A Two-Color Lift of the Shifted $t$-Schur Measure

Abstract: At the specialization $t=-q$, $q\geq0$, the shifted $t$-Schur function associated with the modified odd Greaves--Jing--Zhu operator is $Q_λ[X+qX]$. Instead of merging the two alphabets $X$ and $qX$, we insert an intermediate strict partition between the two corresponding half-vertex operators. This gives a two-color lift of the shifted Schur measure on pairs $μ\subseteqλ$ with weight [ Q_μ(qX)Q_{λ/μ}(X)P_λ(Y). ] We compute the normalization and both marginals, identify an explicit Markov transition kernel, prove a semigroup property, and show that the two color volumes $|μ|$ and $|λ|-|μ|$ are independent. We also realize the model as a two-time shifted Schur process and write its Pfaffian correlation kernel in Vuletić's convention. Rectangular specializations give closed formulas and Gaussian limits for the color volumes.

• The paper introduces a novel two-color extension of the shifted t-Schur measure with an explicit Markov factorization.
• It establishes semigroup properties and provides closed-form expressions for blue and red partition statistics through combinatorial methods.
• The study reveals a Pfaffian point process structure that enables precise computation of gap probabilities and limiting distribution results.

Summary of "A Two-Color Lift of the Shifted $t$-Schur Measure" (2607.02108)

Introduction and Main Construction

The paper presents a new probabilistic measure on pairs of strict partitions, termed a two-color lift of the shifted $t$-Schur measure. At the special value $t=-q$, the authors use a plethystic substitution to relate the shifted $t$-Schur function to the Schur $Q$-function evaluated at $X+qX$. Historically, the shifted Schur measure utilizes a single specialization $X+qX$, but this work introduces an intermediate strict partition, distinguishing the $X$ and $qX$ alphabets.

Formally, the central measure is defined on pairs $(\mu, \lambda)$ with strict partitions $t$0:

$t$1

where $t$2, $t$3, and $t$4 are Schur $t$5- and $t$6-functions, and $t$7 is the Schur Q/P Cauchy kernel. This construction generalizes the one-time shifted Schur measure and provides new combinatorial and probabilistic insights.

Markov Structure and Semigroup Properties

A key result is the Markov factorization of the joint measure:

$t$8

where $t$9 is the marginal on $t=-q$0 and $t=-q$1 is an explicit Markov kernel describing the transition $t=-q$2. The kernel

$t=-q$3

admits a semigroup property:

$t=-q$4

which is the probabilistic counterpart of the half-vertex operator identity for Schur $t=-q$5-functions. This factorization and semigroup structure are critical for explicit computation of marginals and for the analytic tractability of the process.

Color Volumes and Independence

The measure admits a natural interpretation in terms of color volume statistics, where $t=-q$6 (blue volume) and $t=-q$7 (red volume) represent the partition sizes filled by each alphabet. A central finding is that these random variables are independent; their joint probability generating function factors:

$t=-q$8

and their cumulants are given explicitly. For finite alphabets, $t=-q$9 and $t$0 are sums of independent random variables with explicit distributions, leading to explicit central limit theorems in the rectangular case.

Point Process Description and Pfaffian Structure

The framework realizes the joint law as a two-time shifted Schur process in the sense of Vuletić. This leads to a Pfaffian point process structure for the ensemble of blue and red parts, captured by a Pfaffian kernel constructed from the $t$1-symbols of the specializations:

$t$2

The correlation functions can be explicitly computed in terms of this kernel and yield gap probabilities for the largest parts through a Fredholm-Pfaffian formula. The explicit double contour integrals are given for rectangular specializations.

Conditional Law, Combinatorial Interpretation, and Limit Theorems

The conditional law of the blue shape given the final partition $t$3 is expressed simply in terms of the ratio of $t$4-functions:

$t$5

yielding explicit formulas for conditional moments. The measure has a direct combinatorial interpretation in terms of weighted pairs of shifted semistandard tableaux, linking to classical results of Stembridge and Sagan.

In the case of rectangular alphabets, closed formulas for the distributions of $t$6 and $t$7 are derived, including explicit expressions for the mean, variance, and the central limit scaling. For large system size, the joint law converges to independent normal distributions.

Implications and Future Directions

This two-color extension of the shifted Schur measure provides a structurally richer probabilistic model, revealing new independence phenomena and permitting explicit calculation of statistics inaccessible in the original setting. The Markov kernel and semigroup structure strengthen connections between symmetric function theory and integrable probability.

On a theoretical level, the approach demonstrates the utility of preserving "intermediate" data between merged alphabets in plethystic constructions, suggesting broader generalizations for other symmetric function measures and point processes. The explicit Pfaffian description implies potential applications to random strict plane partitions and refined enumerative combinatorics, as well as further exploration of two-time (multi-time) extensions in Schur and shifted Schur processes.

Future developments may include leveraging this measure in asymptotic representation theory, studying universality phenomena of the limiting laws, or formulating analogs for other deformed symmetric function ensembles, such as Hall-Littlewood or Macdonald polynomials, using similar "lifted" Markov factorizations.

Conclusion

The paper systematically constructs and analyzes a two-color lift of the shifted $t$8-Schur measure, unveiling an explicit Markov structure, independence of color volumes, and a Pfaffian point process formulation. The results yield new theoretical insights into the interplay between symmetric functions, probability measures on partitions, and integrable combinatorics, and provide fertile ground for further investigation into multi-parameter and multi-time extensions of Schur measures.