Schur--Weyl duality for diagonalizing a Markov chain on the hypercube

TL;DR

The paper develops an explicit, orthonormal eigenbasis for the Burnside Markov chain on the binary hypercube by leveraging Schur–Weyl duality and an $\mathfrak{sl}_2$-action to decompose $V^{\otimes n}$ into isotypic components labeled by Young tableaux. It introduces the $g_Q^{m,\ell}$ and $f_Q^{m,\ell}$ vectors that span irreducible subspaces and are eigenvectors of the transition operator, with eigenvalues determined by $m+\ell$ and independent of $K_n$. The resulting framework yields exact norm formulas, links to Hahn and Chebyshev polynomials, and sharp mixing-time bounds, including a bounded-time convergence from the one-one state $e_n$ in $\ell^2$ and $\ell^1$. The approach also clarifies eigenvalue multiplicities across irreducible components and opens avenues for generalizations to other $(C_k^n, S_n)$ Burnside processes. Overall, the work provides a principled, algebraic route to precise convergence analyses in highly symmetric Markov chains.

Abstract

We show how the tools of modern algebraic combinatorics -- representation theory, Murphy elements, and particularly Schur--Weyl duality -- can be used to give an explicit orthonormal basis of eigenfunctions for a "curiously slowly mixing Markov chain" on the space of binary $n$-tuples. The basis is used to give sharp rates of convergence to stationarity.

Figures (2)

• Figure 1: The eight vectors $g_Q^{m, \ell}$ for $n = 3$. Observe that we have $g_Q^{m, \ell} \in V^{(m+\ell)}$ in all cases; that is, the vector $g_Q^{m, \ell}$ is only supported on the states $S$ where $|S| = m+\ell$.
• Figure 2: The eight vectors $f_Q^{m, \ell}$ for $n = 3$. Observe that the vector $f_Q^{m, \ell}$ with $m + \ell = 0$ is an eigenvector of $K_3$ with eigenvalue $\beta_0 = 1$, the three vectors with $m + \ell = 2$ are eigenvectors with eigenvalue $\beta_1 = \frac{1}{4}$, and all other vectors are eigenvectors with eigenvalue $0$.

Theorems & Definitions (27)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 2.2
• Remark
• Proposition 2.3: diaconislinram1
• Remark
• Lemma 3.1
• proof
• proof : Proof of \ref{['orthoevthm']}