The Many Faces of Magog Matrices

TL;DR

This paper explores magog matrices, offering new combinatorial representations and bijections that deepen understanding of TSSCPPs and their relation to ASMs.

Contribution

It introduces novel combinatorial models for magog matrices and establishes explicit bijections, enhancing the structural understanding of TSSCPPs.

Findings

Defined magog analogues of corner-sum matrices and height-function matrices.

Established bijections among various combinatorial representations.

Provided new structural insights into the combinatorics of TSSCPPs.

Abstract

Magog matrices, introduced by Holmlund and Striker in 2025, provide a matrix model for totally symmetric self-complementary plane partitions (TSSCPPs), as a natural analogue of alternating sign matrices (ASMs). In this paper, we develop several new combinatorial representations of magog matrices, mirroring classical representations of ASMs. Specifically, we define magog analogues of corner-sum matrices, height-function matrices, fully packed loop configurations, and vertex models, and establish explicit bijections among all of these objects. These constructions provide new structural insight into the combinatorics of TSSCPPs and illuminate parallels and differences between the ASM and TSSCPP frameworks.