[Paper Review] The 144 symmetries of the Littlewood-Richardson coefficients of $SL_3$

Emmanuel Briand, Mercedes Helena Rosas Celis|arXiv (Cornell University)|Apr 10, 2020

Advanced Algebra and Geometry9 references4 citations

TL;DR

This paper computes the complete group of linear symmetries for Littlewood-Richardson coefficients associated with $SL_3$, using SageMath to analyze the chamber complex and piecewise polynomial structure of the coefficients. It discovers 144 symmetries—significantly more than the known 12—showing the group is isomorphic to $S_2 \times (S_3 \wr S_2)$, with an additional symmetry beyond the classical ones.

ABSTRACT

We compute with SageMath the group of all linear symmetries for the Littlewood-Richardson associated to the representations of $SL_3$. We find that there are 144 symmetries, more than the 12 symmetries known for the Littlewood-Richardson coefficients in general.

Motivation & Objective

• To determine the full group of linear symmetries for Littlewood-Richardson coefficients in the $SL_3$ case.
• To investigate whether additional symmetries exist beyond the well-known 12 symmetries from duality and permutation of indices.
• To apply computational algebra and polyhedral geometry to the chamber complex of the coefficient function.
• To verify that the discovered symmetries act transitively on the chambers of the complex.
• To extend the method to other structural coefficients in representation theory, such as plethysm and Kronecker coefficients.

Proposed method

• Utilizes the piecewise polynomial structure of Littlewood-Richardson coefficients $c_{\lambda,\mu}^\nu$ as a function of partition parts.
• Employs the chamber complex—a rational polyhedral fan—derived from the support of the coefficients, as described in Rassart (2004).
• Uses SageMath to compute the group of linear symmetries by analyzing the action on the rays of the chamber complex.
• Identifies symmetries by checking automorphisms of the ray generators and lifting them to linear transformations on $\mathbb{R}^6$.
• Verifies that each candidate symmetry preserves the polynomial formulas in each chamber by comparing transformed polynomials.
• Confirms transitivity of the group action on chambers by analyzing ray configurations per chamber.

Experimental results

Research questions

• RQ1 Are there linear symmetries of the Littlewood-Richardson coefficients for $SL_3$ beyond the known 12?
• RQ2 What is the full group of linear symmetries acting on the coefficients $c_{\lambda,\mu}^\nu$ for $SL_3$?
• RQ3 Can the chamber complex and piecewise polynomial structure be used to systematically compute all symmetries?
• RQ4 Does the group of symmetries act transitively on the chambers of the complex?
• RQ5 How do the symmetries of $SL_3$ compare to those of $SL_N$ for $N \geq 4$?

Key findings

• The full group of linear symmetries for $SL_3$ Littlewood-Richardson coefficients has order 144, significantly exceeding the known 12 symmetries.
• The symmetry group is isomorphic to $S_2 \times (S_3 \wr S_2)$, a non-trivial extension of the classical $S_2 \times S_3$ symmetry group.
• An additional symmetry is identified: $c_{(\lambda_1,\lambda_2),(\mu_1,\mu_2)}^{(\nu_1,\nu_2,\nu_3)} = c_{(\lambda_1+\mu_1-\nu_2,\lambda_2+\mu_1-\nu_2),(\nu_2,\mu_2)}^{(\nu_1,\mu_1,\nu_3)}$.
• The group acts transitively on the 18 chambers of the chamber complex, confirming full symmetry coverage.
• For $GL_3$, the symmetry group doubles to 288 symmetries due to an additional translation-like symmetry involving a shift parameter $m$.
• For $SL_N$ with $N \geq 4$, only the standard 12 symmetries exist, indicating $SL_3$ has exceptional symmetry structure.