[Paper Review] A symmetry property for q-weighted Robinson-Schensted algorithms and other branching insertion algorithms

Yuchen Pei|arXiv (Cornell University)|Jun 10, 2013

Random Matrices and Applications14 references5 citations

TL;DR

This paper establishes a symmetry property for q-weighted Robinson-Schensted algorithms, generalizing the classical symmetry of the standard RS algorithm. Using a generalized growth diagram framework, the author proves that both q-weighted column and row insertion algorithms—defined via branching insertion rules with specific weight functions—satisfy a duality analogous to permutation inversion, extending known results to q-deformed settings with applications in integrable probability and representation theory.

ABSTRACT

In O'Connell-Pei(2013) a q-weighted version of the Robinson-Schensted algorithm was introduced. In this paper we show that this algorithm has a symmetry property analogous to the well known symmetry property of the normal Robinson-Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin(1979,1986,1994,1995). This approach, which uses "growth graphs", can also be applied to a wider class of insertion algorithms which have a branching structure, including some of the other q-weighted versions of the Robinson-Schensted algorithm which have recently been introduced by Borodin-Petrov(2013).

Motivation & Objective

• . The paper aims to establish a symmetry property for q-weighted Robinson-Schensted algorithms analogous to the classical symmetry of the standard RS algorithm.
• . It seeks to generalize Fomin’s growth diagram technique to a broader class of branching insertion algorithms, including q-weighted variants.
• . The research addresses the lack of symmetry results in q-deformed versions of the RS algorithm, which are central to recent developments in integrable probability and stochastic dynamics.
• . The objective includes proving that multiple q-weighted insertion algorithms—both column and row insertion variants—satisfy this symmetry under a unified framework.

Proposed method

• . The paper employs a generalized growth diagram approach, extending Fomin’s framework to handle branching insertion algorithms with q-deformed weights.
• . It defines a branching insertion algorithm via three weight functions: initial (w0), high-level (w1), and low-level (w2), which govern shape transitions during insertion.
• . The symmetry is proven by verifying a set of sufficient conditions on the weight functions: invariance under letter insertion (conditions iii and iv), and shape consistency with insertion order (i and ii).
• . The method uses growth graph rules to encode insertion steps, where the presence or absence of an 'X' in a box corresponds to a specific position in the input sequence.
• . The proof relies on constructing symmetric growth graphs such that the tableau pair output under input reversal is interchanged, mirroring the classical RS symmetry.
• . The framework is applied to two q-weighted algorithms: one from [OP13] (column insertion) and one from [BP13] (row insertion), both shown to satisfy the symmetry property.

Experimental results

Research questions

• RQ1 . Does the q-weighted Robinson-Schensted algorithm with column insertion satisfy a symmetry property analogous to the classical RS algorithm?
• RQ2 . Can Fomin’s growth diagram technique be generalized to handle branching insertion algorithms with q-deformed weights?
• RQ3 . Do q-weighted row insertion algorithms, as introduced in [BP13], also exhibit the same symmetry property as their column insertion counterparts?
• RQ4 . What are the sufficient conditions on weight functions that guarantee the symmetry property in a broad class of branching insertion algorithms?
• RQ5 . How do the q-weighted algorithms relate to classical RS dynamics, and do they preserve key structural properties like duality under input reversal?

Key findings

• . The q-weighted Robinson-Schensted algorithm with column insertion satisfies a symmetry property: reversing the input word results in the interchange of the output tableau pair, just as in the classical RS algorithm.
• . The same symmetry holds for the q-weighted row insertion algorithm introduced in [BP13], confirming a duality under input reversal.
• . A sufficient condition for symmetry is derived in terms of three weight functions (w0, w1, w2), with specific invariance and shape consistency properties under insertion.
• . The growth diagram framework successfully generalizes to q-deformed branching algorithms, enabling a uniform proof of symmetry across multiple q-weighted variants.
• . The weight functions for the [BP13] row insertion algorithm satisfy the symmetry conditions, confirming its duality under reversal.
• . The framework applies to other q-deformed dynamics on Gelfand-Tsetlin patterns, including the 'q-Whittaker-multivariate dynamics' with deterministic long-range interactions, which also exhibit the symmetry property.