum-to-symmetrization property relates the combinatorially defined $\nu=(\nu_1,\dots,\nu_n)$9 to their symmetric sum $$\overline\nu = (q^{\nu_1} t^{-k_1(\nu)},\, \dots,\, q^{\nu_n} t^{-k_n(\nu)}),$$0, reflecting deep connections between inhomogeneous interacting particle systems (such as the $$\overline\nu = (q^{\nu_1} t^{-k_1(\nu)},\, \dots,\, q^{\nu_n} t^{-k_n(\nu)}),$$1-Push TASEP) and modern algebraic combinatorics. Setting all $$\overline\nu = (q^{\nu_1} t^{-k_1(\nu)},\, \dots,\, q^{\nu_n} t^{-k_n(\nu)}),$$2 recovers the homogeneous case, bridging interpolation Macdonald theory and classic symmetric function results.

A significant open direction is to realize a $$\overline\nu = (q^{\nu_1} t^{-k_1(\nu)},\, \dots,\, q^{\nu_n} t^{-k_n(\nu)}),$$3-deformed Markov chain that yields the full two-parameter polynomials $$\overline\nu = (q^{\nu_1} t^{-k_1(\nu)},\, \dots,\, q^{\nu_n} t^{-k_n(\nu)}),$$4 in the stationary distribution, or to find an algebraic structure underlying such dynamics. Such a development would extend the intertwining of probability and symmetric function theory to completely general Macdonald–Torrilhon interpolants.

6. Research Context and Further Directions

The construction and interpretation of Macdonald–Torrilhon interpolants in terms of inhomogeneous Markov dynamics continue an active research trajectory connecting integrable probability, algebraic combinatorics, and representation theory. This approach builds on the work of Knop, Sahi, Okounkov (tableaux-formulas), and recent connections by Ayyer, Martin, Williams, and Ben Dali and Williams, who provide the explicit Markov chain realization and explore its consequences (Dali et al., 13 Feb 2026). The possibility of further generalizing these connections to the full $$\overline\nu = (q^{\nu_1} t^{-k_1(\nu)},\, \dots,\, q^{\nu_n} t^{-k_n(\nu)}),$$5 parameter regime, and to broader classes of stochastic vertex models or quantum integrable systems, remains an inviting direction for current research.