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The $k$-Plancherel measure and a Finite Markov Chain

Svante Linusson, Alperen Özdemir

TL;DR

This work studies a growth process on $(k+1)$-cores, equivalently on $k$-bounded partitions, and introduces the $k$-Plancherel measure as the stationary distribution, connecting to the classical Plancherel measure as $k\to\infty$ and to Rost-type limit shapes when $k$ is fixed. Central to the analysis is the $k$-rectangle property, which reduces the infinite process to a finite Markov chain with $k!$ states that can be realized as a TASEP on a cyclic ring; the transition structure is governed by $k$-Schur function combinatorics (strong/weak tableaux) and their dimensions $d_\lambda^{(k)}$ and $w_\lambda^{(k)}$. The paper proves symmetry results under $k$-conjugation, formulates conjectures about complement symmetry and denominator patterns, and provides a probabilistic framework that suggests a Rost-like limit shape $D_{k+1}$ for fixed $k$, with a concrete program for further analysis via raising operators. Overall, the work links affine symmetric-group combinatorics, $k$-Schur theory, and interacting-particle systems to illuminate partition-growth phenomena in a $k$-bounded setting.

Abstract

Let $\mathcal{P}_k(n)$ denote the set of partitions of $n$ whose largest part is bounded by $k,$ which are in well-known bijection with $(k+1)$-cores $\mathcal{C}_k$. We study a growth process on $\mathcal{C}_k$, whose stationary distribution is the $k$-Plancherel measure, which is a natural extension of the Plancherel measure in the context of $k$-Schur functions. When $k\to\infty$ it converges to the Plancherel measure for partitions, a limit studied first by Vershik-Kerov. However, when $k$ is fixed and $n\to \infty$, we conjecture that it converges to a shape close to the limit shape from the uniform growth of partitions, as studied by Rost. We show that the limiting behavior, for fixed $k$, is governed by a finite Markov chain with $k!$ states over a subset of the $k$-bounded partitions or equivalently as a TASEP over cyclic permutations of length $k+1$. This paper initiates the study of these processes, state some theorems and several intriguing conjectures found by computations of the finite Markov chain.

The $k$-Plancherel measure and a Finite Markov Chain

TL;DR

This work studies a growth process on -cores, equivalently on -bounded partitions, and introduces the -Plancherel measure as the stationary distribution, connecting to the classical Plancherel measure as and to Rost-type limit shapes when is fixed. Central to the analysis is the -rectangle property, which reduces the infinite process to a finite Markov chain with states that can be realized as a TASEP on a cyclic ring; the transition structure is governed by -Schur function combinatorics (strong/weak tableaux) and their dimensions and . The paper proves symmetry results under -conjugation, formulates conjectures about complement symmetry and denominator patterns, and provides a probabilistic framework that suggests a Rost-like limit shape for fixed , with a concrete program for further analysis via raising operators. Overall, the work links affine symmetric-group combinatorics, -Schur theory, and interacting-particle systems to illuminate partition-growth phenomena in a -bounded setting.

Abstract

Let denote the set of partitions of whose largest part is bounded by which are in well-known bijection with -cores . We study a growth process on , whose stationary distribution is the -Plancherel measure, which is a natural extension of the Plancherel measure in the context of -Schur functions. When it converges to the Plancherel measure for partitions, a limit studied first by Vershik-Kerov. However, when is fixed and , we conjecture that it converges to a shape close to the limit shape from the uniform growth of partitions, as studied by Rost. We show that the limiting behavior, for fixed , is governed by a finite Markov chain with states over a subset of the -bounded partitions or equivalently as a TASEP over cyclic permutations of length . This paper initiates the study of these processes, state some theorems and several intriguing conjectures found by computations of the finite Markov chain.
Paper Structure (14 sections, 14 theorems, 87 equations, 7 figures)

This paper contains 14 sections, 14 theorems, 87 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background
  3. $k$-Schur functions
  4. Core partitions
  5. Weak and strong orders
  6. k-tableaux
  7. A growth process on bounded partitions and a finite Markov chain
  8. Cauchy identity and the $k$-Plancherel measure
  9. Pieri rule and the growth process
  10. $k$-rectangle property and a finite Markov chain
  11. TASEP formulation of the finite chain
  12. Conjugates and complements of partitions
  13. Limit shape of core partitions
  14. Appendix: Raising Operators

Key Result

Theorem 3.2

Let $\lambda \in \mathcal{R}_k$ with $l_1(\lambda)=k-1$ and $\mu=\lambda \backslash (1^{k-1}).$ Then, we have $P(\lambda,\mu)=1/k.$ That is to say, adding a box in the first column of a partition with $(k-1)$ parts with size $1$, so that we remove $k$ 1's and $l_1$ becomes 0, is always $1/k$.

Figures (7)

  • Figure 1: The Young diagram of a $4$-bounded partition and the map to a $5$-core partition $\kappa$ with hook lengths. The red boxes have hook length greater than 5 and are removed by $\mathfrak{p}$ and added by $\mathfrak{c}$.
  • Figure 2: A state in the $\pmod5$-process with the corresponding $5$-core $(7,3,1)$ and cyclic permutation $1 4 2 3 5$. Followed by the 4-bounded partition image $(4,3,1)$ under $\mathfrak{p}$ and its projection to $(3,1)\in\mathcal{R}_k$ using the $k$-rectangle property.
  • Figure 3: The Markov chain on $3$-bounded partitions. The cyclic permutation $\alpha^{-1}(\lambda)$ is given for each partition and the stationary probability $\pi(\lambda)$.
  • Figure 4: The stationary distribution of the Markov chain for $4$-bounded and $5$-bounded partitions from data. The partitions in $\mathcal{R}_k$ are indexed according to factorial base on the $x$-axis, see Section 4.1. of K14, where $l_i(\lambda)$ is the number of parts of size i in $\lambda \in \mathcal{R}_k$.
  • Figure 5: The limiting piecewise-linear curve $D_4$ for random 4-cores.
  • ...and 2 more figures

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 3.1
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • ...and 28 more