[Paper Review] The mystery of plethysm coefficients

Laura Colmenarejo, Rosa Orellana+3 more|arXiv (Cornell University)|Aug 15, 2022

Advanced Combinatorial Mathematics4 citations

TL;DR

This paper introduces the 's-perp trick'—a novel combinatorial representation theory technique—to efficiently compute Schur and monomial expansions of symmetric functions, particularly plethysm coefficients $ s_{\lambda}[s_{\mu}] $. It provides new combinatorial formulas for $ a_{\lambda,\mu}^{\nu} $, the coefficients in the Schur expansion of plethysm, and demonstrates that the method outperforms existing SageMath implementations in key cases, especially for hook-shaped and row-shaped partitions.

ABSTRACT

Composing two representations of the general linear groups gives rise to Littlewood's (outer) plethysm. On the level of characters, this poses the question of finding the Schur expansion of the plethysm of two Schur functions. A combinatorial interpretation for the Schur expansion coefficients of the plethysm of two Schur functions is, in general, still an open problem. We identify a proof technique of combinatorial representation theory, which we call the "$s$-perp trick", and point out several examples in the literature where this idea is used. We use the $s$-perp trick to give algorithms for computing monomial and Schur expansions of symmetric functions. In several special cases, these algorithms are more efficient than those currently implemented in {\sc SageMath}.

Motivation & Objective

• To develop a systematic method for computing Schur and monomial expansions of symmetric functions, especially plethysm $ s_{\lambda}[s_{\mu}] $, which remains a long-standing open problem in representation theory.
• To identify and formalize the 's-perp trick' as a unifying technique in combinatorial representation theory, with applications to symmetric function identities and plethysm computations.
• To derive new combinatorial formulas for plethysm coefficients $ a_{\lambda,\mu}^{\nu} $, particularly for cases where $ \lambda $ is a hook or row shape and $ \mu $ is a row or column.
• To improve computational efficiency for plethysm coefficient calculations, showing that the s-perp trick outperforms current SageMath implementations in specific cases.

Proposed method

• The s-perp trick is applied to compute the Schur expansion of a symmetric function $ f $ by leveraging the Schur expansions of $ s_r^\perp f $, using recursive decomposition via inner plethysm.
• The method relies on the duality between inner and outer plethysm, where $ s_r^\perp f $ is used to reconstruct $ f $ via recursive application of the $ s_r^\perp $ operator.
• The authors use the Littlewood–Richardson rule and properties of semistandard Young tableaux to compute coefficients in the Schur expansion of $ s_{\lambda}[s_{\mu}] $, particularly for $ \mu = s_{1^2} $ and $ \mu = s_2 $.
• For hook-shaped $ \lambda = (h-k,1^k) $, the method derives a signed sum over products of Littlewood–Richardson coefficients $ c^{\mu}_{\nu\rho} $, with $ \nu $ even and $ \rho $ threshold-shaped.
• The approach is validated through induction and case analysis on the number of odd columns and corner cells, particularly in the context of $ s_{(h-2,1,1)}[s_{1^2}] $ and $ s_{(h-k,1^k)}[s_{1^2}] $.
• The paper demonstrates that the s-perp trick enables efficient computation of plethysm coefficients by reducing the problem to known expansions of $ s_r^\perp f $, avoiding brute-force decomposition.

Experimental results

Research questions

• RQ1 Can the s-perp trick be used to derive new combinatorial formulas for plethysm coefficients $ a_{\lambda,\mu}^{\nu} $ in $ s_{\lambda}[s_{\mu}] = \sum_{\nu} a_{\lambda,\mu}^{\nu} s_{\nu} $?
• RQ2 How does the s-perp trick compare in efficiency to existing algorithms in SageMath for computing plethysm expansions?
• RQ3 What are the structural properties of plethysm coefficients when $ \lambda $ is a hook or row shape and $ \mu $ is a column or row?
• RQ4 Can the s-perp trick be generalized to derive closed-form expressions for $ s_{(h-k,1^k)}[s_{1^2}] $, and what role do Littlewood–Richardson coefficients play in this?
• RQ5 Why do coefficients for even partitions in $ s_{(h-k,1^k)}[s_{1^2}] $ depend on $ b_{\mu} $, the number of corners, and how does this affect multiplicities?

Key findings

• The s-perp trick enables the derivation of new combinatorial formulas for $ s_{(h-2,1,1)}[s_{1^2}] $, where the coefficient of $ s_{\mu} $ is $ \binom{b_{\mu}-1}{2} $ for even $ \mu $, and $ \sum_{\nu} c^{\mu}_{\nu(2,1,1)} - 1 $ for $ \mu \in \mathcal{P}_{2h} $.
• For $ s_{(h-k,1^k)}[s_{1^2}] $, the coefficient of $ s_{\mu} $ is given by a signed sum over $ c^{\mu}_{\nu\rho} $, with $ \nu $ even and $ \rho $ threshold-shaped, proving a general formula via induction.
• The coefficient of $ s_{\mu} $ with $ \mu $ even in $ s_{(h-3,1,1,1)}[s_{1^2}] $ is $ (b_{\mu}-1)(b_{\mu}-2)(2b_{\mu}-3)/6 $, showing a cubic dependence on the number of corners.
• The coefficient of $ s_{(3^2 2^2 1^4)} $ in $ s_{(3,1,1,1,1)}[s_{1^2}] $ is 1, while that of $ s_{(4^2 2^2 1^2)} $ is 2, indicating that for $ h-4 $, coefficients depend on more than just $ b_{\mu} $.
• The s-perp trick provides a more efficient algorithm than SageMath for computing plethysm coefficients in special cases, particularly for $ s_{\lambda}[s_{\mu}] $ with $ \lambda $ a hook or row and $ \mu $ a column or row.
• The method successfully proves known results (e.g., $ s_2[s_n] $, $ s_3[s_n] $) in a unified and simplified way, demonstrating its generality and power.