Block-Toeplitz determinants, chess tableaux, and the type $\hat{A_1}$ Geiss-Leclerc-Schroer $\phi$-map

TL;DR

This paper connects algebraic structures from type _1 with combinatorial objects like tableaux and matrix minors, providing explicit evaluations of the S map and insights into cluster variables.

Contribution

It offers a new combinatorial interpretation of the S map for shape modules over a preprojective algebra of type _1, linking it to block-Toeplitz minors and cluster algebra conjectures.

Findings

Explicit evaluation of the S map in terms of matrix minors.

Counting standard and chess tableaux computes Euler characteristics of flag varieties.

Proposes a conjecture relating minors to cluster variables.

Abstract

We evaluate the Geiss-Leclerc-Schroer $\phi$-map for shape modules over the preprojective algebra $\Lambda$ of type $\hat{A_1}$ in terms of matrix minors arising from the block-Toeplitz representation of the loop group $\SL_2(\mathcal{L})$. Conjecturally these minors are among the cluster variables for coordinate rings of unipotent cells within $\SL_2(\mathcal{L})$. In so doing we compute the Euler characteristic of any generalized flag variety attached to a shape module by counting standard tableaux of requisite shape and parity; alternatively by counting chess tableaux of requisite shape and content.